Let $U_{\mathcal{P}}({\mathfrak{n}}_{+}),U_{\mathcal{P}}({\mathfrak{n}}_{-})$
($U_{q}^{res}({\mathfrak{n}}_{+}),U_{q}^{res}({\mathfrak{n}}_{-})$) be the
$\mathcal{P}(\mathbb{C}[q,q^{-1}])$–subalgebras of $U_{h}({\mathfrak{g}})$
generated by the $X_{i}^{+}$ and by the $X_{i}^{-}$, $i=1,\ldots,l$ (by the
$(X_{i}^{+})^{(r)}$ and by the $(X_{i}^{-})^{(r)}$, $i=1,\ldots,l$, $r\geq
0$), respectively. Using the root vectors $X_{\beta}^{\pm}$,
$(X_{\beta}^{\pm})^{(r)}$ we can construct bases for these subalgebras Namely,
let $(X^{\pm})^{\bf
r}=(X^{\pm}_{\beta_{1}})^{r_{1}}\ldots(X^{\pm}_{\beta_{D}})^{r_{D}}$,
$(X^{\pm})^{(\bf
r)}=(X^{\pm}_{\beta_{1}})^{(r_{1})}\ldots(X^{\pm}_{\beta_{D}})^{(r_{D})}$,
${\bf r}=(r_{1},\ldots r_{D})\in{\mathbb{N}}^{D}$, $H^{\bf
k}=H_{1}^{k_{1}}\ldots H_{l}^{k_{l}}$, ${\bf
k}=(k_{1},\ldots,k_{l})\in{\mathbb{N}}^{l}$.

Commutation relations (2.3), (2.3), (2.3.1) and (2.3.1) between quantum root
vectors imply the following lemma.

######  Lemma 2.4.2.

The elements $(X^{+})^{\bf r}$, $(X^{-})^{\bf t}$ ($(X^{+})^{\bf(r)}$,
$(X^{-})^{\bf(t)}$) for ${\bf r},~{}{\bf t}\in{\mathbb{N}}^{D}$ form bases of
$U_{\mathcal{P}}({\mathfrak{n}}_{+})$, $U_{\mathcal{P}}({\mathfrak{n}}_{-})$
($U_{q}^{res}({\mathfrak{n}}_{+})$, $U_{q}^{res}({\mathfrak{n}}_{-})$),
respectively.

The elements $(X^{+})^{\bf r}$, $(X^{-})^{\bf t}$ and $H^{\bf k}$ form
topological bases of $U_{h}({\mathfrak{n}}_{+}),U_{h}({\mathfrak{n}}_{-})$ and
$U_{h}(\mathfrak{h})$, respectively.

The multiplication defines an isomorphisms of
${\mathbb{C}[q,q^{-1}]}$–modules:

$U_{q}^{res}({\mathfrak{n}}_{-})\otimes U_{q}^{res}(H)\otimes
U_{q}^{res}({\mathfrak{n}}_{+})\rightarrow U_{q}^{res}({\mathfrak{g}}),$
(2.4.1)

and of complete ${\mathbb{C}}[[h]]$–modules

$U_{h}({\mathfrak{n}}_{-})\otimes U_{h}(\mathfrak{h})\otimes
U_{h}({\mathfrak{n}}_{+})\rightarrow U_{h}({\mathfrak{g}}),$

where the tensor products in the leftt hand side are completed in the $h$–adic
topology.

Let $[\alpha,\beta]$=$\\{\beta_{p},\ldots,\beta_{q}\\}$ be a minimal segment
in $\Delta_{+}$, $U_{\mathcal{P}}([\alpha,\beta])$,
$U_{\mathcal{P}}([-\alpha,-\beta])$ ($U_{q}^{res}([\alpha,\beta])$,
$U_{q}^{res}([-\alpha,-\beta])$) the
$\mathcal{P}(\mathbb{C}[q,q^{-1}])$–subalgebras of $U_{h}({\mathfrak{g}})$
generated by the $X_{\gamma}^{+}$ and by the $X_{\gamma}^{-}$,
$\gamma\in[\alpha,\beta]$ (by the $(X_{\gamma}^{+})^{(r)}$ and by the
$(X_{\gamma}^{-})^{(r)}$, $\gamma\in[\alpha,\beta]$, $r\geq 0$), respectively.
Then the elements
$(X^{\pm}_{\beta_{p}})^{r_{p}}\ldots(X^{\pm}_{\beta_{q}})^{r_{q}}$
($(X^{\pm}_{\beta_{p}})^{(r_{p})}\ldots(X^{\pm}_{\beta_{q}})^{(r_{q})}$),
${r_{i}}\in{\mathbb{N}}$ form bases of $U_{\mathcal{P}}([\alpha,\beta])$,
$U_{\mathcal{P}}([-\alpha,-\beta])$ ($U_{q}^{res}([\alpha,\beta])$,
$U_{q}^{res}([-\alpha,-\beta])$), respectively.

Let $\overline{U}_{\mathcal{P}}([\alpha,\beta])$,
$\overline{U}_{\mathcal{P}}([-\alpha,-\beta])$
($\overline{U}_{q}^{res}([\alpha,\beta])$,
$\overline{U}_{q}^{res}([-\alpha,-\beta])$) be the
$\mathcal{P}(\mathbb{C}[q,q^{-1}])$–subalgebras of $U_{h}({\mathfrak{g}})$
generated by the $\overline{X}_{\gamma}^{+}$ and by the
$\overline{X}_{\gamma}^{-}$, $\gamma\in[\alpha,\beta]$ (by the
$(\overline{X}_{\gamma}^{+})^{(r)}$ and by the
$(\overline{X}_{\gamma}^{-})^{(r)}$, $\gamma\in[\alpha,\beta]$, $r\geq 0$),
respectively. Then the elements
$(\overline{X}^{\pm}_{\beta_{p}})^{r_{p}}\ldots(\overline{X}^{\pm}_{\beta_{q}})^{r_{q}}$
($(\overline{X}^{\pm}_{\beta_{p}})^{(r_{p})}\ldots(\overline{X}^{\pm}_{\beta_{q}})^{(r_{q})}$),
${r_{i}}\in{\mathbb{N}}$ form bases of
$\overline{U}_{\mathcal{P}}([\alpha,\beta])$,
$\overline{U}_{\mathcal{P}}([-\alpha,-\beta])$
($\overline{U}_{q}^{res}([\alpha,\beta])$,
$\overline{U}_{q}^{res}([-\alpha,-\beta])$), respectively.

Let $[\alpha,-\beta]$=$\\{\beta_{p},\ldots,\beta_{q}\\}$,
$\alpha,\beta\in\Delta_{+}$ be a minimal segment in $\Delta$,
$U_{\mathcal{P}^{\prime}}([\alpha,-\beta])$,
$U_{\mathcal{P}^{\prime}}([-\alpha,\beta])$
($U_{U_{q}^{res}(H)}^{res}([\alpha,-\beta])$,
$U_{U_{q}^{res}(H)}^{res}([-\alpha,\beta])$) the
$\mathcal{P}^{\prime}(U_{q}^{res}(H))$–subalgebras of $U_{h}({\mathfrak{g}})$
generated by the $X_{\gamma}$ ($(X_{\gamma})^{(r)}$), where
$\gamma\in[\alpha,-\beta]$ or $\gamma\in[-\alpha,\beta]$, respectively, and
$X_{\gamma}=X_{\gamma}^{+}$ if $\gamma\in\Delta_{+}$,
$X_{\gamma}=X_{\gamma}^{-}$ if $\gamma\in\Delta_{-}$. Then the elements
$(X_{\beta_{p}})^{r_{p}}\ldots(X_{\beta_{q}})^{r_{q}}$
($(X_{\beta_{p}})^{(r_{p})}\ldots(X_{\beta_{q}})^{(r_{q})}$),
${r_{i}}\in{\mathbb{N}}$ form bases of
$U_{\mathcal{P}^{\prime}}([\alpha,-\beta])$
($U_{U_{q}^{res}(H)}^{res}([\alpha,-\beta])$), and the elements
$(X_{-\beta_{p}})^{r_{p}}\ldots(X_{-\beta_{q}})^{r_{q}}$
($(X_{-\beta_{p}})^{(r_{p})}\ldots(X_{-\beta_{q}})^{(r_{q})}$),
${r_{i}}\in{\mathbb{N}}$ form bases of
$U_{\mathcal{P}^{\prime}}([-\alpha,\beta])$
($U_{U_{q}^{res}(H)}^{res}([-\alpha,\beta])$), respectively.

Let $\overline{U}_{\mathcal{P}^{\prime}}([\alpha,-\beta])$,
$\overline{U}_{\mathcal{P}^{\prime}}([-\alpha,\beta])$
($\overline{U}_{U_{q}^{res}(H)}^{res}([\alpha,-\beta])$,
$\overline{U}_{U_{q}^{res}(H)}^{res}([-\alpha,\beta])$) be the
$\mathcal{P}^{\prime}(U_{q}^{res}(H))$–subalgebras of $U_{h}({\mathfrak{g}})$
generated by the $\overline{X}_{\gamma}$ ($(\overline{X}_{\gamma})^{(r)}$),
where $\gamma\in[\alpha,-\beta]$ or $\gamma\in[-\alpha,\beta]$, respectively,
and $\overline{X}_{\gamma}=\overline{X}_{\gamma}^{+}$ if
$\gamma\in\Delta_{+}$, $\overline{X}_{\gamma}=\overline{X}_{\gamma}^{-}$ if
$\gamma\in\Delta_{-}$. Then the elements
$(\overline{X}_{\beta_{p}})^{r_{p}}\ldots(\overline{X}_{\beta_{q}})^{r_{q}}$
($(\overline{X}_{\beta_{p}})^{(r_{p})}\ldots(\overline{X}_{\beta_{q}})^{(r_{q})}$),
${r_{i}}\in{\mathbb{N}}$ form bases of
$\overline{U}_{\mathcal{P}^{\prime}}([\alpha,-\beta])$
($\overline{U}_{U_{q}^{res}(H)}^{res}([\alpha,-\beta])$), and the elements
$(\overline{X}_{-\beta_{p}})^{r_{p}}\ldots(\overline{X}_{-\beta_{q}})^{r_{q}}$
($(\overline{X}_{-\beta_{p}})^{(r_{p})}\ldots(\overline{X}_{-\beta_{q}})^{(r_{q})}$),
${r_{i}}\in{\mathbb{N}}$ form bases of
$\overline{U}_{\mathcal{P}^{\prime}}([-\alpha,\beta])$
($\overline{U}_{U_{q}^{res}(H)}^{res}([-\alpha,\beta])$), respectively.

###### Proof.

The first four statements of this lemma are just Propositions 8.1.7, 9.1.3 and
9.3.3 in [20]. The proofs of the other claims are similar to each other.
Consider, for instance, the case of the algebra
$U_{U_{q}^{res}(H)}^{res}([\alpha,-\beta])$.

Let $[\alpha,-\beta]$=$\\{\beta_{p},\ldots,\beta_{q}\\}$,
$\alpha,\beta\in\Delta_{+}$ be a minimal segment in $\Delta$,
$U_{U_{q}^{res}(H)}^{res}([\alpha,-\beta])$ the $U_{q}^{res}(H)$–subalgebra of
$U_{h}({\mathfrak{g}})$ generated by the $(X_{\gamma})^{(r)}$, where
$\gamma\in[\alpha,-\beta]$, and $X_{\gamma}=X_{\gamma}^{+}$ if
$\gamma\in\Delta_{+}$, $X_{\gamma}=X_{\gamma}^{-}$ if $\gamma\in\Delta_{-}$.
We show that the elements
$(X_{\beta_{p}})^{(r_{p})}\ldots(X_{\beta_{q}})^{(r_{q})}$,
${r_{i}}\in{\mathbb{N}}$ form a basis of
$U_{U_{q}^{res}(H)}^{res}([\alpha,-\beta])$.

Consider the algebra
$U_{q}(\mathfrak{g})=U_{q}^{res}(\mathfrak{g})\otimes_{\mathbb{C}[q,q^{-1}]}\mathbb{C}(q)$.
If $x\in U_{U_{q}^{res}(H)}^{res}([\alpha,-\beta])\subset U_{q}(\mathfrak{g})$
then using commutation relations (2.3) and (2.3.1) one can represent $x$ as a
$U_{q}^{res}(H)\otimes_{\mathbb{C}[q,q^{-1}]}\mathbb{C}(q)$–linear combination
of the elements $(X_{\beta_{p}})^{(r_{p})}\ldots(X_{\beta_{q}})^{(r_{q})}$,
${r_{i}}\in{\mathbb{N}}$. We can also consider $x$ as an element of
$U_{q}(\mathfrak{g})$ and by the Poincaré–Birkhoff–Witt theorem for
$U_{q}(\mathfrak{g})$ (see Proposition 9.1.3 in [20]) the above mentioned
presentation of $x$ is unique. Now by the Poincaré–Birkhoff–Witt theorem for
$U_{q}^{res}(\mathfrak{g})$ (see Proposition 9.3.3 in [20] or the third claim
of this lemma) the coefficients in this presentation must belong to
$U_{q}^{res}(H)$. This completes the proof in the considered case.

∎

A basis for $U_{q}^{res}(H)$ is a little bit more difficult to describe. We do
not need its explicit description.

###  2.5 The universal R–matrix

$U_{h}({\mathfrak{g}})$ is a quasitriangular Hopf algebra, i.e. there exists
an invertible element ${\mathcal{R}}\in U_{h}({\mathfrak{g}})\otimes
U_{h}({\mathfrak{g}})$ (completed tensor product), called a universal
R–matrix, such that

$\Delta^{opp}_{h}(a)={\mathcal{R}}\Delta_{h}(a){\mathcal{R}}^{-1}\mbox{ for
all }a\in U_{h}({\mathfrak{g}}),$ (2.5.1)

and

$\begin{array}[]{l}(\Delta_{h}\otimes
id){\mathcal{R}}={\mathcal{R}}_{13}{\mathcal{R}}_{23},\\\ \\\
(id\otimes\Delta_{h}){\mathcal{R}}={\mathcal{R}}_{13}{\mathcal{R}}_{12},\end{array}$
(2.5.2)

where ${\mathcal{R}}_{12}={\mathcal{R}}\otimes
1,~{}{\mathcal{R}}_{23}=1\otimes{\mathcal{R}},~{}{\mathcal{R}}_{13}=(\sigma\otimes
id){\mathcal{R}}_{23}$.

From (2.5.1) and (2.5.2) it follows that $\mathcal{R}$ satisfies the quantum
Yang–Baxter equation

${\mathcal{R}}_{12}{\mathcal{R}}_{13}{\mathcal{R}}_{23}={\mathcal{R}}_{23}{\mathcal{R}}_{13}{\mathcal{R}}_{12}.$
(2.5.3)

For every quasitriangular Hopf algebra we also have

$(S\otimes id){\mathcal{R}}=(id\otimes
S^{-1}){\mathcal{R}}={\mathcal{R}}^{-1},$

and

$(S\otimes S){\mathcal{R}}={\mathcal{R}}.$ (2.5.4)

An explicit expression for $\mathcal{R}$ may be written by making use of the
q–exponential

$exp_{q}(x)=exp_{q}^{\prime}(qx)=\sum_{k=0}^{\infty}q^{\frac{1}{2}k(k+1)}{x^{k}\over[k]_{q}!}$

in terms of which the element $\mathcal{R}$ takes the form

${\mathcal{R}}=\prod_{\beta}exp_{q_{\beta}}[(1-q_{\beta}^{-2})X_{\beta}^{-}\otimes
X_{\beta}^{+}]exp\left[h\sum_{i=1}^{l}(Y_{i}\otimes H_{i})\right],$ (2.5.5)

where the product is over all the positive roots of $\mathfrak{g}$, and the
order of the terms is such that the $\alpha$–term appears to the left of the
$\beta$–term if $\alpha<\beta$ with respect to the normal ordering

$\beta_{1}=\alpha_{i_{1}},\beta_{2}=s_{i_{1}}\alpha_{i_{2}},\ldots,\beta_{D}=s_{i_{1}}\ldots
s_{i_{D-1}}\alpha_{i_{D}}$

of $\Delta_{+}$ which is used in the definition of the quantum root vectors
$X_{\beta}^{\pm}$.

One can calculate the action of the comultiplication on the root vectors
$X_{\beta_{k}}^{\pm}$ in terms of the universal R–matrix. For instance for
$\Delta_{h}(X_{\beta_{k}}^{-})$ one has

$\Delta_{h}(X_{\beta_{k}}^{-})=\widetilde{\mathcal{R}}_{<\beta_{k}}(X_{\beta_{k}}^{-}\otimes
1+e^{h\beta^{\vee}}\otimes
X_{\beta_{k}}^{-})\widetilde{\mathcal{R}}^{-1}_{<\beta_{k}},$ (2.5.6)

where

$\widetilde{\mathcal{R}}_{<\beta_{k}}=\widetilde{\mathcal{R}}_{\beta_{1}}\ldots\widetilde{\mathcal{R}}_{\beta_{k-1}},~{}\widetilde{\mathcal{R}}_{\beta_{r}}=exp_{q_{\beta_{r}}}[(1-q_{\beta_{r}}^{-2})X_{\beta_{r}}^{+}\otimes
X_{\beta_{r}}^{-}].$

The r–matrix $r_{-}=-\frac{1}{2}h^{-1}({\mathcal{R}}-1\otimes
1)~{}~{}(\mbox{mod }h)$, which is the classical limit of $\mathcal{R}$,
coincides with the classical r–matrix (2.1.3).

###  2.6 Realizations of quantum groups associated to Weyl group elements

q-W–algebras will be defined in terms of certain integral forms of
non–standard realizations of quantum groups associated to Weyl group elements.

Let $s$ be an element of the Weyl group $W$ of the pair
$(\mathfrak{g},\mathfrak{h})$, and $\mathfrak{h}^{\prime}$ the orthogonal
complement, with respect to the symmetric bilinear form, to the subspace of
$\mathfrak{h}$ fixed by the natural action of $s$ on $\mathfrak{h}$. Let
$\mathfrak{h}^{\prime*}$ be the image of $\mathfrak{h}^{\prime}$ in
$\mathfrak{h}^{*}$ under the identification
$\mathfrak{h}^{*}\simeq\mathfrak{h}$ induced by the symmetric bilinear form on
$\mathfrak{g}$. The restriction of the natural action of $s$ on
$\mathfrak{h}^{*}$ to the subspace $\mathfrak{h}^{\prime*}$ has no fixed
points. Therefore one can define the Cayley transform ${1+s\over 1-s}$ of the
restriction of $s$ to $\mathfrak{h}^{\prime*}$. Denote by
$P_{\mathfrak{h}^{\prime*}}$ the orthogonal projection operator onto
${\mathfrak{h}^{\prime*}}$ in $\mathfrak{h}^{*}$, with respect to the bilinear
form.

Let $\kappa\in\mathbb{Z}$ be an integer number and $U_{h}^{s}({\mathfrak{g}})$
the topological algebra over ${\mathbb{C}}[[h]]$ topologically generated by
elements $e_{i},f_{i},H_{i},~{}i=1,\ldots l$ subject to the relations:

$\begin{array}[]{l}[H_{i},H_{j}]=0,~{}~{}[H_{i},e_{j}]=a_{ij}e_{j},~{}~{}[H_{i},f_{j}]=-a_{ij}f_{j},~{}~{}e_{i}f_{j}-q^{c_{ij}}f_{j}e_{i}=\delta_{i,j}{K_{i}-K_{i}^{-1}\over
q_{i}-q_{i}^{-1}},\\\ \\\ c_{ij}=\kappa\left({1+s\over
1-s}P_{\mathfrak{h}^{\prime*}}\alpha_{i},\alpha_{j}\right),~{}~{}K_{i}=e^{d_{i}hH_{i}},\\\
\\\
\sum_{r=0}^{1-a_{ij}}(-1)^{r}q^{rc_{ij}}\left[\begin{array}[]{c}1-a_{ij}\\\
r\end{array}\right]_{q_{i}}(e_{i})^{1-a_{ij}-r}e_{j}(e_{i})^{r}=0,~{}i\neq
j,\\\ \\\
\sum_{r=0}^{1-a_{ij}}(-1)^{r}q^{rc_{ij}}\left[\begin{array}[]{c}1-a_{ij}\\\
r\end{array}\right]_{q_{i}}(f_{i})^{1-a_{ij}-r}f_{j}(f_{i})^{r}=0,~{}i\neq
j.\end{array}$ (2.6.1)

######  Proposition 2.6.1.

For every solution $n_{ij}\in{\mathbb{C}},~{}i,j=1,\ldots,l$ of equations

$d_{j}n_{ij}-d_{i}n_{ji}=c_{ij}$ (2.6.2)

there exists an algebra isomorphism
$\psi_{\\{n_{ij}\\}}:U_{h}^{s}({\mathfrak{g}})\rightarrow
U_{h}({\mathfrak{g}})$ defined by the formulas:

$\psi_{\\{n_{ij}\\}}(e_{i})=X_{i}^{+}\prod_{p=1}^{l}L_{p}^{n_{ip}},~{}~{}\psi_{\\{n_{ij}\\}}(f_{i})=\prod_{p=1}^{l}L_{p}^{-n_{ip}}X_{i}^{-},~{}~{}\psi_{\\{n_{ij}\\}}(H_{i})=H_{i}.$

###### Proof.

The proof of this proposition is by direct verification of defining relations
(2.6.1). The most nontrivial part is to verify the deformed quantum Serre
relations, i.e. the last two relations in (2.6.1). For instance, the defining
relations of $U_{h}({\mathfrak{g}})$ imply the following relations for
$\psi_{\\{n_{ij}\\}}(e_{i})$,

$\sum_{k=0}^{1-a_{ij}}(-1)^{k}\left[\begin{array}[]{c}1-a_{ij}\\\
k\end{array}\right]_{q_{i}}q^{k({d_{j}}n_{ij}-d_{i}n_{ji})}\psi_{\\{n_{ij}\\}}(e_{i})^{1-a_{ij}-k}\psi_{\\{n_{ij}\\}}(e_{j})\psi_{\\{n_{ij}\\}}(e_{i})^{k}=0,$

for any $i\neq j$. Now using equation (2.6.2) we arrive to the quantum Serre
relations for $e_{i}$ in (2.6.1).

∎

The general solution of equation (2.6.2) is given by

$n_{ij}=\frac{1}{2d_{j}}(c_{ij}+{s_{ij}}),$ (2.6.3)

where $s_{ij}=s_{ji}$.

We shall only use the solution for which $s_{ij}=0$ for all $i,j=1,\ldots l$.
Then

$n_{ij}=\frac{1}{2d_{j}}c_{ij}$ (2.6.4)

From now on we assume that solution (2.6.4) is used to identify
$U_{h}^{s}({\mathfrak{g}})$ and $U_{h}({\mathfrak{g}})$.

The algebra $U_{h}^{s}({\mathfrak{g}})$ is called the realization of the
quantum group $U_{h}({\mathfrak{g}})$ corresponding to the element $s\in W$.
Denote by $U_{h}^{s}({\mathfrak{n}}_{\pm})$ the subalgebra in
$U_{h}^{s}({\mathfrak{g}})$ generated by $e_{i}~{}(f_{i}),i=1,\ldots,l$. Let
$U_{h}^{s}({\mathfrak{h}})$ be the subalgebra in $U_{h}^{s}({\mathfrak{g}})$
generated by $H_{i},~{}i=1,\ldots,l$.

We shall construct analogues of quantum root vectors for
$U_{h}^{s}({\mathfrak{g}})$. It is convenient to introduce an operator
$K\in{\rm End}~{}{\mathfrak{h}}$ defined by

$KH_{i}=\sum_{j=1}^{l}{n_{ij}\over d_{i}}Y_{j}.$ (2.6.5)

From (2.6.4) we obtain that

$Kh=\frac{\kappa}{2}{1+s\over
1-s}P_{\mathfrak{h}^{\prime}}h,h\in\mathfrak{h}.$

######  Proposition 2.6.2.

Let $s\in W$ be an element of the Weyl group $W$ of the pair
$(\mathfrak{g},\mathfrak{h})$, $\Delta$ the root system of the pair
$(\mathfrak{g},\mathfrak{h})$. Let $U_{h}^{s}({\mathfrak{g}})$ be the
realization of the quantum group $U_{h}({\mathfrak{g}})$ associated to $s$.

For any normal ordering of the root system $\Delta_{+}$ the elements

$e_{\beta}=\psi_{\\{n_{ij}\\}}^{-1}(X_{\beta}^{+}e^{hK\beta^{\vee}})~{}\text{and}~{}f_{\beta}=\psi_{\\{n_{ij}\\}}^{-1}(e^{-hK\beta^{\vee}}X_{\beta}^{-}),~{}\beta\in\Delta_{+}$

lie in the subalgebras $U_{h}^{s}({\mathfrak{n}}_{+})$ and
$U_{h}^{s}({\mathfrak{n}}_{-})$, respectively.

The elements $f_{\beta}\in U_{h}^{s}({\mathfrak{n}_{-}})$,
$\beta\in\Delta_{\mathfrak{m}_{+}}$ generate a subalgebra
$U_{h}^{s}({\mathfrak{m}}_{-})\subset U_{h}^{s}({\mathfrak{g}})$ such that

$U_{h}^{s}({\mathfrak{m}}_{-})/hU_{h}^{s}({\mathfrak{m}}_{-})\simeq
U({\mathfrak{m}_{-}}),$

where ${\mathfrak{m}_{-}}$ is the Lie subalgebra of $\mathfrak{g}$ generated
by the root vectors $X_{-\alpha}$, $\alpha\in\Delta_{\mathfrak{m}_{+}}$.

###### Proof.

Fix a normal ordering of the root system $\Delta_{+}$. Let
$\beta=\sum_{i=1}^{l}m_{i}\alpha_{i}\in\Delta_{+}$ be a positive root,
$X_{\beta}^{+}\in U_{h}({\mathfrak{g}})$ the corresponding quantum root vector
constructed with the help of the fixed normal ordering of $\Delta_{+}$. Then
$\beta^{\vee}=\sum_{i=1}^{l}m_{i}d_{i}H_{i}$, and so
$K\beta^{\vee}=\sum_{i,j=1}^{l}m_{i}n_{ij}Y_{j}$. Now the proof of the first
statement follows immediately from Proposition 2.4.1, commutation relations
(2.1.1) and the definition of the isomorphism $\psi_{\\{n_{ij}\\}}$.

The second assertion is a consequence of (2.3.4).

∎

The realizations $U_{h}^{s}({\mathfrak{g}})$ of the quantum group
$U_{h}({\mathfrak{g}})$ are related to quantizations of some nonstandard
bialgebra structures on $\mathfrak{g}$. At the quantum level changing
bialgebra structure corresponds to the so–called Drinfeld twist. The relevant
class of such twists described in the following proposition.

######  Proposition 2.6.3.

Let $(A,\mu,\imath,\Delta,\varepsilon,S)$ be a Hopf algebra over a commutative
ring with multiplication $\mu$, unit $\imath$, comultiplication $\Delta$,
counit $\varepsilon$ and antipode $S$. Let $\mathcal{F}$ be an invertible
element of $A\otimes A$ such that

$\begin{array}[]{l}{\mathcal{F}}_{12}(\Delta\otimes
id)({\mathcal{F}})={\mathcal{F}}_{23}(id\otimes\Delta)({\mathcal{F}}),\\\ \\\
(\varepsilon\otimes
id)({\mathcal{F}})=(id\otimes\varepsilon)({\mathcal{F}})=1.\end{array}$
(2.6.6)

Then, $v=\mu(id\otimes S)({\mathcal{F}})$ is an invertible element of $A$ with

$v^{-1}=\mu(S\otimes id)({\mathcal{F}}^{-1}).$

Moreover, if we define $\Delta^{\mathcal{F}}:A\rightarrow A\otimes A$ and
$S^{\mathcal{F}}:A\rightarrow A$ by

$\Delta^{\mathcal{F}}(a)={\mathcal{F}}\Delta(a){\mathcal{F}}^{-1},~{}~{}S^{\mathcal{F}}(a)=vS(a)v^{-1},$

then $(A,\mu,\imath,\Delta^{\mathcal{F}},\varepsilon,S^{\mathcal{F}})$ is a
Hopf algebra denoted by $A^{\mathcal{F}}$ and called the twist of $A$ by
${\mathcal{F}}$.

######  Corollary 2.6.4.

Suppose that $A$ and ${\mathcal{F}}$ are as in Proposition 2.6.3, but assume
in addition that $A$ is quasitriangular with universal R–matrix $\mathcal{R}$.
Then $A^{\mathcal{F}}$ is quasitriangular with universal R–matrix

${\mathcal{R}}^{\mathcal{F}}={\mathcal{F}}_{21}{\mathcal{R}}{\mathcal{F}}^{-1},$
(2.6.7)

where ${\mathcal{F}}_{21}=\sigma{\mathcal{F}}$.

Let $P_{\mathfrak{h}^{\prime}}$ be the orthogonal projection operator onto
$\mathfrak{h}^{\prime}$ in $\mathfrak{h}$ with respect to the bilinear form on
$\mathfrak{h}$. Equip $U_{h}^{s}({\mathfrak{g}})$ with the comultiplication
$\Delta_{s}$ given by

$\begin{array}[]{c}\Delta_{s}(H_{i})=H_{i}\otimes 1+1\otimes H_{i},\\\ \\\
\Delta_{s}(e_{i})=e_{i}\otimes e^{-hd_{i}H_{i}}+e^{h\kappa d_{i}{1+s\over
1-s}P_{\mathfrak{h}^{\prime}}H_{i}}\otimes
e_{i},~{}~{}\Delta_{s}(f_{i})=f_{i}\otimes 1+e^{-h\kappa d_{i}{1+s\over
1-s}P_{\mathfrak{h}^{\prime}}H_{i}+hd_{i}H_{i}}\otimes f_{i},\end{array}$

the antipode $S_{s}(x)$ given by

$S_{s}(e_{i})=-e^{-h\kappa d_{i}{1+s\over
1-s}P_{\mathfrak{h}^{\prime}}H_{i}}e_{i}e^{hd_{i}H_{i}},~{}S_{s}(f_{i})=-e^{h\kappa
d_{i}{1+s\over
1-s}P_{\mathfrak{h}^{\prime}}H_{i}-hd_{i}H_{i}}f_{i},~{}S_{s}(H_{i})=-H_{i},$

and counit defined by

$\varepsilon_{s}(H_{i})=\varepsilon_{s}(X_{i}^{\pm})=0.$

The comultiplication $\Delta_{s}$ is obtained from the standard
comultiplication by a Drinfeld twist. Namely, let

${\mathcal{F}}=exp(-h\sum_{i,j=1}^{l}{n_{ij}\over d_{i}}Y_{i}\otimes Y_{j})\in
U_{h}({\mathfrak{h}})\otimes U_{h}({\mathfrak{h}}).$ (2.6.8)

Then

$\Delta_{s}(a)=(\psi_{\\{n_{ij}\\}}^{-1}\otimes\psi_{\\{n_{ij}\\}}^{-1}){\mathcal{F}}\Delta_{h}(\psi_{\\{n_{ij}\\}}(a)){\mathcal{F}}^{-1}.$
(2.6.9)

Note that the Hopf algebra $U_{h}^{s}({\mathfrak{g}})$ is a quantization of
the bialgebra structure on $\mathfrak{g}$ defined by the cocycle

$\delta(x)=({\rm ad}_{x}\otimes 1+1\otimes{\rm
ad}_{x})2r^{s}_{\pm},~{}~{}r^{s}_{\pm}\in{\mathfrak{g}}\otimes{\mathfrak{g}},$
(2.6.10)

where $r^{s}_{\pm}=r_{\pm}+\frac{1}{2}\sum_{i=1}^{l}\kappa{1+s\over
1-s}P_{\mathfrak{h}^{\prime}}H_{i}\otimes Y_{i}$, and $r_{\pm}$ is given by
(2.1.3).

$U_{h}^{s}({\mathfrak{g}})$ is a quasitriangular topological Hopf algebra with
the universal R–matrix
${\mathcal{R}}^{s}={\mathcal{F}}_{21}{\mathcal{R}}{\mathcal{F}}^{-1}$,

$\begin{array}[]{l}{\mathcal{R}}^{s}=\prod_{\beta}exp_{q_{\beta}}[(1-q_{\beta}^{-2})f_{\beta}\otimes
e_{\beta}e^{-h\kappa{1+s\over
1-s}P_{\mathfrak{h}^{\prime}}\beta^{\vee}}]\times\\\
exp\left[h(\sum_{i=1}^{l}(Y_{i}\otimes H_{i})-\sum_{i=1}^{l}\kappa{1+s\over
1-s}P_{\mathfrak{h}^{\prime}}H_{i}\otimes Y_{i})\right],\end{array}$ (2.6.11)

where the order of the terms is such that the $\alpha$–term appears to the
left of the $\beta$–term if $\alpha<\beta$ with respect to the normal ordering
of $\Delta_{+}$ with the help of which the quantum root vectors
$e_{\beta},f_{\beta}$ are defined in Proposition 2.6.2.

Similarly to (2.2.8) one obtains that for a reduced decomposition
$w=s_{i_{1}}\ldots s_{i_{k}}$ and $T_{w}=T_{i_{1}}\ldots T_{i_{k}}$ only
depending on $w$ one has from (2.2.8) and (2.6.9)

$\displaystyle\Delta_{s}(T_{w})=\prod^{k}_{p=1}\theta_{\beta_{p}}^{s}q^{\sum_{i=1}^{l}(-Y_{i}\otimes
KH_{i}+T_{w}Y_{i}\otimes T_{w}KH_{i})}T_{w}\otimes T_{w}=$ (2.6.12)
$\displaystyle=T_{w}\otimes T_{w}q^{\sum_{i=1}^{l}(-T_{w^{-1}}Y_{i}\otimes
T_{w^{-1}}KH_{i}+Y_{i}\otimes
KH_{i})}\prod_{p=1}^{k}\overline{\theta}_{\beta_{p}^{\prime}}^{s},$

where for $p=1,\ldots,k$

$e_{\beta_{p}}=\psi_{\\{n_{ij}\\}}^{-1}(X_{\beta_{p}}^{+}e^{hK\beta_{p}^{\vee}}),f_{\beta}=\psi_{\\{n_{ij}\\}}^{-1}(e^{-hK\beta_{p}^{\vee}}X_{\beta_{p}}^{-}),\beta_{p}=s_{i_{1}}\ldots
s_{i_{p-1}}\alpha_{i_{p}},$ $X_{\beta_{p}}^{\pm}=T_{i_{1}}\ldots
T_{i_{p-1}}X_{i_{p}}^{\pm},~{}{\overline{X}_{\beta_{p}^{\prime}}^{\pm}}=T_{i_{k}}^{-1}\ldots
T_{i_{p+1}}^{-1}X_{i_{p}}^{\pm},$
$\overline{e}_{\beta_{p}^{\prime}}=\psi_{\\{n_{ij}\\}}^{-1}(\overline{X}_{\beta_{p}^{\prime}}^{+}e^{hK{\beta_{p}^{\prime}}^{\vee}}),\overline{f}_{\beta_{p}^{\prime}}=\psi_{\\{n_{ij}\\}}^{-1}(e^{-hK{\beta_{p}^{\prime}}^{\vee}}\overline{X}_{\beta_{p}^{\prime}}^{-}),~{}K_{\beta_{p}^{\prime}}=T_{i_{k}}^{-1}\ldots
T_{i_{p+1}}^{-1}K_{i_{p}},\beta_{p}^{\prime}=s_{i_{k}}\ldots
s_{i_{p+1}}\alpha_{i_{p}}$
$\theta_{\beta_{p}}^{s}=exp_{q_{\beta_{p}}}[(1-q_{\beta_{p}}^{-2})e_{\beta_{p}}e^{-h\kappa{1+s\over
1-s}P_{\mathfrak{h}^{\prime}}\beta_{p}^{\vee}}\otimes
f_{\beta_{p}}],~{}\overline{\theta}_{\beta_{p}^{\prime}}^{s}=exp_{q_{\beta_{p}^{\prime}}}[(1-q_{\beta_{p}^{\prime}}^{-2})K_{\beta_{p}^{\prime}}^{-1}e^{h\kappa{1+s\over
1-s}P_{\mathfrak{h}^{\prime}}{\beta_{p}^{\prime}}^{\vee}}{\overline{f}_{\beta_{p}^{\prime}}}\otimes{\overline{e}_{\beta_{p}^{\prime}}}K_{\beta_{p}^{\prime}}].$

In the same way, for $\overline{T}_{w}=T_{i_{1}}^{-1}\ldots T_{i_{k}}^{-1}$
only depending on $w$ one has from (2.2.9) and (2.6.9)

$\displaystyle\Delta_{s}(\overline{T}_{w})=\prod^{k}_{p=1}{\overline{\theta}_{\beta_{p}}^{s}}^{\prime}q^{\sum_{i=1}^{l}(-Y_{i}\otimes
KH_{i}+T_{w}Y_{i}\otimes
T_{w}KH_{i})}\overline{T}_{w}\otimes\overline{T}_{w}=$ (2.6.13)
$\displaystyle=\overline{T}_{w}\otimes\overline{T}_{w}q^{\sum_{i=1}^{l}(-T_{w^{-1}}Y_{i}\otimes
T_{w^{-1}}KH_{i}+Y_{i}\otimes
KH_{i})}\prod_{p=1}^{k}{\theta_{\beta_{p}^{\prime}}^{s}}^{\prime},$

where for $p=1,\ldots,k$

$\displaystyle\overline{e}_{\beta_{p}}=\psi_{\\{n_{ij}\\}}^{-1}(\overline{X}_{\beta_{p}}^{+}e^{hK\beta_{p}^{\vee}}),\overline{f}_{\beta_{p}}=\psi_{\\{n_{ij}\\}}^{-1}(e^{-hK\beta_{p}^{\vee}}\overline{X}_{\beta_{p}}^{-}),\overline{K}_{\beta_{p}}=T_{i_{1}}^{-1}\ldots
T_{i_{p-1}}^{-1}K_{i_{p}},$
$\displaystyle{\overline{X}_{\beta_{p}}^{\pm}}=T_{i_{1}}^{-1}\ldots
T_{i_{p-1}}^{-1}X_{i_{p}}^{\pm},\beta_{p}=s_{i_{1}}\ldots
s_{i_{p-1}}\alpha_{i_{p}},$
$e_{\beta_{p}^{\prime}}^{\prime}=\psi_{\\{n_{ij}\\}}^{-1}({X_{\beta_{p}^{\prime}}^{+}}^{\prime}e^{hK{\beta_{p}^{\prime}}^{\vee}}),f_{\beta_{p}^{\prime}}^{\prime}=\psi_{\\{n_{ij}\\}}^{-1}(e^{-hK{\beta_{p}^{\prime}}^{\vee}}{X_{\beta_{p}^{\prime}}^{-}}^{\prime}),\beta_{p}^{\prime}=s_{i_{k}}\ldots
s_{i_{p+1}}\alpha_{i_{p}},{X_{\beta_{p}^{\prime}}^{\pm}}^{\prime}=T_{i_{k}}\ldots
T_{i_{p+1}}X_{i_{p}}^{\pm},$
${\theta_{\beta_{p}^{\prime}}^{s}}^{\prime}=exp_{q_{\beta_{p}^{\prime}}^{-1}}[(1-q_{\beta_{p}^{\prime}}^{2})e_{\beta_{p}^{\prime}}^{\prime}e^{-h\kappa{1+s\over
1-s}P_{\mathfrak{h}^{\prime}}{\beta_{p}^{\prime}}^{\vee}}\otimes
f_{\beta_{p}^{\prime}}^{\prime}],~{}{\overline{\theta}_{\beta_{p}}^{s}}^{\prime}=exp_{q_{\beta_{p}}^{-1}}[(1-q_{\beta_{p}}^{2})\overline{K}_{\beta_{p}}^{-1}e^{h\kappa{1+s\over
1-s}P_{\mathfrak{h}^{\prime}}\beta_{p}^{\vee}}\overline{f}_{\beta_{p}}\otimes\overline{e}_{\beta_{p}}\overline{K}_{\beta_{p}}].$

###  2.7 Some forms and specializations of quantum groups

In order to define q-W–algebras we shall actually need not the algebras
$U_{h}^{s}({\mathfrak{g}})$ themselves but some their forms defined over
certain rings. They are similar to the rational form and the restricted
integral form for the standard quantum group $U_{h}(\mathfrak{g})$. The
motivations of the definitions given below will be clear in Section 3.2. The
results below are slight modifications of similar statements for
$U_{h}(\mathfrak{g})$.

We start with a very important technical lemma which will play the key role in
the definition of q-W–algebras. Below we keep the notation introduced in
Section 1.2.

Let $s\in W$ be an element of the Weyl group. Recall that $s$ can be
represented as a product of two involutions, $s=s^{1}s^{2}$, where
$s^{1}=s_{\gamma_{1}}\ldots s_{\gamma_{n}}$, $s^{2}=s_{\gamma_{n+1}}\ldots
s_{\gamma_{l^{\prime}}}$, and the roots
$\gamma_{1},\ldots,\gamma_{l^{\prime}}$ form a basis of a subspace
${\mathfrak{h}^{\prime}}^{*}\subset\mathfrak{h}^{*}$ on which $s$ acts without
fixed points. We shall study the matrix elements of the Cayley transform of
the restriction of $s$ to ${\mathfrak{h}^{\prime}}^{*}$ with respect to this
basis.

######  Lemma 2.7.1.

Let $P_{{\mathfrak{h}^{\prime}}^{*}}$ be the orthogonal projection operator
onto ${{\mathfrak{h}^{\prime}}^{*}}$ in $\mathfrak{h}^{*}$, with respect to
the bilinear form. Then the matrix elements of the operator ${1+s\over
1-s}P_{{\mathfrak{h}^{\prime}}^{*}}$ in the basis
$\gamma_{1},\ldots,\gamma_{l^{\prime}}$ are of the form

$\left({1+s\over
1-s}P_{{\mathfrak{h}^{\prime}}^{*}}\gamma_{i},\gamma_{j}\right)=\varepsilon_{ij}(\gamma_{i},\gamma_{j}),$
(2.7.1)

where

$\varepsilon_{ij}=\left\\{\begin{array}[]{ll}-1&i<j\\\ 0&i=j\\\
1&i>j\end{array}\right..$

###### Proof.

First we calculate the matrix of the element $s$ with respect to the basis
$\gamma_{1},\ldots,\gamma_{l^{\prime}}$. We obtain this matrix in the form of
the Gauss decomposition of the operator
$s:{\mathfrak{h}^{\prime}}^{*}\rightarrow{\mathfrak{h}^{\prime}}^{*}$.

Let $z_{i}=s\gamma_{i}$. Recall that
$s_{\gamma_{i}}(\gamma_{j})=\gamma_{j}-A_{ij}\gamma_{i}$,
$A_{ij}=(\gamma_{i}^{\vee},\gamma_{j})$. Using this definition the elements
$z_{i}$ may be represented as

$z_{i}=y_{i}-\sum_{k\geq i}A_{ki}y_{k},$

where

$y_{i}=s_{\gamma_{1}}\ldots s_{\gamma_{i-1}}\gamma_{i}.$ (2.7.2)

Using the matrix notation we can rewrite the last formula as follows

$\begin{array}[]{l}z_{i}=(I+V)_{ki}y_{k},\\\ \\\ \mbox{ where
}V_{ki}=\left\\{\begin{array}[]{ll}A_{ki}&k\geq i\\\
0&k<i\end{array}\right.\end{array}$ (2.7.3)

To calculate the matrix of the operator
$s:{\mathfrak{h}^{\prime}}^{*}\rightarrow{\mathfrak{h}^{\prime}}^{*}$ with
respect to the basis $\gamma_{1},\ldots,\gamma_{l^{\prime}}$ we have to
express the elements $y_{i}$ via $\gamma_{1},\ldots,\gamma_{l^{\prime}}$.
Applying the definition of reflections to (2.7.2) we can pull out the element
$\gamma_{i}$ to the right,

$y_{i}=\gamma_{i}-\sum_{k<i}A_{ki}y_{k}.$

Therefore

$\gamma_{i}=(I+U)_{ki}y_{k}~{},\mbox{ where
}U_{ki}=\left\\{\begin{array}[]{ll}A_{ki}&k<i\\\ 0&k\geq i\end{array}\right.$

Thus

$y_{k}=(I+U)^{-1}_{jk}\gamma_{j}.$ (2.7.4)

Summarizing (2.7.4) and (2.7.3) we obtain

$s\gamma_{i}=\left((I+U)^{-1}(I-V)\right)_{ki}\gamma_{k}.$ (2.7.5)

This implies

${1+s\over 1-s}P_{{\mathfrak{h}^{\prime}}^{*}}\gamma_{i}=\left({2I+U-V\over
U+V}\right)_{ki}\gamma_{k}.$ (2.7.6)

Observe that $(U+V)_{ki}=A_{ki}$ and $(2I+U-V)_{ij}=-A_{ij}\varepsilon_{ij}$.
Substituting these expressions into (2.7.6) we get

$\displaystyle\left({1+s\over
1-s}P_{{\mathfrak{h}^{\prime}}^{*}}\gamma_{i},\gamma_{j}\right)=-(A^{-1})_{kp}\varepsilon_{pi}A_{pi}(\gamma_{j},\gamma_{k})=\varepsilon_{ij}(\gamma_{i},\gamma_{j}).$
(2.7.7)

This completes the proof of the lemma.

∎

Let $\gamma_{i}^{*}$, $i=1,\ldots,l^{\prime}$ be the basis of
$\mathfrak{h}^{\prime*}$ dual to $\gamma_{i}$, $i=1,\ldots,l^{\prime}$ with
respect to the restriction of the bilinear form $(\cdot,\cdot)$ to
$\mathfrak{h}^{\prime*}$. Since the numbers $(\gamma_{i},\gamma_{j})$ are
integer each element $\gamma_{i}^{*}$ has the form
$\gamma_{i}^{*}=\sum_{j=1}^{l^{\prime}}m_{ij}\gamma_{j}$, where
$m_{ij}\in\mathbb{Q}$. Therefore by Lemma 2.7.1 and using for simple roots
$\alpha_{i}$ the decomposition of the form
$P_{{\mathfrak{h}^{\prime}}^{*}}\alpha_{i}=\sum_{p=1}^{l^{\prime}}(\alpha_{i},\gamma_{p})\gamma_{p}^{*}=\sum_{p,q=1}^{l^{\prime}}(\alpha_{i},\gamma_{p})m_{pq}\gamma_{q}$
we deduce that the numbers

$\displaystyle p_{ij}=\frac{1}{2d_{j}}\left({1+s\over
1-s}P_{{\mathfrak{h}^{\prime}}^{*}}\alpha_{i},\alpha_{j}\right)=\qquad\qquad\qquad\qquad\qquad\qquad\qquad$
(2.7.8)
$\displaystyle\qquad\qquad=\frac{1}{2d_{j}}\sum_{k,l,p,q=1}^{l^{\prime}}(\gamma_{k},\alpha_{i})(\gamma_{l},\alpha_{j})\left({1+s\over
1-s}P_{{\mathfrak{h}^{\prime}}^{*}}\gamma_{p},\gamma_{q}\right)m_{kp}m_{lq},~{}i,j=1,\ldots,l$

are rational, $p_{ij}\in\mathbb{Q}$, as all factors in the products in the sum
in the right hand side are rational. Denote by $d$ an integer number divisible
by all the denominators of the rational numbers $p_{ij}$, $i,j=1,\ldots,l$.

Let $r\in\mathbb{N}$ be such that $a^{-1}_{ij}\in\frac{1}{r}\mathbb{Z}$,
$i,j=1,\ldots,l$. Let $U_{q}^{s}({\mathfrak{g}})$ be the
$\mathbb{C}(q^{\frac{1}{dr^{2}}})$-algebra generated by the elements
$e_{i},f_{i},L_{i}^{\pm 1},t_{i}^{\pm 1},~{}i=1,\ldots,l$ with the same
relations as the relations in $U_{h}^{s}({\mathfrak{g}})$ for the generators
denoted by the same symbols, where we assume that $t_{i}^{\pm
1}=\exp(\pm\frac{h\kappa}{d}Y_{i})$. The coefficients of these relations
indeed belong to $\mathbb{C}(q^{\frac{1}{dr^{2}}})$, where
$q^{\frac{1}{dr^{2}}}=e^{h\frac{1}{dr^{2}}}$.

Let $U_{q}({\mathfrak{g}})$ be the $\mathbb{C}(q^{\frac{1}{dr^{2}}})$-algebra
generated by the elements $X_{i}^{\pm},L_{i}^{\pm 1},t_{i}^{\pm
1},~{}i=1,\ldots,l$ subject to the same relations as the relations in
$U_{h}({\mathfrak{g}})$ for the generators denoted by the same symbols, where
we assume that $t_{i}^{\pm 1}=\exp(\pm\frac{h\kappa}{d}Y_{i})$. The
coefficients of these relations indeed belong to
$\mathbb{C}(q^{\frac{1}{dr^{2}}})$, where
$q^{\frac{1}{dr^{2}}}=e^{h\frac{1}{dr^{2}}}$.

Note that by the choice of $d$ we have
$q^{c_{ij}}\in\mathbb{C}[q^{\frac{\kappa}{d}},q^{-\frac{\kappa}{d}}]$.

The second form of $U_{h}^{s}({\mathfrak{g}})$ is a subalgebra
$U_{\mathcal{A}}^{s}(\mathfrak{g})$ in $U_{q}^{s}(\mathfrak{g})$ over the ring
$\mathcal{A}=\mathcal{P}[q^{\frac{1}{dr^{2}}},q^{-\frac{1}{dr^{2}}}]$.
$U_{\mathcal{A}}^{s}(\mathfrak{g})$ is the subalgebra in
$U_{q}^{s}(\mathfrak{g})$ generated over $\mathcal{A}$ by the elements

$L_{i}^{\pm 1},t_{i}^{\pm 1},~{}{K_{i}-K_{i}^{-1}\over
q_{i}-q_{i}^{-1}},~{}e_{i},~{}f_{i},~{}i=1,\ldots,l.$

Denote also by $U_{\mathcal{A}}(\mathfrak{g})$ the subalgebra in
$U_{q}(\mathfrak{g})$ generated over $\mathcal{A}$ by the elements

$L_{i}^{\pm 1},t_{i}^{\pm 1},~{}{K_{i}-K_{i}^{-1}\over
q_{i}-q_{i}^{-1}},~{}X_{i}^{\pm},~{}i=1,\ldots,l.$

For the solution $n_{ij}=\frac{1}{2d_{j}}c_{ij}$ to equations (2.6.2) the root
vectors $e_{\beta},f_{\beta}$ belong to all the above introduced
specializations of $U_{h}({\mathfrak{g}})$.

For any normal ordering of $\Delta_{+}$ we denote
$e_{\beta}^{(k)}=\frac{e_{\beta}^{k}}{[k]_{q_{\alpha}}!},f_{\beta}^{(k)}=\frac{f_{\beta}^{k}}{[k]_{q_{\alpha}}!}$,
where $e_{\beta},f_{\beta}$ are the corresponding quantum root vectors from
Proposition 2.6.2. Let $U_{\mathcal{B}}^{s,res}(\mathfrak{g})$ be the
subalgebra in $U_{q}^{s}(\mathfrak{g})$ generated over
$\mathcal{B}=\mathbb{C}[q^{\frac{1}{dr^{2}}},q^{-\frac{1}{dr^{2}}}]$ by the
elements

$L_{i}^{\pm 1},t_{i}^{\pm
1},~{}e_{i}^{(k)},~{}f_{i}^{(k)},~{}i=1,\ldots,l,k\geq 1.$

Denote also by $U_{\mathcal{B}}^{res}(\mathfrak{g})$ the subalgebra in
$U_{q}(\mathfrak{g})$ generated over $\mathcal{B}$ by the elements

$L_{i}^{\pm 1},t_{i}^{\pm 1},~{}(X_{i}^{\pm})^{(k)},~{}i=1,\ldots,l,k\geq 1.$

Let $\varepsilon\in\mathbb{C}^{*}$. Fix a root of $\varepsilon$ of degree
$r^{2}d$, $\varepsilon^{\frac{1}{dr^{2}}}$ and if $\varepsilon=1$ put
$\varepsilon^{\frac{1}{r^{2}d}}=1$. Then we define the specialization
$U_{\varepsilon}^{s}(\mathfrak{g})$ of $U_{\mathcal{A}}^{s}(\mathfrak{g})$,
$U_{\varepsilon}^{s}(\mathfrak{g})=U_{\mathcal{A}}^{s}(\mathfrak{g})/(q^{\frac{1}{dr^{2}}}-\varepsilon^{\frac{1}{dr^{2}}})U_{\mathcal{A}}^{s}(\mathfrak{g})$,
and

$U_{\varepsilon}^{s,res}(\mathfrak{g})=U_{\mathcal{B}}^{s,res}(\mathfrak{g})/(q^{\frac{1}{dr^{2}}}-\varepsilon^{\frac{1}{dr^{2}}})U_{\mathcal{B}}^{s,res}(\mathfrak{g})$

$U_{q}^{s}(\mathfrak{g})$, $U_{\mathcal{A}}^{s}(\mathfrak{g})$,
$U_{\mathcal{B}}^{s,res}(\mathfrak{g})$,
$U_{\varepsilon}^{s,res}(\mathfrak{g})$ and
$U_{\varepsilon}^{s}(\mathfrak{g})$ are Hopf algebras with the
comultiplication given on generators by the same formulas as in
$U_{h}^{s}(\mathfrak{g})$ with $q^{\frac{1}{dr^{2}}}=e^{h\frac{1}{dr^{2}}}$.

The elements $t_{i}$ and $L_{i}$ are central in the algebra
$U_{1}^{s}(\mathfrak{g})$, and the quotient of $U_{1}^{s}(\mathfrak{g})$ by
the two–sided ideal generated by $t_{i}-1$ and $L_{i}-1$ is isomorphic to
$U(\mathfrak{g})$. Note that none of the specializations of
$U_{h}^{s}(\mathfrak{g})$ introduced above is quasitriangular.

The algebra isomorphism $\psi_{\\{n_{ij}\\}}$ with
$n_{ij}=\frac{1}{2d_{j}}c_{ij}$ induces isomorphisms of $U_{h}(\mathfrak{g})$
and $U_{h}^{s}(\mathfrak{g})$ and of the forms and specializations of
$U_{h}(\mathfrak{g})$ and $U_{h}^{s}(\mathfrak{g})$ with the superscript $s$
defined above and of their counterparts with the superscript s dropped. We
shall always identify them using these isomorphisms.

Note that the structure constants in the commutation relations for
$U_{\mathcal{A}}^{s}(\mathfrak{g})=U_{\mathcal{A}}(\mathfrak{g})$ actually
belong to $\mathcal{P}[q^{\frac{\kappa}{d}},q^{-\frac{\kappa}{d}}]$, so if
$\kappa$ is divisible by $d$ the specialization
$U_{\varepsilon}^{s}(\mathfrak{g})$ actually depends on
$\varepsilon\in\mathbb{C}^{*}$ but not on its root
$\varepsilon^{\frac{1}{dr^{2}}}$. As we shall see below one can define an
action of the universal R-matrix $\mathcal{R}^{s}$ on tensor products of
finite rank $U_{\mathcal{B}}^{s,res}(\mathfrak{g})$–modules. This action will
play a crucial role in subsequent considerations.

$U_{\mathcal{A}}(\mathfrak{g})$, $U_{\mathcal{B}}^{res}(\mathfrak{g})$,
$U_{\mathcal{A}}^{s}(\mathfrak{g})$ and
$U_{\mathcal{B}}^{s,res}(\mathfrak{g})$ can be also regarded as subalgebras of
$U_{h}^{s}(\mathfrak{g})\simeq U_{h}(\mathfrak{g})$, and
$U_{q}^{res}(\mathfrak{g})$ can be regarded as a
$\mathbb{C}[q,q^{-1}]$–subalgebra in
$U_{\mathcal{B}}^{s,res}(\mathfrak{g})\simeq
U_{\mathcal{B}}^{res}(\mathfrak{g})$.

Denote by $U_{q}^{s}({\mathfrak{n}}_{+}),U_{q}^{s}({\mathfrak{n}}_{-})$ and
$U_{q}^{s}({\mathfrak{h}})$ the subalgebras of $U_{q}^{s}({\mathfrak{g}})$
generated by the $e_{i}$, $f_{i}$ and by the $t_{i},L_{i}$, respectively, and
let $U_{q}^{s}({\mathfrak{b}}_{\pm})$ be the subalgebra in
$U_{q}^{s}({\mathfrak{g}})$ generated by $U_{q}^{s}({\mathfrak{n}}_{\pm})$ and
by $U_{q}^{s}({\mathfrak{h}})$,
$U_{q}^{s}({\mathfrak{b}}_{\pm})=U_{q}^{s}({\mathfrak{n}}_{+})U_{q}^{s}({\mathfrak{h}})$.

Let $U_{\mathcal{A}}^{s}({\mathfrak{n}}_{+})$,
$U_{\mathcal{A}}^{s}({\mathfrak{n}}_{-})$
($U_{\mathcal{B}}^{s,res}({\mathfrak{n}}_{+})$,
$U_{\mathcal{B}}^{s,res}({\mathfrak{n}}_{-})$) be the subalgebras of
$U_{\mathcal{A}}^{s}({\mathfrak{g}})$
($U_{\mathcal{B}}^{s,res}(\mathfrak{g})$) generated by the $e_{i}$ and by the
$f_{i}$, $i=1,\ldots,l$ (by the $e_{i}^{(r)}$ and by the $f_{i}^{(r)}$,
$i=1,\ldots,l$, $r\geq 0$), respectively, and
$U_{\mathcal{A}}(\mathfrak{h})=U_{\mathcal{A}}^{s}(\mathfrak{h})$ the
subalgebra in
$U_{\mathcal{A}}({\mathfrak{g}})=U_{\mathcal{A}}^{s}({\mathfrak{g}})$
generated by $t_{i},L_{i}$, $i=1,\ldots,l$.

The elements

$\left[\begin{array}[]{l}K_{i};c\\\
r\end{array}\right]_{q_{i}}=\prod_{s=1}^{r}\frac{K_{i}q_{i}^{c+1-s}-K_{i}^{-1}q_{i}^{s-1-c}}{q_{i}^{s}-q_{i}^{-s}}~{},~{}i=1,\ldots,l,~{}c\in\mathbb{Z},~{}r\in\mathbb{N}$

belong to $U_{\mathcal{B}}^{s,res}(\mathfrak{g})$. Denote by
$U_{\mathcal{B}}^{s,res}(\mathfrak{h})$ the subalgebra of
$U_{\mathcal{B}}^{s,res}(\mathfrak{g})$ generated by those elements and by
$t_{i}^{\pm 1},L_{i}^{\pm 1}$, $i=1,\ldots,l$.

Define an algebra antiautomorphism $\omega$ of
$U_{h}^{s}({\mathfrak{g}})\simeq U_{h}({\mathfrak{g}})$ by

$\omega(X_{i}^{\pm})=X_{i}^{\mp},\omega(H_{i})=H_{i},\omega(h)=-h.$

It commutes with the braid group action and for any $\alpha\in\Delta_{+}$
satisfies

$\omega(X_{\alpha}^{+})=X_{\alpha}^{-},\omega(f_{\alpha})=e_{\alpha},\omega(e_{\alpha})=f_{\alpha}.$

Define also an algebra antiautomorphism $\omega_{0}$ of
$U_{h}^{s}({\mathfrak{g}})\simeq U_{h}({\mathfrak{g}})$ by

$\omega_{0}(e_{i})=e_{i},\omega_{0}(f_{i})=f_{i},\omega_{0}(H_{i})=-H_{i},\omega_{0}(h)=-h.$

It satisfies

$\omega_{0}(X_{i}^{\pm})=q_{i}^{\mp
n_{ii}}X_{i}^{\pm}=X_{i}^{\pm},\omega\omega_{0}=\omega_{0}\omega,$
$\omega_{0}(T_{i}X_{j}^{\pm})=(-1)^{a_{ij}}q_{i}^{\pm
a_{ij}(n_{ii}-1)}T_{i}(\omega_{0}X_{j}^{\pm})=(-1)^{a_{ij}}q_{i}^{\mp
a_{ij}}T_{i}(\omega_{0}X_{j}^{\pm}),i\neq j,$
$\omega_{0}(T_{i}X_{i}^{+})=q_{i}^{-2}q_{i}^{2n_{ii}}T_{i}(\omega_{0}X_{i}^{+})=q_{i}^{-2}T_{i}(\omega_{0}X_{i}^{+}),\omega_{0}(T_{i}X_{i}^{-})=q_{i}^{2}T_{i}(\omega_{0}X_{i}^{-}).$

As a consequence we obtain that if $X$ is a homogeneous polynomial in quantum
simple root vectors then

$T_{i}(\omega_{0}X)=c_{X}\omega_{0}(T_{i}X),$

where $c_{X}=\epsilon p$, $\epsilon=\pm 1$, $p\in q^{\mathbb{Z}}$, and hence

$\omega_{0}(e_{\alpha})=c_{\alpha}e_{\alpha},$ (2.7.9)

where $c_{\alpha}=\epsilon_{\alpha}p_{\alpha}$, $\epsilon_{\alpha}=\pm 1$,
$p_{\alpha}\in q^{\mathbb{Z}}$. We also have

$\omega_{0}(f_{\alpha})=\omega_{0}\omega(e_{\alpha})=\omega\omega_{0}(e_{\alpha})=\omega(c_{\alpha}e_{\alpha})=c_{\alpha}^{-1}\omega(e_{\alpha})=c_{\alpha}^{-1}f_{\alpha}$
(2.7.10)

Note also that $\omega_{0}$ is a coalgebra homomorphism.

The antiautomorphisms $\omega$ and $\omega_{0}$ give rise to antiautomorphisms
of $U_{q}^{s}({\mathfrak{g}})$, $U_{\mathcal{B}}^{s,res}(\mathfrak{g})$ and
$U_{\mathcal{A}}^{s}(\mathfrak{g})$ which we denote by the same letters.

Using the root vectors $e_{\beta}$ and $f_{\beta}$ we can construct bases for
the algebras introduced above.

######  Lemma 2.7.2.

Fix a normal ordering of the system of positive roots $\Delta_{+}$ and let
$e_{\beta},f_{\beta}$ be the corresponding quantum root vectors defined in
Proposition 2.6.2. Then the elements $f_{\beta}$ satisfy the following
commutation relations

${f}_{\alpha}{f}_{\beta}-q^{(\alpha,\beta)+\kappa({1+s\over
1-s}P_{\mathfrak{h}^{\prime*}}\alpha,\beta)}{f}_{\beta}{f}_{\alpha}=\sum_{\alpha<\delta_{1}<\ldots<\delta_{n}<\beta}C(p_{1},\ldots,p_{n}){f}_{\delta_{1}}^{(p_{1})}{f}_{\delta_{2}}^{(p_{2})}\ldots{f}_{\delta_{n}}^{(p_{n})}=$
(2.7.11)
$=\sum_{\alpha<\delta_{1}<\ldots<\delta_{n}<\beta}C^{\prime}(p_{1},\ldots,p_{n}){f}_{\delta_{1}}^{p_{1}}{f}_{\delta_{2}}^{p_{2}}\ldots{f}_{\delta_{n}}^{p_{n}},~{}\alpha<\beta,$

where $C(p_{1},\ldots,p_{n})\in\mathcal{B}$,
$C^{\prime}(p_{1},\ldots,p_{n})\in\mathcal{A}$.

The elements $e^{\bf r}=e_{\beta_{1}}^{r_{1}}\ldots
e_{\beta_{D}}^{r_{D}},~{}~{}f^{\bf t}=f_{\beta_{D}}^{t_{D}}\ldots
f_{\beta_{1}}^{t_{1}}$, for ${\bf r}=(r_{1},\ldots r_{D}),~{}{\bf
t}=(t_{1},\ldots t_{D})\in{\mathbb{N}}^{D}$, form bases of
$U_{q}^{s}({\mathfrak{n}}_{+}),U_{q}^{s}({\mathfrak{n}}_{-})$, respectively,
and the multiplication defines an isomorphism of
$\mathbb{C}(q^{\frac{1}{dr^{2}}})$–modules:

$U_{q}^{s}({\mathfrak{n}}_{-})\otimes U_{q}^{s}({\mathfrak{h}})\otimes
U_{q}^{s}({\mathfrak{n}}_{+})\rightarrow U_{q}^{s}({\mathfrak{g}}).$

The elements $e^{\bf r}$, $f^{\bf t}$
($e^{\bf(r)}=e_{\beta_{1}}^{(r_{1})}\ldots e_{\beta_{D}}^{(r_{D})}$,
$f^{\bf(t)}=f_{\beta_{D}}^{(t_{D})}\ldots f_{\beta_{1}}^{(t_{1})}$) for ${\bf
r},~{}{\bf t}\in{\mathbb{N}}^{D}$ form bases of
$U_{\mathcal{A}}^{s}({\mathfrak{n}}_{+}),U_{\mathcal{A}}^{s}({\mathfrak{n}}_{-})$
($U_{\mathcal{B}}^{s,res}({\mathfrak{n}}_{+}),U_{\mathcal{B}}^{s,res}({\mathfrak{n}}_{-})$),
respectively.

The multiplication defines an isomorphisms of $\mathcal{B}$–modules:

$U_{\mathcal{B}}^{s,res}({\mathfrak{n}}_{-})\otimes
U_{\mathcal{B}}^{s,res}({\mathfrak{h}})\otimes
U_{\mathcal{B}}^{s,res}({\mathfrak{n}}_{+})\rightarrow
U_{\mathcal{B}}^{s,res}({\mathfrak{g}}).$

Let $[\alpha,\beta]=\\{\beta_{p},\ldots,\beta_{q}\\}$ be a minimal segment in
$\Delta_{+}$, $U_{\mathcal{A}}^{s}([\alpha,\beta])$,
$U_{\mathcal{A}}^{s}([-\alpha,-\beta])$
($U_{\mathcal{B}}^{s,res}([\alpha,\beta])$,
$U_{\mathcal{B}}^{s,res}([-\alpha,-\beta])$) the
$\mathcal{A}(\mathcal{B})$–subalgebras of $U_{h}^{s}({\mathfrak{g}})$
generated by the $e_{\gamma}$ and by the $f_{\gamma}$,
$\gamma\in[\alpha,\beta]$ (by the $(e_{\gamma})^{(r)}$ and by the
$(f_{\gamma})^{(r)}$, $\gamma\in[\alpha,\beta]$, $r\in\mathbb{N}$),
respectively. Then the elements
$(e_{\beta_{p}})^{r_{p}}\ldots(e_{\beta_{q}})^{r_{q}}$,
$(f_{\beta_{q}})^{r_{q}}\ldots(f_{\beta_{p}})^{r_{p}}$
($(e_{\beta_{p}})^{(r_{p})}\ldots(e_{\beta_{q}})^{(r_{q})}$,
$(f_{\beta_{q}})^{(r_{q})}\ldots(f_{\beta_{p}})^{(r_{p})}$),
$r_{i}\in{\mathbb{N}}$ form bases of $U_{\mathcal{A}}^{s}([\alpha,\beta])$,
$U_{\mathcal{A}}^{s}([-\alpha,-\beta])$
($U_{\mathcal{B}}^{s,res}([\alpha,\beta])$,
$U_{\mathcal{B}}^{s,res}([-\alpha,-\beta])$), respectively.

The elements $f^{\bf t}$ ($f^{\bf(t)}=f_{\beta_{D}}^{(t_{D})}\ldots
f_{\beta_{1}}^{(t_{1})}$) for ${\bf t}\in{\mathbb{N}}^{D}$ with $t_{i}>0$ for
at least one $i\geq p$ form a basis in the right ideal of
$U_{\mathcal{A}}^{s}({\mathfrak{n}}_{-})$
($U_{\mathcal{B}}^{s,res}({\mathfrak{n}}_{-})$) generated by $f_{\gamma}$,
$\gamma\in[\beta_{p},\beta_{D}]$ (by $(f_{\gamma})^{(r)}$,
$\gamma\in[\beta_{p},\beta_{D}]$, $r>0$).

###### Proof.

Commutation relations (2.7.11) follow from commutation relations (2.3),
(2.1.1), (2.1.2), Proposition 2.4.1, the definition of the elements
$e_{\beta},f_{\beta}$ and the definition of the isomorphism
$\psi_{\\{n_{ij}\\}}$.

The other statements of this lemma, except for the last one, follow
straightforwardly from Lemma 2.4.2 and Propositions 2.6.1 and 2.6.2.

For the last statement, using commutation relations (2.7.11) we can represent
any element of the right ideal of $U_{\mathcal{A}}^{s}({\mathfrak{n}}_{-})$
generated by $f_{\gamma}$, $\gamma\in[\beta_{p},\beta_{D}]$ as an
$\mathcal{A}$–linear combination of the elements $f^{\bf t}$ for ${\bf
t}\in{\mathbb{N}}^{D}$ with $t_{i}>0$ for at least one $i\geq p$. This
presentation is unique by the Poincaré–Birkhoff–Witt decomposition for
$U_{\mathcal{A}}^{s}({\mathfrak{n}}_{-})$ stated above.

Note that a similar result holds for the algebra
$U_{q}^{s}({\mathfrak{n}}_{-})=U_{\mathcal{A}}^{s}({\mathfrak{n}}_{-})\otimes_{\mathcal{A}}\mathbb{C}(q^{\frac{1}{dr^{2}}})$
for the same reasons.

We can apply it to represent any element of the right ideal of
$U_{\mathcal{B}}^{s,res}({\mathfrak{n}}_{-})\subset
U_{q}^{s}({\mathfrak{n}}_{-})$ generated by $(f_{\gamma})^{(r)}$,
$\gamma\in[\beta_{p},\beta_{D}]$, $r>0$ as a
$\mathbb{C}(q^{\frac{1}{dr^{2}}})$–linear combination of the elements
$f^{\bf(t)}$ for ${\bf t}\in{\mathbb{N}}^{D}$ with $t_{i}>0$ for at least one
$i\geq p$. This presentation is unique and by the uniqueness of the
Poincaré–Birkhoff–Witt decomposition for
$U_{\mathcal{B}}^{s,res}({\mathfrak{n}}_{-})$ stated above the coefficients in
this decomposition belong to $\mathcal{B}$. This completes the proof.

∎

A basis for $U_{\mathcal{B}}^{s,res}(\mathfrak{h})$ is a little bit more
difficult to describe. We do not need its explicit description.

######  Remark 2.7.3.

Applying the antiautomorphism $\omega_{0}$ to the elements of the bases
constructed in Lemma 2.7.2 and using (2.7.9) and (2.7.10) we obtain other
bases of similar types where the order of the quantum root vectors in the
products defining the elements of the bases is reversed.

By specializing the above constructed bases for
$q^{\frac{1}{rd^{2}}}=\varepsilon^{\frac{1}{rd^{2}}}$ one can obtain similar
bases and similar subalgebras for $U_{\varepsilon}^{s}(\mathfrak{g})$ and
$U_{\varepsilon}^{s,res}(\mathfrak{g})$.

Using formulas (2.5.6) and (2.6.9) one can also find that

$\Delta_{s}(f_{\beta_{k}})=\widetilde{\mathcal{R}}_{<\beta_{k}}^{s}(e^{-h\kappa{1+s\over
1-s}P_{\mathfrak{h}^{\prime}}\beta_{k}^{\vee}+h\beta_{k}^{\vee}}\otimes
f_{\beta_{k}}+f_{\beta_{k}}\otimes
1)(\widetilde{\mathcal{R}}^{s}_{<\beta_{k}})^{-1}=$ (2.7.12)
$=G_{\beta_{k}}^{-1}\otimes f_{\beta_{k}}+f_{\beta_{k}}\otimes
1+\sum_{i}y_{i}\otimes x_{i},$

where

$G_{\beta}=e^{h\kappa{1+s\over
1-s}P_{\mathfrak{h}^{\prime}}\beta^{\vee}-h\beta^{\vee}},y_{i}=e^{-h\kappa{1+s\over
1-s}P_{\mathfrak{h}^{\prime}}\gamma_{x_{i}}^{\vee}+h\gamma_{x_{i}}^{\vee}}\overline{y}_{i},$
$\overline{y}_{i}\in U_{\mathcal{A}}^{s}([-\beta_{k+1},-\beta_{D}])\cap
U_{\mathcal{B}}^{s,res}([-\beta_{k+1},-\beta_{D}]),$ $x_{i}\in
U_{\mathcal{A}}^{s}([-\beta_{1},-\beta_{k-1}])\cap
U_{\mathcal{B}}^{s,res}([-\beta_{1},-\beta_{k-1}]),$

$\overline{y}_{i},x_{i}$ belong to weight components and have non-zero
weights, $\gamma_{x_{i}}$ is the weight of $x_{i}$,

$\widetilde{\mathcal{R}}^{s}_{<\beta_{k}}=\widetilde{\mathcal{R}}^{s}_{\beta_{1}}\ldots\widetilde{\mathcal{R}}^{s}_{\beta_{k-1}},~{}\widetilde{\mathcal{R}}^{s}_{\beta_{r}}=exp_{q_{\beta_{r}}}[(1-q_{\beta_{r}}^{-2})e_{\beta_{r}}e^{-h\kappa{1+s\over
1-s}P_{\mathfrak{h}^{\prime}}\beta^{\vee}}\otimes f_{\beta_{r}}],$

and

$(\widetilde{\mathcal{R}}_{<\beta_{k}}^{s})^{-1}=(\widetilde{\mathcal{R}}_{\beta_{k-1}}^{s})^{-1}\ldots(\widetilde{\mathcal{R}}^{s}_{\beta_{1}})^{-1},~{}(\widetilde{\mathcal{R}}^{s}_{\beta_{r}})^{-1}=exp_{q_{\beta_{r}}^{-1}}[(1-q_{\beta_{r}}^{2})e_{\beta_{r}}e^{-h\kappa{1+s\over
1-s}P_{\mathfrak{h}^{\prime}}\beta^{\vee}}\otimes f_{\beta_{r}}].$

From (2.7.12) we also obtain

$\Delta_{s}(f_{\beta_{k}}^{(n)})=\frac{1}{[n]_{q_{\beta_{k}}}!}\widetilde{\mathcal{R}}_{<\beta_{k}}^{s}(G_{\beta_{k}}^{-1}\otimes
f_{\beta_{k}}+f_{\beta_{k}}\otimes
1)^{n}(\widetilde{\mathcal{R}}^{s}_{<\beta_{k}})^{-1}=$ (2.7.13)
$=\widetilde{\mathcal{R}}_{<\beta_{k}}^{s}(\sum_{k=0}^{n}q_{\beta_{k}}^{k(n-k)}G_{\beta_{k}}^{-k}f_{\beta_{k}}^{(n-k)}\otimes
f_{\beta_{k}}^{(k)})(\widetilde{\mathcal{R}}^{s}_{<\beta_{k}})^{-1}=$
$=\sum_{k=0}^{n}q_{\beta_{k}}^{k(n-k)}G_{\beta_{k}}^{-k}f_{\beta_{k}}^{(n-k)}\otimes
f_{\beta_{k}}^{(k)}+\sum_{i}y_{i}^{(n)}\otimes x_{i}^{(n)},$

where

$y_{i}^{(n)}=e^{-h\kappa{1+s\over
1-s}P_{\mathfrak{h}^{\prime}}\gamma_{x_{i}^{(n)}}^{\vee}+h\gamma_{x_{i}^{(n)}}^{\vee}}\overline{y}_{i}^{(n)},$

$\overline{y}_{i}^{(n)}\in I_{k}^{>}$, $x_{i}^{(n)}\in I_{k}^{<}$ belong to
weight components and have non-zero weights, $\gamma_{x_{i}^{(n)}}$ is the
weight of $x_{i}^{(n)}$, $I_{k}^{>}$ is the ideal in
$U_{\mathcal{B}}^{s,res}([-\beta_{k},-\beta_{D}])$ generated by
$f_{\beta_{i}}^{(p)}$, $i=k+1,\ldots,D$, $p>0$, and $I_{k}^{<}$ is the ideal
in $U_{\mathcal{B}}^{s,res}([-\beta_{1},-\beta_{k}])$ generated by
$f_{\beta_{i}}^{(p)}$, $i=1,\ldots,k-1$, $p>0$.

Similarly

$\Delta_{s}(e_{\beta_{k}})=\widetilde{\mathcal{R}}_{<\beta_{k}}^{s}(e_{\beta_{k}}\otimes
e^{-h\beta_{k}^{\vee}}+e^{h\kappa{1+s\over
1-s}P_{\mathfrak{h}^{\prime}}\beta_{k}^{\vee}}\otimes
e_{\beta_{k}})(\widetilde{\mathcal{R}}^{s}_{<\beta_{k}})^{-1}=$ (2.7.14)
$=e_{\beta_{k}}\otimes e^{-h\beta_{k}^{\vee}}+e^{h\kappa{1+s\over
1-s}P_{\mathfrak{h}^{\prime}}\beta_{k}^{\vee}}\otimes
e_{\beta_{k}}+\sum_{i}x_{i}^{\prime}\otimes y_{i}^{\prime},$

where

$y_{i}^{\prime}\in
U_{\mathcal{A}}^{s}([\beta_{k+1},\beta_{D}])U_{\mathcal{A}}^{s}(\mathfrak{h})\cap
U_{\mathcal{B}}^{s,res}([\beta_{k+1},\beta_{D}])U_{\mathcal{B}}^{s,res}(\mathfrak{h}),$
$x_{i}^{\prime}\in
U_{\mathcal{A}}^{s}([\beta_{1},\beta_{k-1}])U_{\mathcal{A}}^{s}(\mathfrak{h})\cap
U_{\mathcal{B}}^{s,res}([\beta_{1},\beta_{k-1}])U_{\mathcal{B}}^{s,res}(\mathfrak{h}),$

and ${y}_{i}^{\prime},x_{i}^{\prime}$ belong to weight components and have
non-zero weights.

From these formulas we deduce

$S_{s}(f_{\beta})=-G_{\beta}f_{\beta}-\sum_{i}S_{s}(y_{i})x_{i}=-G_{\beta}(f_{\beta}+\sum_{i}y_{i}S_{s}(x_{i})),$
(2.7.15)
$S_{s}^{-1}(f_{\beta})=-(f_{\beta}+\sum_{i}S_{s}^{-1}(x_{i})y_{i})G_{\beta}=-f_{\beta}G_{\beta}-\sum_{i}x_{i}S_{s}^{-1}(y_{i}).$
(2.7.16)

We also have

$\omega_{0}S_{s}^{-1}(f_{\beta})=-G_{\beta}(\omega_{0}f_{\beta}+\sum_{i}\omega_{0}(y_{i})\omega_{0}S_{s}^{-1}(x_{i}))=$
$=-G_{\beta}\omega_{0}f_{\beta}-\sum_{i}\omega_{0}S_{s}^{-1}(y_{i})\omega_{0}(x_{i}),$
$\omega_{0}f_{\beta}=-G_{\beta}^{-1}(\omega_{0}S_{s}^{-1}(f_{\beta})+\sum_{i}\omega_{0}S_{s}^{-1}(y_{i})).$

As usual, one can define highest weight, Verma and finite-dimensional modules
for all forms and specializations of the quantum group
$U_{h}^{s}({\mathfrak{g}})$ introduced above. We recall that by Propositions
6.5.5 and 6.5.7 in [20] $U_{h}(\mathfrak{g})\simeq U(\mathfrak{g})[[h]]$, and
this isomorphism restricts to the identity map on $U(\mathfrak{h})$ and
induces a canonical isomorphism of the center of $U_{h}(\mathfrak{g})$ and of
$Z(U(\mathfrak{g}))[[h]]$, where $Z(U(\mathfrak{g}))$ is the center of
$U(\mathfrak{g})$. Therefore if $V$ is a $U_{h}(\mathfrak{g})$–module
topologically free and of finite rank over $\mathbb{C}[[h]]$ then $V_{1}=V/hV$
is a finite-dimensional $U(\mathfrak{g})$–module which is completely
reducible, and its irreducible components are highest weight irreducible
finite-dimensional representations of $U(\mathfrak{g})$. Since, as observed in
Section 6.5 B in [20], such modules have no non–trivial deformations one has
$V\simeq V_{1}[[h]]$.

If $V_{1}$ is a highest weight irreducible representation of $U(\mathfrak{g})$
with highest weight $\lambda\in P_{+}$ we call the corresponding
representation $V=V_{\lambda}$ a highest weight indecomposable representation
of highest weight $\lambda$. $V_{\lambda}$ is generated by a highest weight
vector with respect to the action of $\mathfrak{h}\subset
U_{h}(\mathfrak{h})$. All indecomposable $U_{h}(\mathfrak{g})$–modules
topologically free and of finite rank over $\mathbb{C}[[h]]$ can be obtained
this way.

Recall that $U_{\mathcal{B}}^{s,res}(\mathfrak{g})\simeq
U_{\mathcal{B}}^{res}(\mathfrak{g})$ can be regarded as a subalgebra of
$U_{h}^{s}({\mathfrak{g}})\simeq U_{h}(\mathfrak{g})$. Let $V$ be a
$U_{h}(\mathfrak{g})$–module topologically free and of finite rank over
$\mathbb{C}[[h]]$ (for brevity we shall call such modules finite rank
$U_{h}(\mathfrak{g})$–modules). Then a
$U_{\mathcal{B}}^{s,res}(\mathfrak{g})$–module $V^{res}$ is called a
$U_{\mathcal{B}}^{s,res}(\mathfrak{g})$–lattice in $V$ if
$V^{res}\otimes_{\mathcal{B}}\mathbb{C}[[h]]\simeq V$.

For any $U_{h}(\mathfrak{g})$–module $V$ topologically free and of finite rank
over $\mathbb{C}[[h]]$ a $U_{\mathcal{B}}^{s,res}(\mathfrak{g})$–lattice
$V^{res}$ exists. Indeed, by the above discussion it suffices to consider the
case of indecomposable $V=V_{\lambda}$ generated by a highest weight vector
$v$. In this case, similarly to the proof of Proposition 4.2 in [80] (see also
Proposition 10.1.4 in [20]) one can show that
$V^{res}_{\lambda}=U_{\mathcal{B}}^{s,res}(\mathfrak{g})v$ is a
$U_{\mathcal{B}}^{s,res}(\mathfrak{g})$–lattice, $V^{res}_{\lambda}$ is the
direct sum of its intersections with the weight spaces of $V_{\lambda}$ and
each such intersection is a finitely generated free $\mathcal{B}$–module of
finite rank. Moreover, using the arguments from the proof of Proposition 4.2
in Section 4.9 of [80] one can see that the last two properties hold for any
$U_{\mathcal{B}}^{s,res}(\mathfrak{g})$–lattice in any topologically free
$U_{h}(\mathfrak{g})$–module $V$ of finite rank over $\mathbb{C}[[h]]$.

The specialization of $V^{res}_{\lambda}$ at
$q^{\frac{1}{dr^{2}}}=\varepsilon^{\frac{1}{dr^{2}}}$ is a highest weight
$U_{\varepsilon}^{s,res}(\mathfrak{g})$–module. In general this module is not
irreducible even if $V$ is indecomposable.

The R-matrix $\mathcal{R}^{s}$ acts in tensor products of
$U_{\mathcal{B}}^{s,res}(\mathfrak{g})$–lattices in
$U_{h}(\mathfrak{g})$–modules topologically free and of finite rank over
$\mathbb{C}[[h]]$. Namely, recalling (2.6.11) we can represent
$\mathcal{R}^{s}$ in the form
$\mathcal{R}^{s}=\mathcal{E}^{s}\mathcal{R}^{s}_{0}$, where

$\displaystyle\mathcal{R}^{s}_{0}=exp\left[h(\sum_{i=1}^{l}(Y_{i}\otimes
H_{i})-\sum_{i=1}^{l}\kappa{1+s\over 1-s}P_{\mathfrak{h}^{\prime}}H_{i}\otimes
Y_{i})\right],$ (2.7.17)
$\displaystyle{\mathcal{E}}^{s}=\prod_{\beta}\sum_{k=0}^{\infty}q_{\beta}^{{k(k+1)}\over
2}(1-q_{\beta}^{-2})^{k}f_{\beta}^{(k)}\otimes
e_{\beta}^{k}e^{-kh\kappa{1+s\over
1-s}P_{\mathfrak{h}^{\prime}}\beta^{\vee}}=$
$\displaystyle=\prod_{\beta}\sum_{k=0}^{\infty}q_{\beta}^{{k(k+1)}\over
2}(1-q_{\beta}^{-2})^{k}f_{\beta}^{k}\otimes
e_{\beta}^{(k)}e^{-kh\kappa{1+s\over
1-s}P_{\mathfrak{h}^{\prime}}\beta^{\vee}},$

and for $\beta=\sum_{i=1}^{l}c_{i}\alpha_{i}$

$e^{-kh\kappa{1+s\over
1-s}P_{\mathfrak{h}^{\prime}}\beta^{\vee}}=e^{-kh\sum_{i,j=1}^{l}\frac{c_{ij}}{d_{j}}c_{i}Y_{j}}\in
U_{\mathcal{B}}^{s,res}(\mathfrak{h})\cap U_{\mathcal{A}}^{s}(\mathfrak{h}).$

We can define the action of $\mathcal{R}^{s}_{0}$ on tensor products of
modules of the form $V^{res}$, where $V$ is a highest weight finite rank
$U_{h}({\mathfrak{g}})$–module as follows. If $V^{res}$, $W^{res}$ are two
such modules and $v_{\lambda}\in V^{res}$, $w_{\mu}\in W^{res}$ are vectors of
weights $\lambda$ and $\mu$ then we define

$\mathcal{R}^{s}_{0}v_{\lambda}\otimes
w_{\mu}=q^{(\lambda,\mu)-(\kappa{1+s\over
1-s}P_{\mathfrak{h}^{\prime*}}\mu,\lambda)}v_{\lambda}\otimes w_{\mu},$

and $q^{(\lambda,\mu)-(\kappa{1+s\over
1-s}P_{\mathfrak{h}^{\prime*}}\mu,\lambda)}\in\mathcal{B}$.

Since for any $\beta\in\Delta_{+}$ and any module of the form $V^{res}$, where
$V$ is a finite rank $U_{h}({\mathfrak{g}})$–module, $f_{\beta}^{(k)}$ and
$e_{\beta}^{k}$ belong to $U_{\mathcal{B}}^{s,res}(\mathfrak{g})$ and act as
zero operators on $V^{res}_{\mathcal{B}}$ for $k$ large enough, and for any
$k$

$e^{-kh\kappa{1+s\over 1-s}P_{\mathfrak{h}^{\prime}}\beta^{\vee}}\in
U_{\mathcal{B}}^{s,res}(\mathfrak{h})\cap U_{\mathcal{A}}^{s}(\mathfrak{h}),$

$\mathcal{E}^{s}$ naturally acts on tensor products of such modules.

For two modules of the form $V^{res},W^{res}$, where $V$ and $W$ are highest
weight finite rank $U_{h}({\mathfrak{g}})$–modules, we denote by $R^{VW}$ the
operator corresponding to the action of $\mathcal{R}^{s}$ in $V^{res}\otimes
W^{res}$.

With this definition the identity

$\Delta_{s}^{opp}(x)=\mathcal{R}^{s}\Delta_{s}(x){\mathcal{R}^{s}}^{-1},x\in
U_{\mathcal{B}}^{s,res}(\mathfrak{g})$

still holds being evaluated in tensor products of modules of the form
$V^{res}$, where $V$ is a finite rank $U_{h}({\mathfrak{g}})$–module.

From the explicit formulas and the results in [73], Section 5.2 it follows
that the braid group elements given by (2.2.3) and (2.2.4) act in
$U_{h}(\mathfrak{g})$–modules topologically free and of finite rank over
$\mathbb{C}[[h]]$ and in $U_{\mathcal{B}}^{s,res}(\mathfrak{g})$–lattices in
them.

We shall also need the following technical lemma regarding the action of the
elements $T_{w}$ and $\overline{T}_{w}$ on finite rank indecomposable modules.

######  Lemma 2.7.4.

Let $V$ be a finite rank $U_{h}(\mathfrak{g})$–module. If $T,T^{\prime}$ are
two elements of the braid group $\mathcal{B}_{\mathfrak{g}}$ which act as the
same transformation on $\mathfrak{h}\subset U_{h}(\mathfrak{h})$ and $v$ is a
highest weight vector in $V$ then $Tv=tT^{\prime}v$, where $t$ is a non–zero
multiple of a power of $q$.

###### Proof.

Since $v$ generates a highest weight indecomposable submodule
$V_{\lambda}\subset V$ of highest weight $\lambda$ equal to the weight of $v$,
and $V_{\lambda}$ is invariant under the braid group action, we can assume
without loss of generality that $V=V_{\lambda}$.

First observe that similarly to the proof of Proposition 4.2 in [80] (see also
Proposition 10.1.4 in [20]) one can show that
$V^{\prime}=U_{q}^{res}(\mathfrak{g})v$ is a
$U_{q}^{res}(\mathfrak{g})$–lattice in $V_{\lambda}$ in the sense that
$V^{\prime}\otimes_{\mathbb{C}[q,q^{-1}]}\mathbb{C}[[h]]\simeq V_{\lambda}$.
This module coincides with the one defined in Section 4.1 in [75] (see also
Proposition 10.1.4 in [20]).

It is well known that any element $T$ of the braid group acts as an invertible
linear automorphism of $V^{\prime}$ which can be specialized to any non–zero
numeric value of $q$ in the sense that for any $\varepsilon\in\mathbb{C}^{*}$
$T$ gives rise to a linear automorphism of
$U_{q}^{res}(\mathfrak{g})/(q-\varepsilon)U_{q}^{res}(\mathfrak{g})$–module
$V^{\prime}_{\varepsilon}=V^{\prime}/(q-\varepsilon)V^{\prime}$. It suffices
to verify this statement when $T=T_{i}$ for $i=1,\ldots,l$, and in this case
it follows from the explicit formulas and the results in [73], Section 5.2.

Recall that elements of the braid group act as Weyl group elements on
$\mathfrak{h}\subset U_{h}(\mathfrak{h})$. Assume that the action of $T$ and
$T^{\prime}$ on $\mathfrak{h}\subset U_{h}(\mathfrak{h})$ coincides with the
action of a Weyl group element $w$. Since the $\mathbb{C}[q,q^{-1}]$–submodule
of $V^{\prime}$ which consists of elements of weight $w\mu$ has rank one and
$Tv$ and $T^{\prime}v$ must belong to this submodule, the relation
$Tv=t(q)T^{\prime}v$ must hold for some rational function $t(q)$ of $q$ with
poles or zeroes only at zero and infinity. Indeed, if $t(q_{0})=0$, $q_{0}\neq
0,\infty$ then in $V^{\prime}_{q_{0}}$ we have $Tv=0$, i.e. $T$ does not
induce an automorphism of $V^{\prime}_{q_{0}}$, and if
$t^{-1}(q_{0}^{\prime})=0$, $q_{0}^{\prime}\neq 0,\infty$ then in
$V^{\prime}_{q_{0}^{\prime}}$ we have $T^{\prime}v=0$, i.e. $T^{\prime}$ does
not induce an automorphism of $V^{\prime}_{q_{0}^{\prime}}$. In both cases we
arrive at a contradiction. Thus $t(q)$ must be a non–zero multiple of a power
of $q$.

∎

###  2.8 Bibliographic comments

The material presented in Sections 2.1, 2.2, 2.3, 2.4 and 2.5 is mostly
standard and we refer to books [20, 54, 73] for more details and omitted
proofs. Formula (2.2.3) can be found in [93].

Realizations of quantum groups associated to Weyl group elements were
introduces in [98] in the case of Coxeter elements and in [104] in general.

Lemma 2.7.1 is a generalization of the result of Exercise 3 in Chapter V, §6,
[10].

Specializations of quantum groups similar to those which appear in this book
were considered in [106]. In this book we introduce slightly different
specializations of quantum groups in order to use restricted specializations
as well.

##  Chapter 3 q-W–algebras

In this chapter we introduce q–W–algebras and study the structure of their the
quasi–classical versions, Poisson q-W–algebras. In the next chapter similar
results will be obtained for q-W–algebras.

As we briefly mentioned in the introduction the naive definition of
q-W–algebras as Hecke type algebras $Hk(A,B,\chi)$ requires some modification.
In fact the main ingredient of the definition of q-W–algebras is the adjoint
action of the quantum group on itself, and they are defined using a
$\mathcal{B}$–subalgebra $\mathbb{C}_{\mathcal{B}}[G_{*}]$ of the quantum
group the restriction of the adjoint action to which is locally finite. When
$q$ is specialized to $\varepsilon\in\mathbb{C}^{*}$ which is not a root of
unity the algebra $\mathbb{C}_{\mathcal{B}}[G_{*}]$ becomes the locally finite
part of the quantum group with respect to the adjoint action which was
introduced and studied by Joseph.

The algebra $\mathbb{C}_{\mathcal{B}}[G_{*}]$ is a quantization of the algebra
of regular functions on a Poisson manifold $G_{*}$ which is isomorphic to $G$
as a manifold and the Poisson structure of which is closely related to that of
the Poisson–Lie group $G^{*}$ dual to a quasitriangular Poisson–Lie group $G$.

After recalling basic facts on Poisson–Lie groups in Section 3.1 we introduce
an algebra $\mathbb{C}[G^{*}]$ of functions on $G^{*}$ in Section 3.2, its
quantization $\mathbb{C}_{\mathcal{B}}[G^{*}]\subset U_{h}^{s}(\mathfrak{g})$
and the subalgebra
$\mathbb{C}_{\mathcal{B}}[G_{*}]\subset\mathbb{C}_{\mathcal{B}}[G^{*}]$.

A special choice of the bialgebra structure entering the definitions of
$\mathbb{C}[G^{*}]$, $\mathbb{C}_{\mathcal{B}}[G^{*}]\subset
U_{h}^{s}(\mathfrak{g})$ and $\mathbb{C}_{\mathcal{B}}[G_{*}]$ is crucial for
the definition of q-W–algebras. It depends on the choice of a Weyl group
element $s\in W$ and ensures that one can define a subalgebra
$\mathbb{C}_{\mathcal{B}}[M_{+}]\subset\mathbb{C}_{\mathcal{B}}[G^{*}]$
equipped with a non–trivial character, so that the q-W–algebra
$W_{\mathcal{B}}^{s}(G)$ can be defined as the result of a quantum constrained
reduction with respect to the subalgebra $\mathbb{C}_{\mathcal{B}}[M_{+}]$.

Next, in Section 3.4 we proceed with the study of the specialization
$W^{s}(G)$ of the algebra $W_{\mathcal{B}}^{s}(G)$ at
$q^{\frac{1}{rd^{2}}}=1$. We recall that $W^{s}(G)$ is naturally a Poisson
algebra which can be regarded as the algebra of regular functions on a reduced
Poisson manifold which is also an algebraic variety. Poisson reduction works
well for differential Poisson manifolds. Therefore it is easier firstly to
describe the reduced Poisson structure on the algebra of
$C^{\infty}$–functions on the reduced Poisson manifold and then to recover the
structure of the algebraic variety on it. This is done in Proposition 3.4.3
and Theorem 3.4.5.

In Section 3.5 we define a projection operator $\Pi$ into the algebra
$W^{s}(G)$. In Theorem 3.5.6, which is central in this chapter, we obtain a
formula for the operator $\Pi$ suitable for quantization. This formula plays
the key role in the proof of Theorem 4.7.2 describing a localization of the
algebra $W_{\mathcal{B}}^{s}(G)$ in terms of a quantum counterpart of the
operator $\Pi$. Miraculously the formula for $\Pi$ from Theorem 3.5.6 can be
directly extrapolated to the quantum case.

###  3.1 Some facts on Poisson–Lie groups

In this section we recall some notions related to Poisson–Lie groups. These
facts will be needed for the study of Poisson q-W–algebras.

Let $G$ be a finite-dimensional Lie group equipped with a Poisson bracket,
$\mathfrak{g}$ its Lie algebra. $G$ is called a Poisson–Lie group if the
multiplication $G\times G\rightarrow G$ is a Poisson map. A Poisson bracket
satisfying this axiom is degenerate and, in particular, is identically zero at
the unit element of the group. Linearizing this bracket at the unit element
defines the structure of a Lie algebra in the space
$T^{*}_{e}G\simeq{\mathfrak{g}}^{*}$. The pair
(${\mathfrak{g}},{\mathfrak{g}}^{*})$ is called the tangent bialgebra of $G$.

Lie brackets in $\mathfrak{g}$ and $\mathfrak{g}^{*}$ satisfy the following
compatibility condition:

Let $\delta:{\mathfrak{g}}\rightarrow{\mathfrak{g}}\wedge{\mathfrak{g}}$ be
the dual of the commutator map
$[,]_{*}:{\mathfrak{g}}^{*}\wedge{\mathfrak{g}}^{*}\rightarrow{\mathfrak{g}}^{*}$.
Then  $\delta$ is a 1-cocycle on ${\mathfrak{g}}$ (with respect to the adjoint
action of  $\mathfrak{g}$ on ${\mathfrak{g}}\wedge{\mathfrak{g}}$).

Let $c_{ij}^{k},f^{ab}_{c}$ be the structure constants of
${\mathfrak{g}},{\mathfrak{g}}^{*}$ with respect to the dual bases
$\\{e_{i}\\},\\{e^{i}\\}$ in ${\mathfrak{g}},{\mathfrak{g}}^{*}$. The
compatibility condition means that

$c_{ab}^{s}f^{ik}_{s}~{}-~{}c_{as}^{i}f^{sk}_{b}~{}+~{}c_{as}^{k}f^{si}_{b}~{}-~{}c_{bs}^{k}f^{si}_{a}~{}+~{}c_{bs}^{i}f^{sk}_{a}~{}~{}=~{}~{}0.$

This condition is symmetric with respect to exchange of $c$ and $f$. Thus if
$({\mathfrak{g}},{\mathfrak{g}}^{*})$ is a Lie bialgebra, then
$({\mathfrak{g}}^{*},{\mathfrak{g}})$ is also a Lie bialgebra.

The following proposition shows that the category of finite-dimensional Lie
bialgebras is isomorphic to the category of finite-dimensional connected
simply connected Poisson–Lie groups.

######  Proposition 3.1.1.

If $G$ is a connected simply connected finite-dimensional Lie group, every
bialgebra structure on $\mathfrak{g}$ is the tangent bialgebra of a unique
Poisson structure on $G$ which makes $G$ into a Poisson–Lie group.

Let $G$ be a finite-dimensional Poisson–Lie group,
$({\mathfrak{g}},{\mathfrak{g}}^{*})$ the tangent bialgebra of $G$. The
connected simply connected finite-dimensional Poisson–Lie group corresponding
to the Lie bialgebra $({\mathfrak{g}}^{*},{\mathfrak{g}})$ is called the dual
Poisson–Lie group and denoted by $G^{*}$.

$({\mathfrak{g}},{\mathfrak{g}}^{*})$ is called a factorizable Lie bialgebra
if the following conditions are satisfied:

  1. 1.

${\mathfrak{g}}$ is equipped with a non–degenerate invariant scalar product
$\left(\cdot,\cdot\right)$.

We shall always identify ${\mathfrak{g}}^{*}$ and ${\mathfrak{g}}$ by means of
this scalar product.

  2. 2.

The dual Lie bracket on ${\mathfrak{g}}^{*}\simeq{\mathfrak{g}}$ is given by

$\left[X,Y\right]_{*}=\frac{1}{2}\left(\left[rX,Y\right]+\left[X,rY\right]\right),X,Y\in{\mathfrak{g}},$
(3.1.1)

where $r\in{\rm End}\ {\mathfrak{g}}$ is a skew symmetric linear operator
(classical r-matrix).

  3. 3.

$r$ satisfies the modified classical Yang-Baxter identity:

$\left[rX,rY\right]-r\left(\left[rX,Y\right]+\left[X,rY\right]\right)=-\left[X,Y\right],\;X,Y\in{\mathfrak{g}}{\bf.}$
(3.1.2)

Define operators $r_{\pm}\in{\rm End}\ {\mathfrak{g}}$ by

$r_{\pm}=\frac{1}{2}\left(r\pm id\right).$

We shall need some properties of the operators $r_{\pm}$. Denote by
${\mathfrak{b}}_{\pm}$ and ${\mathfrak{n}}_{\mp}$ the image and the kernel of
the operator $r_{\pm}$:

${\mathfrak{i}}_{\pm}=Im~{}r_{\pm},~{}~{}{\mathfrak{k}}_{\mp}=Ker~{}r_{\pm}.$
(3.1.3)

######  Proposition 3.1.2.

Let $({\mathfrak{g}},{\mathfrak{g}}^{*})$ be a factorizable Lie bialgebra.
Then

(i) ${\mathfrak{i}}_{\pm}\subset{\mathfrak{g}}$ is a Lie subalgebra, the
subspace ${\mathfrak{k}}_{\pm}$ is a Lie ideal in
${\mathfrak{i}}_{\pm},~{}{\mathfrak{i}}_{\pm}^{\perp}={\mathfrak{k}}_{\pm}$.

(ii) ${\mathfrak{k}}_{\pm}$ is an ideal in ${\mathfrak{g}}^{*}$.

(iii) ${\mathfrak{i}}_{\pm}$ is a Lie subalgebra in ${\mathfrak{g}}^{*}$.
Moreover ${\mathfrak{i}}_{\pm}={\mathfrak{g}}^{*}/{\mathfrak{k}}_{\pm}$.

(iv) $({\mathfrak{i}}_{\pm},{\mathfrak{i}}_{\pm}^{*})$ is a subbialgebra of
$({\mathfrak{g}},{\mathfrak{g}}^{*})$ and
$({\mathfrak{i}}_{\pm},{\mathfrak{i}}_{\pm}^{*})\simeq({\mathfrak{i}}_{\pm},{\mathfrak{i}}_{\mp})$.
The canonical paring between ${\mathfrak{i}}_{\mp}$ and
${\mathfrak{i}}_{\pm}$is given by

$(X_{\mp},Y_{\pm})_{\pm}=(X_{\mp},r_{\pm}^{-1}Y_{\pm}),~{}X_{\mp}\in{\mathfrak{i}}_{\mp};~{}Y_{\pm}\in{\mathfrak{i}}_{\pm}.$
(3.1.4)

The classical Yang–Baxter equation implies that $r_{\pm}$ , regarded as a
mapping from ${\mathfrak{g}}^{*}$ into ${\mathfrak{g}}$, is a Lie algebra
homomorphism. Moreover, $r_{+}^{*}=-r_{-},$ and $r_{+}-r_{-}=id.$

Put ${\mathfrak{d}}={\mathfrak{g}\oplus{\mathfrak{g}}}$ (direct sum of two
copies). The mapping

$\displaystyle{\mathfrak{g}}^{*}\rightarrow{\mathfrak{d}}~{}~{}~{}:X\mapsto(X_{+},~{}X_{-}),~{}~{}~{}X_{\pm}~{}=~{}r_{\pm}X$
(3.1.5)

is a Lie algebra embedding. Thus we may identify ${\mathfrak{g}^{*}}$ with a
Lie subalgebra in ${\mathfrak{d}}$.

Naturally, embedding (3.1.5) extends to a homomorphism

$G^{*}\rightarrow G\times G,~{}~{}L\mapsto(L_{+},L_{-}).$

We shall identify $G^{*}$ with the corresponding subgroup in $G\times G$.

###  3.2 Quantization of algebraic Poisson–Lie groups and the definition of
q-W–algebras

In this section we introduce the main object of this book, q-W–algebras. We
start by defining the relevant Poisson–Lie groups and their quantizations. We
consider algebras defined over the ring $\mathcal{B}$ since later in our
construction the restricted specialization of the quantum group
$U_{h}^{s}(\mathfrak{g})$ defined over $\mathcal{B}$ will play the key role.

Let $\mathfrak{g}$ be a finite-dimensional complex semisimple Lie algebra,
$\mathfrak{h}\subset\mathfrak{g}$ its Cartan subalgebra. Let $s\in W$ be an
element of the Weyl group $W$ of the pair $(\mathfrak{g},\mathfrak{h})$ and
$\Delta_{+}$ the system of positive roots associated to $s$. Observe that
cocycle (2.6.10) equips $\mathfrak{g}$ with the structure of a factorizable
Lie bialgebra, where the scalar product is given by the symmetric bilinear
form. Using the identification ${\rm
End}~{}{\mathfrak{g}}\cong{\mathfrak{g}}\otimes{\mathfrak{g}}$ the
corresponding r–matrix may be represented as

$r^{s}=P_{+}-P_{-}+\kappa{1+s\over 1-s}P_{\mathfrak{h}^{\prime}},$

where $P_{+},P_{-}$ and $P_{\mathfrak{h}^{\prime}}$ are the orthogonal
projection operators onto the nilradical ${\mathfrak{n}}_{+}$ corresponding to
$\Delta_{+}$, the opposite nilradical ${\mathfrak{n}}_{-}$, and
${\mathfrak{h}}^{\prime}$, respectively, in the direct sum

${\mathfrak{g}}={\mathfrak{n}}_{+}+{\mathfrak{h}}^{\prime}+\mathfrak{h}^{\prime\perp}+{\mathfrak{n}}_{-},$

and $\mathfrak{h}^{\prime\perp}$ is the orthogonal complement to
$\mathfrak{h}^{\prime}$ in $\mathfrak{h}$ with respect to the symmetric
bilinear form.

Let $G$ be the connected simply connected semisimple Poisson–Lie group with
the tangent Lie bialgebra $({\mathfrak{g}},{\mathfrak{g}}^{*})$, $G^{*}$ the
dual Poisson–Lie group.

Observe that $G$ is an algebraic group (see e.g. §104, Theorem 12 in [119]).
Note also that

$r^{s}_{+}=P_{+}+\frac{\kappa}{2}{1+s\over
1-s}P_{\mathfrak{h}^{\prime}}+\frac{1}{2}P_{\mathfrak{h}},~{}~{}r^{s}_{-}=-P_{-}+\frac{\kappa}{2}{1+s\over
1-s}P_{\mathfrak{h}^{\prime}}-\frac{1}{2}P_{\mathfrak{h}},$

where $P_{{\mathfrak{h}}}$ is the orthogonal projection operator onto
${\mathfrak{h}}\subset\mathfrak{g}$ with respect to the symmetric bilinear
form, and hence the subspaces ${\mathfrak{i}}_{\pm}$ and
${\mathfrak{k}}_{\pm}$ defined by (3.1.3) coincide with the Borel subalgebras
${\mathfrak{b}}_{\pm}$ in $\mathfrak{g}$ corresponding to $\Delta_{\pm}$ and
their nilradicals ${\mathfrak{n}}_{\pm}$, respectively. Therefore every
element $(L_{+},L_{-})\in G^{*}$ may be uniquely written as

$(L_{+},L_{-})=(n_{+},n_{-})(h_{+},h_{-}),$ (3.2.1)

where $n_{\pm}\in N_{\pm}$, $h_{+}=exp((\frac{\kappa}{2}{1+s\over
1-s}P_{{\mathfrak{h}^{\prime}}}+\frac{1}{2}id)x),~{}h_{-}=exp((\frac{\kappa}{2}{1+s\over
1-s}P_{{\mathfrak{h}^{\prime}}}-\frac{1}{2}id)x),~{}x\in{\mathfrak{h}}$. In
particular, $G^{*}$ is a solvable subgroup in $G\times G$. In general $G^{*}$
does not need to be algebraic.

Our main object will be a certain algebra of functions on $G^{*}$,
${\mathbb{C}}[G^{*}]$. This algebra may be explicitly described as follows.
Let $\pi_{V}$ be a finite-dimensional representation of $G$. Then matrix
elements of $\pi_{V}(L_{\pm})$ are well–defined functions on $G^{*}$, and
${\mathbb{C}}[G^{*}]$ is the subspace in $C^{\infty}(G^{*})$ generated by
matrix elements of $\pi_{V}(L_{\pm})$, where $V$ runs through all
finite–dimensional representations of $G$. The elements
$L^{\pm,V}=\pi_{V}(L_{\pm})$ may be viewed as elements of the space
${\mathbb{C}}[G^{*}]\otimes{\rm End}V$. For every two finite–dimensional
${\mathfrak{g}}$–modules $V$ and $W$ we denote
${r^{s}_{+}}^{VW}=(\pi_{V}\otimes\pi_{W})r^{s}_{+}$, where $r^{s}_{+}$ is
regarded as an element of ${\mathfrak{g}}\otimes{\mathfrak{g}}$.

######  Proposition 3.2.1.

${\mathbb{C}}[G^{*}]$ is a Poisson subalgebra in the Poisson algebra
$C^{\infty}(G^{*})$, the Poisson brackets of the elements $L^{\pm,V}$ are
given by

$\begin{array}[]{l}\\{L^{\pm,V}_{1},L^{\pm,W}_{2}\\}=-2[{r_{\pm}^{s}}^{VW},L^{\pm,V}_{1}L^{\pm,W}_{2}],\\\
\\\
\\{L^{-,V}_{1},L^{+,W}_{2}\\}~{}=-2[{r_{\pm}^{s}}^{VW},L^{-,V}_{1}L^{+,W}_{2}],\end{array}$
(3.2.2)

where

$L^{\pm,V}_{1}=L^{\pm,V}\otimes I_{W},~{}~{}L^{\pm,W}_{2}=I_{V}\otimes
L^{\pm,W},$

and $I_{X}$ is the unit matrix in $X$.

Moreover, the map
$\Delta:{\mathbb{C}}[G^{*}]\rightarrow{\mathbb{C}}[G^{*}]\otimes{\mathbb{C}}[G^{*}]$
dual to the multiplication in $G^{*}$,

$\Delta(L^{\pm,V}_{ij})=\sum_{k}L^{\pm,V}_{ik}\otimes L^{\pm,V}_{kj},$ (3.2.3)

is a homomorphism of Poisson algebras, and the map
$S:{\mathbb{C}}[G^{*}]\rightarrow{\mathbb{C}}[G^{*}]$,

$S(L^{\pm,V}_{ij})=(L^{\pm,V})^{-1}_{ij}$

is an antihomomorphism of Poisson algebras.

######  Remark 3.2.2.

Recall that a Poisson–Hopf algebra is a Poisson algebra which is also a Hopf
algebra such that the comultiplication is a homomorphism of Poisson algebras
and the antipode is an antihomomorphism of Poisson algebras. According to
Proposition 3.2.1 ${\mathbb{C}}[G^{*}]$ is a Poisson–Hopf algebra.

Now we construct a quantization of the Poisson–Hopf algebra
${\mathbb{C}}[G^{*}]$. For any finite rank representation
$\pi_{V}:U_{\mathcal{B}}^{s,res}({\mathfrak{g}})\rightarrow V^{res}$, where
$V$ is a finite rank representation of $U_{h}({\mathfrak{g}})$, one can define
an action of elements $H_{i}$, $i=1,\ldots,l$ on $V^{res}$ by requiring that
$H_{i}$ acts on weight vectors of weight $\lambda$ by multiplication by
$\lambda(H_{i})$. Then from the definition of the R–matrix ${\mathcal{R}}^{s}$
it follows that ${{}^{q}{L^{\pm,V}}}$ given by

${{}^{q}{L^{-,V}}}=(id\otimes\pi_{V}){{\mathcal{R}}_{21}^{s}}^{-1}=(id\otimes\pi_{V}S^{s}){\mathcal{R}}_{21}^{s},~{}~{}{{}^{q}{L^{+,V}}}=(id\otimes\pi_{V}){\mathcal{R}}^{s}.$

are well–defined invertible elements of $U_{h}^{s}({\mathfrak{g}})\otimes{\rm
End}_{\mathcal{B}}(V^{res})$.

If we fix a basis in $V^{res}$, ${{}^{q}{L^{\pm,V}}}$ may be regarded as
matrices with matrix elements $({{}^{q}{L^{\pm,V}}})_{ij}$ being elements of
$U_{h}^{s}({\mathfrak{g}})$.

We denote by ${\mathbb{C}}_{\mathcal{B}}[G^{*}]$ the $\mathcal{B}$–Hopf
subalgebra in $U_{h}^{s}({\mathfrak{g}})$ generated by matrix elements of
$({{}^{q}{L^{\pm,V}}})^{\pm 1}$, where $V$ runs through all finite rank
representation of $U_{h}({\mathfrak{g}})$.

From the definition of the elements ${{}^{q}{L^{\pm,V}}}$ and from formula
(2.7.17) it follows that ${\mathbb{C}}_{\mathcal{B}}[G^{*}]$ is the
$\mathcal{B}$–subalgebra in $U_{h}^{s}({\mathfrak{g}})$ generated by the
elements
$q^{\pm(Y_{i}-\kappa\frac{1+s}{1-s}P_{\mathfrak{h}^{\prime}}Y_{i})},q^{\pm(-Y_{i}-\kappa\frac{1+s}{1-s}P_{\mathfrak{h}^{\prime}}Y_{i})},~{}i=1,\ldots,l,~{}\tilde{f}_{\beta}=(1-q_{\beta}^{-2})f_{\beta},~{}\tilde{e}_{\beta}=(1-q_{\beta}^{2})e_{\beta}e^{h\beta^{\vee}},~{}\beta\in\Delta_{+}$
(see Section 1.4 in [28] and Theorem 4.6 in [27] for a similar result in case
of the quantum group associated to the standard bialgebra structure).

From the Yang–Baxter equation for ${\mathcal{R}}^{s}$ we get relations between
${{}^{q}{L^{\pm,V}}}$,

$\begin{array}[]{l}R^{VW}{{}^{q}{L^{\pm,W}_{2}}}{{}^{q}{L^{\pm,V}_{1}}}={{}^{q}{L^{\pm,V}_{1}}}{{}^{q}{L^{\pm,W}_{2}}}R^{VW},\end{array}$
(3.2.4)
$R^{VW}{{}^{q}{L^{+,W}_{2}}}{{}^{q}{L^{-,V}_{1}}}={{}^{q}{L^{-,V}_{1}}}{{}^{q}{L^{+,W}_{2}}}R^{VW}.$
(3.2.5)

By ${{}^{q}{L^{\pm,W}_{1}}},~{}{{}^{q}{L^{\pm,V}_{2}}}$ we understand the
following matrices in $V^{res}\otimes W^{res}$ with entries being elements of
$U_{h}^{s}({\mathfrak{g}})$

${{}^{q}{L^{\pm,V}_{1}}}={{}^{q}{L^{\pm,V}}}\otimes
I_{W},~{}~{}{{}^{q}{L^{\pm,W}_{2}}}=I_{V}\otimes{{}^{q}{L^{\pm,W}}},$

where $I_{X}$ is the unit matrix in $X$.

From (2.5.2) we can obtain the action of the comultiplication on the matrices
${{}^{q}{L^{\pm,V}}}$:

$\Delta_{s}({{}^{q}{L^{\pm,V}_{ij}}})=\sum_{k}{{}^{q}{L^{\pm,V}_{ik}}}\otimes{{}^{q}{L^{\pm,V}_{kj}}}$
(3.2.6)

and the antipode,

$S_{s}({{}^{q}{L^{\pm,V}_{ij}}})=({{}^{q}{L^{\pm,V}}})^{-1}_{ij}.$ (3.2.7)

Since ${\mathcal{R}}^{s}=1\otimes 1$ (mod $h$) relations (3.2.4) and (3.2.5)
imply that the quotient algebra
${\mathbb{C}}_{\mathcal{B}}[G^{*}]/(q^{\frac{1}{dr^{2}}}-1){\mathbb{C}}_{\mathcal{B}}[G^{*}]$
is commutative, and one can equip it with a Poisson structure given by

$\\{x_{1},x_{2}\\}=\frac{1}{dr^{2}}{[a_{1},a_{2}]\over
q^{\frac{1}{dr^{2}}}-1}~{}(\mbox{mod }(q^{\frac{1}{dr^{2}}}-1)),$ (3.2.8)

where $a_{1},a_{2}\in{\mathbb{C}}_{\mathcal{B}}[G^{*}]$ reduce to
$x_{1},x_{2}\in{\mathbb{C}}_{\mathcal{B}}[G^{*}]/(q^{\frac{1}{dr^{2}}}-1){\mathbb{C}}_{\mathcal{B}}[G^{*}]$
mod $(q^{\frac{1}{dr^{2}}}-1)$.

Obviously, the comultiplication and the antipode in
${\mathbb{C}}_{\mathcal{B}}[G^{*}]$ induce a comultiplication and an antipode
in
${\mathbb{C}}_{\mathcal{B}}[G^{*}]/(q^{\frac{1}{dr^{2}}}-1){\mathbb{C}}_{\mathcal{B}}[G^{*}]$
compatible with the introduced Poisson structure, and the quotient
${\mathbb{C}}_{\mathcal{B}}[G^{*}]/(q^{\frac{1}{dr^{2}}}-1){\mathbb{C}}_{\mathcal{B}}[G^{*}]$
becomes a Poisson–Hopf algebra.

######  Proposition 3.2.3.

The Poisson–Hopf algebra
${\mathbb{C}}_{\mathcal{B}}[G^{*}]/(q^{\frac{1}{dr^{2}}}-1){\mathbb{C}}_{\mathcal{B}}[G^{*}]$
is isomorphic to ${\mathbb{C}}[G^{*}]$ as a Poisson–Hopf algebra.

###### Proof.

Denote by
$p:{\mathbb{C}}_{\mathcal{B}}[G^{*}]\rightarrow{\mathbb{C}}_{\mathcal{B}}[G^{*}]/(q^{\frac{1}{dr^{2}}}-1){\mathbb{C}}_{\mathcal{B}}[G^{*}]={\mathbb{C}}[G^{*}]^{\prime}$
the canonical projection, and let ${\tilde{L}^{\pm,V}}=(p\otimes
p_{V})({{}^{q}{L^{\pm,V}}})\in{\mathbb{C}}[G^{*}]^{\prime}\otimes{\rm End}V$,
where
$p_{V}:V^{res}\rightarrow\overline{V}=V^{res}/(q^{\frac{1}{dr^{2}}}-1)V^{res}$
is the projection of $V^{res}$ onto the corresponding module $\overline{V}$
over
$U^{res}(\mathfrak{g})=U_{\mathcal{B}}^{s,res}({\mathfrak{g}})/(q^{\frac{1}{dr^{2}}}-1)U_{\mathcal{B}}^{s,res}({\mathfrak{g}})$
equipped also with the natural action of $\mathfrak{h}$ as described above. We
denote by $\pi_{\overline{V}}$ the corresponding representation of
$U(\mathfrak{g})$.

Observe that the map

$\imath:{\mathbb{C}}[G^{*}]^{\prime}\rightarrow{\mathbb{C}}[G^{*}],~{}~{}(\imath\otimes
id){\tilde{L}^{\pm,V}}={L^{\pm,\overline{V}}}$

is a well–defined linear isomorphism. Indeed, consider, for instance, element
${\tilde{L}^{+,V}}$. From (2.6.11) it follows that

$\begin{array}[]{l}\tilde{L}^{+,V}=\prod_{\beta}exp[p((1-q_{\beta}^{-2})f_{\beta})\otimes\pi_{\overline{V}}(X_{\beta})]\times\\\
\times(p\otimes
id)\exp\left[\sum_{i=1}^{l}hH_{i}\otimes\pi_{\overline{V}}((\kappa{1+s\over
1-s}P_{{\mathfrak{h}^{\prime}}}+id)Y_{i})\right].\end{array}$ (3.2.9)

On the other hand (3.2.1) implies that every element $L_{+}$ may be
represented in the form

$L_{+}=\prod_{\beta}\exp[b_{\beta}X_{\beta}]\exp\left[\sum_{i=1}^{l}b_{i}(\kappa{1+s\over
1-s}P_{{\mathfrak{h}^{\prime}}}+id)Y_{i}\right],~{}b_{i},b_{\beta}\in{\mathbb{C}},$

and hence

$\begin{array}[]{l}L^{+,\overline{V}}=\prod_{\beta}exp[b_{\beta}\pi_{\overline{V}}(X_{\beta})]exp\left[\sum_{i=1}^{l}b_{i}\pi_{\overline{V}}((\kappa{1+s\over
1-s}P_{{\mathfrak{h}^{\prime}}}+id)Y_{i})\right].\end{array}$

Comparing this with (3.2.9) and recalling the definition of $\imath$ we deduce
that $\imath$ is a linear isomorphism. We have to prove that $\imath$ is an
isomorphism of Poisson–Hopf algebras.

Observe that ${\mathcal{R}}^{s}=1\otimes 1-2hr_{-}^{s}$ (mod $h^{2}$).
Therefore from commutation relations (3.2.4), (3.2.5) it follows that
${\mathbb{C}}[G^{*}]^{\prime}$ is a commutative algebra, and the Poisson
brackets of matrix elements ${\tilde{L}^{\pm,\overline{V}}}_{ij}$ (see
(3.2.8)) are given by (3.2.2), where $L^{\pm,\overline{V}}$ are replaced by
${\tilde{L}^{\pm,\overline{V}}}$. The factor $\frac{1}{dr^{2}}$ in formula
(3.2.8) normalizes the Poisson bracket in such a way that bracket (3.2.8) is
in agreement with (3.2.2).

From (3.2.6) we also obtain that the action of the comultiplication on the
matrices ${\tilde{L}^{\pm,\overline{V}}}$ is given by (3.2.3), where
$L^{\pm,\overline{V}}$ are replaced by ${\tilde{L}^{\pm,\overline{V}}}$. This
completes the proof.

∎

We shall call the map
$p:{\mathbb{C}}_{\mathcal{B}}[G^{*}]\rightarrow{\mathbb{C}}_{\mathcal{B}}[G^{*}]/(q^{\frac{1}{dr^{2}}}-1){\mathbb{C}}_{\mathcal{B}}[G^{*}]={\mathbb{C}}[G^{*}]$
the quasiclassical limit.

Now using the Hopf algebra ${\mathbb{C}}_{\mathcal{B}}[G^{*}]$ we shall define
q-W–algebras. From the definition of the elements ${{}^{q}{L^{\pm,V}}}$ it
follows that the matrix elements of ${{}^{q}{L^{\pm,V}}}^{\pm 1}$ form Hopf
subalgebras
${\mathbb{C}}_{\mathcal{B}}[B_{\pm}]\subset{\mathbb{C}}_{\mathcal{B}}[G^{*}]$,
and that ${\mathbb{C}}_{\mathcal{B}}[G^{*}]$ contains the subalgebra
${\mathbb{C}}_{\mathcal{B}}[N_{+}]$ generated by elements the
$\tilde{f}_{\beta}=(1-q_{\beta}^{-2})f_{\beta}$, $\beta\in\Delta_{+}$.

Suppose that the positive root system $\Delta_{+}$ and its ordering are
associated to $s$. Denote by ${\mathbb{C}}_{\mathcal{B}}[M_{+}]$ the
subalgebra in ${\mathbb{C}}_{\mathcal{B}}[N_{+}]$ generated by the elements
$\tilde{f}_{\beta}$, $\beta\in{\Delta_{\mathfrak{m}_{+}}}$.

The linear subspace of $\mathfrak{g}$ generated by the root vectors
$X_{\alpha}$ ($X_{-\alpha}$), $\alpha\in\Delta_{\mathfrak{m}_{+}}$ is in fact
a Lie subalgebra ${\mathfrak{m}_{+}}\subset\mathfrak{g}$
(${\mathfrak{m}_{-}}\subset\mathfrak{g}$). By definition
$\Delta_{\mathfrak{m}_{+}}\subset\Delta_{+}$, and hence
${\mathfrak{m}_{\pm}}\subset\mathfrak{n}_{\pm}$.

Note that one can consider $\mathfrak{n}_{+}$ and $\mathfrak{m}_{\pm}$ as Lie
subalgebras in $\mathfrak{g}^{*}$ via embeddings

$\mathfrak{n}_{+}\rightarrow\mathfrak{g}^{*}\subset\mathfrak{g}\oplus\mathfrak{g},~{}x\mapsto(x,0),$
$\mathfrak{m}_{+}\rightarrow\mathfrak{g}^{*}\subset\mathfrak{g}\oplus\mathfrak{g},~{}x\mapsto(x,0),$
$\mathfrak{m}_{-}\rightarrow\mathfrak{g}^{*}\subset\mathfrak{g}\oplus\mathfrak{g},~{}x\mapsto(0,x),$

Using these embeddings the algebraic subgroups $N_{+},M_{\pm}\subset G$
corresponding to the algebraic Lie subalgebras
$\mathfrak{n}_{+},\mathfrak{m}_{\pm}\subset\mathfrak{g}$ can be regarded as
Lie subgroups in $G^{*}$ corresponding to the Lie subalgebras
$\mathfrak{n}_{+},\mathfrak{m}_{\pm}\subset\mathfrak{g}^{*}$. This way the
algebra ${\mathbb{C}}_{\mathcal{B}}[N_{+}]$ becomes naturally a quantization
of the algebra of regular functions on the subgroup $N_{+}\subset G^{*}$, and
${\mathbb{C}}_{\mathcal{B}}[M_{+}]$ becomes a quantization of the algebra of
regular functions on the subgroup $M_{+}\subset G^{*}$ in the sense that
$p({\mathbb{C}}_{\mathcal{B}}[N_{+}])={\mathbb{C}}[N_{+}]$ and
$p({\mathbb{C}}_{\mathcal{B}}[M_{+}])={\mathbb{C}}[M_{+}]$. Here and below for
every algebraic variety $V$ we denote by ${\mathbb{C}}[V]$ the algebra of
regular functions on $V$.

Note that $M_{-}$ can also be regarded as a subgroup in $G^{*}$ corresponding
to the Lie subalgebra $\mathfrak{m}_{-}\subset\mathfrak{g}^{*}$.

The following proposition gives the most important property of the subalgebra
${\mathbb{C}}_{\mathcal{B}}[M_{+}]$ which plays the key role in the definition
of q-W–algebras.

######  Proposition 3.2.4.

The defining relations in the subalgebra ${\mathbb{C}}_{\mathcal{B}}[M_{+}]$
for the generators $\tilde{f}_{\beta}=(1-q_{\beta}^{-2})f_{\beta}$,
$\beta\in\Delta_{\mathfrak{m}_{+}}=\\{\beta_{1},\ldots,\beta_{c}\\}$ are of
the form

$\tilde{f}_{\alpha}\tilde{f}_{\beta}-q^{(\alpha,\beta)+\kappa({1+s\over
1-s}P_{\mathfrak{h}^{\prime*}}\alpha,\beta)}\tilde{f}_{\beta}\tilde{f}_{\alpha}=\sum_{\alpha<\delta_{1}<\ldots<\delta_{n}<\beta}C(k_{1},\ldots,k_{n}){\tilde{f}_{\delta_{1}}}^{k_{1}}{\tilde{f}_{\delta_{2}}}^{k_{2}}\ldots{\tilde{f}_{\delta_{n}}}^{k_{n}},~{}\alpha<\beta,$
(3.2.10)

where $C(k_{1},\ldots,k_{n})\in\mathcal{B}$, the products
${\tilde{f}_{\beta_{1}}}^{k_{1}}{\tilde{f}_{\beta_{2}}}^{k_{2}}\ldots{\tilde{f}_{\beta_{c}}}^{k_{c}}$
form a $\mathcal{B}$–basis of ${\mathbb{C}}_{\mathcal{B}}[M_{+}]$ and the
products
${\tilde{f}_{\beta_{1}}}^{k_{1}}{\tilde{f}_{\beta_{2}}}^{k_{2}}\ldots{\tilde{f}_{\beta_{D}}}^{k_{D}}$
form a $\mathcal{B}$–basis of ${\mathbb{C}}_{\mathcal{B}}[N_{+}]$.

If $\kappa=1$ then for any $k_{i}\in\mathcal{B}$, $i=1,\ldots,l^{\prime}$ the
map $\chi_{q}^{s}:{\mathbb{C}}_{\mathcal{B}}[M_{+}]\rightarrow\mathcal{B}$,

$\chi_{q}^{s}(\tilde{f}_{\beta})=\left\\{\begin{array}[]{ll}0&\beta\not\in\\{\gamma_{1},\ldots,\gamma_{l^{\prime}}\\}\\\
k_{i}&\beta=\gamma_{i}\end{array}\right.,$ (3.2.11)

is a character of ${\mathbb{C}}_{\mathcal{B}}[M_{+}]$ vanishing on the r.h.s.
and on the l.h.s. of relations (3.2.10).

Assume that $\varepsilon^{2d_{i}}\neq 1$, and $\varepsilon^{4}\neq 1$ if
$\mathfrak{g}$ is of type $G_{2}$. Suppose also that there exists
$n\in\mathbb{Z}$ such that $\varepsilon^{nd-1}=1$. Let $\kappa=nd$. Then the
algebra
${\mathbb{C}}_{\varepsilon}[M_{+}]={\mathbb{C}}_{\mathcal{B}}[M_{+}]/(q^{\frac{1}{dr^{2}}}-\varepsilon^{\frac{1}{dr^{2}}}){\mathbb{C}}_{\mathcal{B}}[M_{+}]$,
where $\varepsilon^{\frac{1}{dr^{2}}}$ is a root of $\varepsilon$ of degree
$\frac{1}{dr^{2}}$, is isomorphic to
$U_{\varepsilon}^{s}({\mathfrak{m}_{-}})$, the elements $f^{\bf
r}=f_{\beta_{1}}^{r_{1}}\ldots f_{\beta_{c}}^{r_{c}}$, $r_{i}\in\mathbb{N}$,
$i=1,\ldots d$ form a linear basis of
$U_{\varepsilon}^{s}({\mathfrak{m}}_{-})$, and for any $c_{i}\in\mathbb{C}$,
$i=1,\ldots,l^{\prime}$ the map
$\chi_{\varepsilon}^{s}:U_{\varepsilon}^{s}({\mathfrak{m}_{-}})\rightarrow\mathbb{C}$,

$\chi_{\varepsilon}^{s}(f_{\beta})=\left\\{\begin{array}[]{ll}0&\beta\not\in\\{\gamma_{1},\ldots,\gamma_{l^{\prime}}\\}\\\
c_{i}&\beta=\gamma_{i}\end{array}\right.,$ (3.2.12)

is a character of $U_{\varepsilon}^{s}({\mathfrak{m}_{-}})$.

###### Proof.

By Step 1, Theorem 21.1 in [29] the products
${\tilde{f}_{\beta_{1}}}^{k_{1}}{\tilde{f}_{\beta_{2}}}^{k_{2}}\ldots{\tilde{f}_{\beta_{D}}}^{k_{D}}$
form a $\mathcal{B}$–basis of ${\mathbb{C}}_{\mathcal{B}}[N_{+}]$.

By Lemma 2.7.2 and Remark 2.7.3 any element of
${\mathbb{C}}_{\mathcal{B}}[M_{+}]$ can be uniquely represented as a
$\mathbb{C}(q^{\frac{1}{dr^{2}}})$–linear combination of the elements
${\tilde{f}_{\beta_{1}}}^{k_{1}}{\tilde{f}_{\beta_{2}}}^{k_{2}}\ldots{\tilde{f}_{\beta_{c}}}^{k_{c}}$.
By the uniqueness of the Poincaré–Birkhoff–Witt decomposition for
${\mathbb{C}}_{\mathcal{B}}[N_{+}]$ established above the coefficients of this
decomposition must belong to $\mathcal{B}$.

From (2.7.11) we also obtain commutation relations (3.2.10) with
$C(k_{1},\ldots,k_{n})\in\mathbb{C}(q^{\frac{1}{dr^{2}}})$. As we already
proved the products
${\tilde{f}_{\beta_{1}}}^{k_{1}}{\tilde{f}_{\beta_{2}}}^{k_{2}}\ldots{\tilde{f}_{\beta_{c}}}^{k_{c}}$
form a $\mathcal{B}$–basis of ${\mathbb{C}}_{\mathcal{B}}[M_{+}]$. Therefore
the coefficients $C(k_{1},\ldots,k_{n})$ in (3.2.10) belong to $\mathcal{B}$.

Assume that $\kappa=1$. In order to prove that the map
$\chi_{q}^{s}:{\mathbb{C}}_{\mathcal{B}}[M_{+}]\rightarrow\mathcal{B}$ defined
by (3.2.11) is a character we show that all relations (3.2.10) for
$\tilde{f}_{\alpha},\tilde{f}_{\beta}$ with
$\alpha,\beta\in\Delta_{\mathfrak{m}_{+}}$, which are obviously defining
relations in the subalgebra ${\mathbb{C}}_{\mathcal{B}}[M_{+}]$, belong to the
kernel of $\chi_{q}^{s}$. By definition the only generators of
${\mathbb{C}}_{\mathcal{B}}[M_{+}]$ on which $\chi_{q}^{s}$ may not vanish are
$\tilde{f}_{\gamma_{i}}$, $i=1,\ldots,l^{\prime}$. By part (v) of Proposition
1.6.1 for any two roots $\alpha,\beta\in\Delta_{\mathfrak{m}_{+}}$ such that
$\alpha<\beta$ the sum $\alpha+\beta$ cannot be represented as a linear
combination $\sum_{k=1}^{t}c_{k}\gamma_{i_{k}}$, where $c_{k}\in\mathbb{N}$
and $\alpha<\gamma_{i_{1}}<\ldots<\gamma_{i_{t}}<\beta$. Hence for any two
roots $\alpha,\beta\in\Delta_{\mathfrak{m}_{+}}$ such that $\alpha<\beta$ the
value of the map $\chi_{q}^{s}$ on the right hand side of the corresponding
commutation relation (3.2.10) is equal to zero.

Therefore it suffices to prove that

$\chi_{q}^{s}(\tilde{f}_{\gamma_{i}}\tilde{f}_{\gamma_{j}}-q^{(\gamma_{i},\gamma_{j})+({1+s\over
1-s}P_{{\mathfrak{h}^{\prime}}^{*}}\gamma_{i},\gamma_{j})}\tilde{f}_{\gamma_{j}}\tilde{f}_{\gamma_{j}})=k_{i}k_{j}(1-q^{(\gamma_{i},\gamma_{j})+({1+s\over
1-s}P_{{\mathfrak{h}^{\prime}}^{*}}\gamma_{i},\gamma_{j})})=0,~{}i<j.$

The last identity holds provided $(\gamma_{i},\gamma_{j})+({1+s\over
1-s}P_{{\mathfrak{h}^{\prime}}}^{*}\gamma_{i},\gamma_{j})=0$ for $i<j$ which
is indeed the case by Lemma 2.7.1.

Assume now that $\varepsilon^{2d_{i}}\neq 1$, and $\varepsilon^{4}\neq 1$ if
$\mathfrak{g}$ is of type $G_{2}$. Suppose also that there exists
$n\in\mathbb{Z}$ such that $\varepsilon^{nd-1}=1$. Let $\kappa=nd$.

Under these conditions imposed on $\varepsilon$ the map
${\mathbb{C}}_{\varepsilon}[M_{+}]\to
U_{\varepsilon}^{s}({\mathfrak{m}_{-}})$,
$\tilde{f}_{\alpha}\mapsto(1-q_{\alpha}^{-2})f_{\alpha}$,
$\alpha\in\Delta_{\mathfrak{m}_{+}}$ is obviously an algebra isomorphism.

By Lemma 2.7.2 and Remark 2.7.3 the elements $f^{\bf
r}=f_{\beta_{1}}^{r_{1}}\ldots f_{\beta_{c}}^{r_{c}}$, $r_{i}\in\mathbb{N}$,
$i=1,\ldots d$ form a linear basis of
$U_{\varepsilon}^{s}({\mathfrak{m}}_{-})$.

From (2.7.11) we obtain the following commutation relations

$f_{\alpha}f_{\beta}-\varepsilon^{(\alpha,\beta)+nd({1+s\over
1-s}P_{\mathfrak{h}^{\prime*}}\alpha,\beta)}f_{\beta}f_{\alpha}=\sum_{\alpha<\delta_{1}<\ldots<\delta_{n}<\beta}D(k_{1},\ldots,k_{n})f_{\delta_{1}}^{k_{1}}f_{\delta_{2}}^{k_{2}}\ldots
f_{\delta_{n}}^{k_{n}},~{}\alpha<\beta,$ (3.2.13)

where $D(k_{1},\ldots,k_{n})\in\mathbb{C}$.

In order to show that the map
$\chi_{\varepsilon}^{s}:U_{\varepsilon}^{s}({\mathfrak{m}_{-}})\rightarrow\mathbb{C}$
defined by (3.2.12) is a character we verify that all relations (3.2.13) for
$f_{\alpha},f_{\beta}$ with $\alpha,\beta\in\Delta_{\mathfrak{m}_{+}}$, which
are obviously defining relations in the subalgebra
$U_{\varepsilon}^{s}({\mathfrak{m}}_{-})$, belong to the kernel of
$\chi_{\varepsilon}^{s}$. By definition the only generators of
$U_{\varepsilon}^{s}({\mathfrak{m}}_{-})$ on which $\chi_{\varepsilon}^{s}$
may not vanish are $f_{\gamma_{i}}$, $i=1,\ldots,l^{\prime}$. By part (v) of
Proposition 1.6.1 for any two roots $\alpha,\beta\in\Delta_{\mathfrak{m}_{+}}$
such that $\alpha<\beta$ the sum $\alpha+\beta$ cannot be represented as a
linear combination $\sum_{k=1}^{t}c_{k}\gamma_{i_{k}}$, where
$c_{k}\in\mathbb{N}$ and $\alpha<\gamma_{i_{1}}<\ldots<\gamma_{i_{t}}<\beta$.
Hence for any two roots $\alpha,\beta\in\Delta_{\mathfrak{m}_{+}}$ such that
$\alpha<\beta$ the value of the map $\chi_{\varepsilon}^{s}$ on the right hand
side of the corresponding commutation relation (3.2.10) is equal to zero.

Therefore it suffices to prove that

$\chi_{\varepsilon}^{s}(f_{\gamma_{i}}f_{\gamma_{j}}-\varepsilon^{(\gamma_{i},\gamma_{j})+nd({1+s\over
1-s}P_{{\mathfrak{h}^{\prime}}^{*}}\gamma_{i},\gamma_{j})}f_{\gamma_{j}}f_{\gamma_{j}})=c_{i}c_{j}(1-\varepsilon^{(\gamma_{i},\gamma_{j})+nd({1+s\over
1-s}P_{{\mathfrak{h}^{\prime}}^{*}}\gamma_{i},\gamma_{j})})=0,~{}i<j.$

By Lemma 2.7.1 $({1+s\over
1-s}P_{{\mathfrak{h}^{\prime}}}^{*}\gamma_{i},\gamma_{j})=-(\gamma_{i},\gamma_{j})$
for $i<j$, and hence

$\chi_{\varepsilon}^{s}(f_{\gamma_{i}}f_{\gamma_{j}}-\varepsilon^{(\gamma_{i},\gamma_{j})+nd({1+s\over
1-s}P_{{\mathfrak{h}^{\prime}}^{*}}\gamma_{i},\gamma_{j})}f_{\gamma_{j}}f_{\gamma_{j}})=c_{i}c_{j}(1-\varepsilon^{(\gamma_{i},\gamma_{j})(1-nd)})=0$

for $i<j$ as by the assumption $\varepsilon^{nd-1}=1$. This completes the
proof.

∎

Next we define the algebra ${\mathbb{C}}_{\mathcal{B}}[{G}_{*}]$ and discuss
its properties. For any finite rank representation $V$ of
$U_{h}({\mathfrak{g}})$, let
${{}^{q}{L^{V}}}={{}^{q}{L^{-,V}}}^{-1}{{}^{q}{L^{+,V}}}=(id\otimes\pi_{V}){\mathcal{R}}_{21}^{s}{{\mathcal{R}}^{s}}$.
Let ${\mathbb{C}}_{\mathcal{B}}[{G}_{*}]$ be the $\mathcal{B}$–subalgebra in
${\mathbb{C}}_{\mathcal{B}}[G^{*}]$ generated by the matrix entries of
${{}^{q}{{L}^{V}}}$, where $V$ runs over all finite rank representations of
$U_{h}^{s}({\mathfrak{g}})$. From the definition of $\mathcal{R}^{s}$ we have

$\begin{array}[]{l}{\mathcal{R}}^{s}_{21}{\mathcal{R}}^{s}=\prod_{\beta}\sum_{k=0}^{\infty}q_{\beta}^{{k(k+1)}\over
2}[(1-q_{\beta}^{-2})e_{\beta}^{k}e^{-hk\kappa{1+s\over
1-s}P_{{\mathfrak{h}^{\prime}}}\beta^{\vee}}\otimes f_{\beta}^{(k)}]\times\\\
\times\exp\left[2h\sum_{i=1}^{l}Y_{i}\otimes
H_{i}\right]\prod_{\beta}\sum_{k=0}^{\infty}q_{\beta}^{{k(k+1)}\over
2}[(1-q_{\beta}^{-2})e^{hk\kappa{1+s\over
1-s}P_{{\mathfrak{h}^{\prime}}}\beta^{\vee}-hk\beta^{\vee}}f_{\beta}^{k}\otimes
e_{\beta}^{(k)}q^{k\beta^{\vee}}].\end{array}$ (3.2.14)

Using this formula one immediately checks that actually
${\mathbb{C}}_{\mathcal{B}}[{G}_{*}]\subset
U_{\mathcal{A}}^{s}(\mathfrak{g})\cap{\mathbb{C}}_{\mathcal{B}}[G^{*}]$.

Define the right adjoint action of $U_{q}^{s}(\mathfrak{g})$ on
$U_{q}^{s}(\mathfrak{g})$ by the formula

${\rm Ad}x(w)=S_{s}(x^{1})wx^{2},$ (3.2.15)

and the left adjoint action of $U_{q}^{s}(\mathfrak{g})$ on
$U_{q}^{s}(\mathfrak{g})$ by

${\rm Ad}^{\prime}x(w)=x^{1}wS_{s}(x^{2}),$ (3.2.16)

where we use the abbreviated Sweedler notation for the coproduct
$\Delta_{s}(x)=x^{1}\otimes x^{2}$, $x,w\in U_{q}^{s}(\mathfrak{g})$.

Let $\mathbb{C}_{\mathcal{B}}[G]$ be the restricted Hopf algebra dual to
$U_{\mathcal{B}}^{s,res}(\mathfrak{g})$ which is generated by the matrix
elements of finite rank representations of
$U_{\mathcal{B}}^{s,res}(\mathfrak{g})$ of the form $V^{res}$, where $V$ is a
finite rank representation of $U_{h}^{s}({\mathfrak{g}})$. Action (3.2.16)
induces a left adjoint action ${\rm Ad}$ of
$U_{\mathcal{B}}^{s,res}(\mathfrak{g})$ on $\mathbb{C}_{\mathcal{B}}[G]$
defined by

$({\rm Ad}xf)(w)=f({\rm
Ad}^{\prime}x(w)),f\in\mathbb{C}_{\mathcal{B}}[G],x,w\in
U_{\mathcal{B}}^{s,res}(\mathfrak{g}).$ (3.2.17)

One can also equip finite rank representations of
$U_{\mathcal{B}}^{s,res}(\mathfrak{g})$ of the form $V^{res}$, where $V$ is a
finite rank representation of $U_{h}^{s}({\mathfrak{g}})$, with a natural
action of $\mathbb{C}_{\mathcal{B}}[G^{*}]$, where the elements

$q^{\pm(Y_{i}-\kappa\frac{1+s}{1-s}P_{\mathfrak{h}^{\prime}}Y_{i})},q^{\pm(-Y_{i}-\kappa\frac{1+s}{1-s}P_{\mathfrak{h}^{\prime}}Y_{i})},~{}i=1,\ldots
l$

act on a weight vector $v_{\lambda}$ of weight $\lambda$ by multiplication by
the elements of

$q^{\pm((Y_{i},\lambda^{\vee})-\kappa(\frac{1+s}{1-s}P_{\mathfrak{h}^{\prime}}Y_{i},\lambda^{\vee}))},q^{\pm(-(Y_{i},\lambda^{\vee})-\kappa(\frac{1+s}{1-s}P_{\mathfrak{h}^{\prime}}Y_{i},\lambda^{\vee}))}\in\mathcal{B},~{}i=1,\ldots
l,$

respectively, and all the other generators of
$\mathbb{C}_{\mathcal{B}}[G^{*}]$ which belong to
$U_{\mathcal{B}}^{s,res}(\mathfrak{g})$ act in a natural way. Therefore
adjoint action (3.2.17) can be extended to an action of
$\mathbb{C}_{\mathcal{B}}[G^{*}]$, where elements
$x\in\mathbb{C}_{\mathcal{B}}[G^{*}]$ act by the same formula (3.2.17).

Note that by Lemma 2.2 in [56]

${\rm Ad}x(wz)={\rm Ad}x^{1}(w){\rm Ad}x^{2}(z).$ (3.2.18)

For $\varepsilon\in\mathbb{C}^{*}$ we define
${\mathbb{C}}_{\varepsilon}[G]={\mathbb{C}}_{\mathcal{B}}[G]/(q^{\frac{1}{dr^{2}}}-\varepsilon^{\frac{1}{dr^{2}}}){\mathbb{C}}_{\mathcal{B}}[G]$,
${\mathbb{C}}_{\varepsilon}[G_{*}]={\mathbb{C}}_{\mathcal{B}}[G_{*}]/(q^{\frac{1}{dr^{2}}}-\varepsilon^{\frac{1}{dr^{2}}}){\mathbb{C}}_{\mathcal{B}}[G_{*}]$,
where $\varepsilon^{\frac{1}{dr^{2}}}$ is a root of $\varepsilon$ of degree
$\frac{1}{dr^{2}}$.

######  Proposition 3.2.5.

If $\varepsilon$ is not a root of unity the algebra
${\mathbb{C}}_{\varepsilon}[{G}_{*}]$ can be identified with the ${\rm Ad}$
locally finite part $U_{\varepsilon}^{s}(\mathfrak{g})^{fin}$ of
$U_{\varepsilon}^{s}(\mathfrak{g})$,

$U_{\varepsilon}^{s}(\mathfrak{g})^{fin}=\\{x\in
U_{\varepsilon}^{s}(\mathfrak{g}):{\rm dim}({\rm
Ad}U_{\varepsilon}^{s}(\mathfrak{g})(x))<+\infty\\},$

where the adjoint action of the algebra $U_{\varepsilon}^{s}(\mathfrak{g})$ on
itself is defined by formula (3.2.15).

Moreover, the map

$\mathbb{C}_{\mathcal{B}}[G]\rightarrow{\mathbb{C}}_{\mathcal{B}}[{G}_{*}],f\mapsto(id\otimes
f)(\mathcal{R}^{s}_{21}\mathcal{R}^{s})$ (3.2.19)

is an isomorphism of $U_{\mathcal{B}}^{s,res}(\mathfrak{g})$ and
$\mathbb{C}_{\mathcal{B}}[G^{*}]$–modules with respect to the adjoint actions
${\rm Ad}$ defined by (3.2.15) and (3.2.17), respectively. In particular,
${\mathbb{C}}_{\mathcal{B}}[{G}_{*}]$ is stable under the adjoint action of
$U_{\mathcal{B}}^{s,res}(\mathfrak{g})$ and $\mathbb{C}_{\mathcal{B}}[G^{*}]$.

###### Proof.

All statements except for the isomorphism
${\mathbb{C}}_{\varepsilon}[{G}_{*}]=U_{\varepsilon}^{s}(\mathfrak{g})^{fin}$
follow from the definitions and the discussion above.

It remains to establish the isomorphism
${\mathbb{C}}_{\varepsilon}[{G}_{*}]=U_{\varepsilon}^{s}(\mathfrak{g})^{fin}$.
Let $V_{\omega_{i}}$, $i=1,\ldots,l$ be the finite rank representation of
$U_{h}(\mathfrak{g})$ with highest weight $\omega_{i}$, $i=1,\ldots,l$. From
formula (3.2.14) and from the definition of
${{}^{q}{L^{V}}}=(id\otimes\pi_{V}){\mathcal{R}}_{21}^{s}{\mathcal{R}}^{s}$ it
follows that the matrix element $(id\otimes
v_{i}^{*}){\mathcal{R}}_{21}^{s}{\mathcal{R}}^{s}(id\otimes v_{i})$ of
${}^{q}{L^{V_{\omega_{i}}}}$ corresponding to the highest weight vector
$v_{i}$ of $V_{\omega_{i}}^{res}$ and to the lowest weight vector
$v_{i}^{*}\in{V_{\omega_{i}}^{res}}^{*}$ of the dual representation
${V_{\omega_{i}}^{res}}^{*}$, normalized in such a way that
$v_{i}^{*}(v_{i})=1$, coincides with $L_{i}^{2}$. This implies that
$L_{i}^{2}$ are elements of the algebra
${\mathbb{C}}_{\varepsilon}[G_{*}]\subset U_{\varepsilon}^{s}(\mathfrak{g})$
as well.

Denote by $\mathfrak{H}\subset{\mathbb{C}}_{\varepsilon}[{G}_{*}]\subset
U_{\varepsilon}^{s}(\mathfrak{g})$ the subalgebra generated by the elements
$L_{i}^{2}\in{\mathbb{C}}_{\varepsilon}[{G}_{*}]$, $i=1,\ldots,l$. Similarly
to Theorem 7.1.6 and Lemma 7.1.16 in [55] one can obtain that
$U_{\varepsilon}^{s}(\mathfrak{g})^{fin}={\rm
Ad}U_{\varepsilon}^{s}(\mathfrak{g})\mathfrak{H}$. Since
${\mathbb{C}}_{\varepsilon}[{G}_{*}]$ is stable under the adjoint action we
have an inclusion,
$U_{\varepsilon}^{s}(\mathfrak{g})^{fin}\subset{\mathbb{C}}_{\varepsilon}[{G}_{*}]$.

On the other hand the adjoint action of $U_{\varepsilon}^{s}(\mathfrak{g})$ on
$\mathbb{C}_{\varepsilon}[G]$ is locally finite by the definition of
$\mathbb{C}_{\varepsilon}[G]$ and of the adjoint action. Using isomorphism
(3.2.19) we deduce that the adjoint action of
$U_{\varepsilon}^{s}(\mathfrak{g})$ on ${\mathbb{C}}_{\varepsilon}[{G}_{*}]$
is locally finite as well. Hence ${\mathbb{C}}_{\varepsilon}[{G}_{*}]\subset
U_{\varepsilon}^{s}(\mathfrak{g})^{fin}$, and
${\mathbb{C}}_{\varepsilon}[{G}_{*}]=U_{\varepsilon}^{s}(\mathfrak{g})^{fin}$.

∎

Now we are ready to define q-W–algebras. In the rest of this section we assume
that $\kappa=1$. Denote by $I_{\mathcal{B}}$ the left ideal in
${\mathbb{C}}_{\mathcal{B}}[G^{*}]$ generated by the kernel of $\chi_{q}^{s}$,
and by $\rho_{\chi^{s}_{q}}$ the canonical projection
${\mathbb{C}}_{\mathcal{B}}[G^{*}]\rightarrow{\mathbb{C}}_{\mathcal{B}}[G^{*}]/I_{\mathcal{B}}=Q_{\mathcal{B}}^{\prime}$.
Let $Q_{\mathcal{B}}$ be the image of ${\mathbb{C}}_{\mathcal{B}}[G_{*}]$
under the projection $\rho_{\chi^{s}_{q}}$.

We shall need the following property of the algebras
${\mathbb{C}}_{\mathcal{B}}[{G}^{*}]$, ${\mathbb{C}}_{\mathcal{B}}[{G}_{*}]$
and $\mathbb{C}_{\mathcal{B}}[G]$ and of
${\mathbb{C}}_{\mathcal{B}}[G^{*}]/I_{\mathcal{B}}$ and $Q_{\mathcal{B}}$.

######  Proposition 3.2.6.

${\mathbb{C}}_{\mathcal{B}}[{G}^{*}]$, ${\mathbb{C}}_{\mathcal{B}}[{G}_{*}]$,
$\mathbb{C}_{\mathcal{B}}[G]$,
${\mathbb{C}}_{\mathcal{B}}[G^{*}]/I_{\mathcal{B}}$ and $Q_{\mathcal{B}}$ are
free $\mathcal{B}$–modules.

###### Proof.

Recall that ${\mathbb{C}}_{\mathcal{B}}[G^{*}]$ is the
$\mathcal{B}$–subalgebra in $U_{h}^{s}({\mathfrak{g}})$ generated by the
elements $q^{\pm(Y_{i}-\kappa\frac{1+s}{1-s}P_{\mathfrak{h}^{\prime}}Y_{i})}$,
$q^{\pm(-Y_{i}-\kappa\frac{1+s}{1-s}P_{\mathfrak{h}^{\prime}}Y_{i})}$,
$i=1,\ldots,l$, $\tilde{f}_{\beta}=(1-q_{\beta}^{-2})f_{\beta}$,
$\tilde{e}_{\beta}=(1-q_{\beta}^{2})e_{\beta}e^{h\beta^{\vee}}$,
$\beta\in\Delta_{+}$. The subalgebra of ${\mathbb{C}}_{\mathcal{B}}[G^{*}]$
generated by the elements
$q^{\pm(Y_{i}-\kappa\frac{1+s}{1-s}P_{\mathfrak{h}^{\prime}}Y_{i})},q^{\pm(-Y_{i}-\kappa\frac{1+s}{1-s}P_{\mathfrak{h}^{\prime}}Y_{i})}$,
$i=1,\ldots,l$ is in turn a subalgebra of the $\mathcal{B}$–subalgebra
$U_{\mathcal{B}}^{\prime}(\mathfrak{h})\subset U_{h}^{s}({\mathfrak{g}})$
generated by the elements $U_{i}=q^{\frac{1}{dr^{2}}Y_{i}},U_{i}^{-1}$,
$i=1,\ldots,l$. The last algebra is obviously $\mathcal{B}$–free with a basis
consisting of the products $U_{1}^{n_{1}}\ldots U_{l}^{n_{l}}$,
$n_{1},\ldots,n_{l}\in\mathbb{Z}$. Since $\mathcal{B}$ is a principal ideal
domain the subalgebra in $U_{\mathcal{B}}^{\prime}(\mathfrak{h})$ generated by
the elements
$q^{\pm(Y_{i}-\kappa\frac{1+s}{1-s}P_{\mathfrak{h}^{\prime}}Y_{i})},q^{\pm(-Y_{i}-\kappa\frac{1+s}{1-s}P_{\mathfrak{h}^{\prime}}Y_{i})},~{}i=1,\ldots,l$
is also $\mathcal{B}$–free by Theorem 6.5 in [92]. Denote by $V_{i}$,
$i\in\mathbb{N}$ elements of some $\mathcal{B}$–basis of this subalgebra.

Now from Step 1 of the proof of the Theorem in Section 12.1 in [29] it follows
that the elements

$\tilde{e}_{\beta_{1}}^{n_{1}}\ldots\tilde{e}_{\beta_{D}}^{n_{D}}V_{i}\tilde{f}_{\beta_{D}}^{k_{D}}\ldots\tilde{f}_{\beta_{1}}^{k_{1}}$

with $n_{j},k_{j},i\in\mathbb{N}$, $j=1,\ldots,D$ form a $\mathcal{B}$–basis
in ${\mathbb{C}}_{\mathcal{B}}[G^{*}]$. Using this basis and the definition of
${\mathbb{C}}_{\mathcal{B}}[G^{*}]/I_{\mathcal{B}}$ one immediately sees that
the classes of the elements
$\tilde{e}_{\beta_{1}}^{n_{1}}\ldots\tilde{e}_{\beta_{D}}^{n_{D}}V_{i}\tilde{f}_{\beta_{D}}^{k_{D}}\ldots\tilde{f}_{\beta_{c+1}}^{k_{c+1}}$
with $n_{j},k_{m},i\in\mathbb{N}$, $j=1,\ldots,D$, $m=c+1,\ldots,D$ form a
$\mathcal{B}$–basis in ${\mathbb{C}}_{\mathcal{B}}[G^{*}]/I_{\mathcal{B}}$.

Since $\mathcal{B}$ is a principal ideal domain the subalgebra
${\mathbb{C}}_{\mathcal{B}}[{G}_{*}]\subset{\mathbb{C}}_{\mathcal{B}}[{G}^{*}]$
is also $\mathcal{B}$–free by Theorem 6.5 in [92], and isomorphism (3.2.19)
implies that $\mathbb{C}_{\mathcal{B}}[G]$ is $\mathcal{B}$–free.

Finally, since $\mathcal{B}$ is a principal ideal domain the
$\mathcal{B}$–submodule
$Q_{\mathcal{B}}\subset{\mathbb{C}}_{\mathcal{B}}[G^{*}]/I_{\mathcal{B}}$ is
$\mathcal{B}$–free.

∎

######  Remark 3.2.7.

The fact that the algebra
$\mathbb{C}_{q}[G]=\mathbb{C}_{\mathcal{B}}[G]\otimes_{\mathcal{B}}\mathbb{C}(q^{\frac{1}{dr^{2}}})$
is $\mathbb{C}(q^{\frac{1}{dr^{2}}})$–free can be proved in a more
straightforward way. Namely, according to the results of Section 7 in [74]
$\mathbb{C}_{q}[G]$ is generated by the matrix elements of finite-dimensional
$U_{q}^{s}(\mathfrak{g})\simeq U_{q}(\mathfrak{g})$–modules. By Theorem 10.1.7
in [20] all such modules are completely reducible, the irreducible components
being highest weight modules $V_{\lambda}^{q}$ generated by highest weight
vectors of integral dominant weights $\lambda\in P_{+}$. Therefore

$\mathbb{C}_{q}[G]\simeq\bigoplus_{\lambda\in
P_{+}}{V_{\lambda}^{q}}^{*}\otimes V_{\lambda}^{q}.$

This decomposition and the fact that all $V_{\lambda}^{q}$ are
$\mathbb{C}(q^{\frac{1}{dr^{2}}})$–free imply that $\mathbb{C}_{q}[G]$ is
$\mathbb{C}(q^{\frac{1}{dr^{2}}})$–free.

Now observe that we have an inclusion $[{\mathbb{C}}_{\mathcal{B}}[M_{+}],{\rm
Ker}\chi_{q}^{s}]\subset{\rm Ker}\chi_{q}^{s}$. Using this inclusion, formula
(3.2.15), the fact that
$\Delta_{s}({\mathbb{C}}_{\mathcal{B}}[M_{+}])\subset{\mathbb{C}}_{\mathcal{B}}[B_{+}]\otimes{\mathbb{C}}_{\mathcal{B}}[M_{+}]$
(see formula (2.7.12)) we deduce that the adjoint action of
${\mathbb{C}}_{\mathcal{B}}[M_{+}]$ on ${\mathbb{C}}_{\mathcal{B}}[G_{*}]$
induces an action on $Q_{\mathcal{B}}^{\prime}$ and on $Q_{\mathcal{B}}$ which
we also call the adjoint action and denote it by $\rm Ad$.

Let $\mathcal{B}_{\varepsilon_{s}}$ be the trivial representation of
${\mathbb{C}}_{\mathcal{B}}[M_{+}]$ given by the counit. Consider the space
$W_{\mathcal{B}}^{s}(G)$ of $\rm Ad$–invariants in $Q_{\mathcal{B}}$,

$W_{\mathcal{B}}^{s}(G)={\rm
Hom}_{{\mathbb{C}}_{\mathcal{B}}[M_{+}]}(\mathcal{B}_{\varepsilon_{s}},Q_{\mathcal{B}}).$
(3.2.20)

######  Proposition 3.2.8.

$W_{\mathcal{B}}^{s}(G)$ is isomorphic to the subspace of all
$v+I_{\mathcal{B}}\in Q_{\mathcal{B}}$ such that $mv\in I_{\mathcal{B}}$ (or
$[m,v]\in I_{\mathcal{B}}$) in ${\mathbb{C}}_{\mathcal{B}}[G^{*}]$ for any
$m\in I_{\mathcal{B}}$, where $v\in{\mathbb{C}}_{\mathcal{B}}[G^{*}]$ is any
representative of $v+I_{\mathcal{B}}\in Q_{\mathcal{B}}$.

Multiplication in ${\mathbb{C}}_{\mathcal{B}}[G^{*}]$ induces a multiplication
on the space $W_{\mathcal{B}}^{s}(G)$.

###### Proof.

For the proof we shall firstly derive the formula for the adjoint action of
the generators $\tilde{f}_{\beta}$. From (2.7.12) using linear independence of
weight components and the fact that ${\mathbb{C}}_{\mathcal{B}}[B_{+}]$ is a
Hopf algebra we obtain

$\Delta_{s}(\tilde{f}_{\beta_{k}})=G_{\beta_{k}}^{-1}\otimes\tilde{f}_{\beta_{k}}+\tilde{f}_{\beta_{k}}\otimes
1+\sum_{i}\tilde{y}_{i}\otimes\tilde{x}_{i},$ (3.2.21)

where

$G_{\beta}=e^{h\kappa{1+s\over
1-s}P_{\mathfrak{h}^{\prime}}\beta^{\vee}-h\beta^{\vee}}\in{\mathbb{C}}_{\mathcal{B}}[B_{+}],\tilde{y}_{i}=e^{-h\kappa{1+s\over
1-s}P_{\mathfrak{h}^{\prime}}\gamma_{x_{i}}^{\vee}+h\gamma_{x_{i}}^{\vee}}\overline{\tilde{y}}_{i},$
$\overline{\tilde{y}}_{i}\in\mathbb{C}_{\mathcal{B}}([-\beta_{k+1},-\beta_{D}]),$
$\tilde{x}_{i}\in\mathbb{C}_{\mathcal{B}}([-\beta_{1},-\beta_{k-1}]),$

$\overline{\tilde{y}}_{i},\tilde{x}_{i}$ belong to weight components and have
non-zero weights, $\gamma_{x_{i}}$ is the weight of $\tilde{x}_{i}$,

$\mathbb{C}_{\mathcal{B}}([-\beta_{k+1},-\beta_{D}])(\mathbb{C}_{\mathcal{B}}([-\beta_{1},-\beta_{k-1}]))$

is the subalgebra in $\mathbb{C}_{\mathcal{B}}[N_{+}]$ generated by
$\tilde{f}_{\beta_{k+1}},\ldots,\tilde{f}_{\beta_{D}}$
($\tilde{f}_{\beta_{1}},\ldots,\tilde{f}_{\beta_{k-1}}$).

By (3.2.21) we also have

$S_{s}(\tilde{f}_{\beta})=-G_{\beta}\tilde{f}_{\beta}-\sum_{i}S_{s}(\tilde{y}_{i})\tilde{x}_{i}.$

Combining this formula with (3.2.21) and using (3.2.15) we deduce

${\rm
Ad}\tilde{f}_{\beta_{k}}w=-G_{\beta_{k}}[\tilde{f}_{\beta_{k}},w]-\sum_{i}S_{s}(\tilde{y}_{i})[\tilde{x}_{i},w]$

The induced action of the elements
$\tilde{f}_{\beta_{k}}\in{\mathbb{C}}_{\mathcal{B}}[M_{+}]$,
$\beta_{k}\in\Delta_{\mathfrak{m}_{+}}$, on $Q_{\mathcal{B}}^{\prime}$ takes
the form

${\rm
Ad}\tilde{f}_{\beta_{k}}v=-G_{\beta_{k}}(\tilde{f}_{\beta_{k}}-\chi_{q}^{s}(\tilde{f}_{\beta_{k}}))v-\sum_{i}S_{s}(\tilde{y}_{i})(\tilde{x}_{i}-\chi_{q}^{s}(\tilde{x}_{i}))v,\beta_{k}\in\Delta_{\mathfrak{m}_{+}}.$
(3.2.22)

We have to show that $W_{\mathcal{B}}^{s}(G)$ is isomorphic to the subspace of
all $v\in Q_{\mathcal{B}}\subset Q_{\mathcal{B}}^{\prime}$ such that $mv=0$ in
$Q_{\mathcal{B}}^{\prime}$ for any $m\in I_{\mathcal{B}}$.

The left ideal $I_{\mathcal{B}}$ is generated by elements $x-\chi_{q}^{s}(x)$,
$x\in{\mathbb{C}}_{\mathcal{B}}[M_{+}]$. Therefore by (3.2.22) if for some
$v\in Q_{\mathcal{B}}$ $mv=0$ in $Q_{\mathcal{B}}^{\prime}$ for any $m\in
I_{\mathcal{B}}$ then $v$ is invariant with respect to the adjoint action of
all generators $\tilde{f}_{\beta_{k}}$ of ${\mathbb{C}}_{\mathcal{B}}[M_{+}]$,
and hence $v\in W_{\mathcal{B}}^{s}(G)$.

Now assume that $v\in W_{\mathcal{B}}^{s}(G)$. We shall prove that $mv=0$ in
$Q_{\mathcal{B}}^{\prime}$ for any $m\in I_{\mathcal{B}}$.

Since the left ideal $I_{\mathcal{B}}$ is generated by elements
$x-\chi_{q}^{s}(x)$, $x\in{\mathbb{C}}_{\mathcal{B}}[M_{+}]$ it suffices to
show that $(x-\chi_{q}^{s}(x))v=0$ for any
$x\in{\mathbb{C}}_{\mathcal{B}}[M_{+}]$. We shall prove this statement by
induction using the subalgebras
$\mathbb{C}_{\mathcal{B}}([-\beta_{1},-\beta_{k}])$, $k=1,\ldots,c$, so that
$\mathbb{C}_{\mathcal{B}}([-\beta_{1},-\beta_{c}])={\mathbb{C}}_{\mathcal{B}}[M_{+}]$.

Observe that $\beta_{1}$ is a simple root, and hence from (3.2.22) we obtain

$0={\rm
Ad}\tilde{f}_{\beta_{1}}v=-G_{\beta_{1}}(\tilde{f}_{\beta_{1}}-\chi_{q}^{s}(\tilde{f}_{\beta_{1}}))v.$

Since the element $G_{\beta_{1}}\in\mathbb{C}_{\mathcal{B}}[N_{+}]$ is
invertible this implies

$(\tilde{f}_{\beta_{1}}-\chi_{q}^{s}(\tilde{f}_{\beta_{1}}))v=0,$

i.e. $(x-\chi_{q}^{s}(x))v=0$ in $Q_{\mathcal{B}}^{\prime}$ for any
$x\in\mathbb{C}_{\mathcal{B}}([-\beta_{1},-\beta_{1}])$ as the subalgebra
$\mathbb{C}_{\mathcal{B}}([-\beta_{1},-\beta_{1}])$ is generated by
$\tilde{f}_{\beta_{1}}$.

Now suppose that for some $1<k\leq c$ $(x-\chi_{q}^{s}(x))v=0$ in
$Q_{\mathcal{B}}^{\prime}$ for any
$x\in\mathbb{C}_{\mathcal{B}}([-\beta_{1},-\beta_{k-1}])$. Then from (3.2.22)
we obtain

$0={\rm
Ad}\tilde{f}_{\beta_{k}}v=-G_{\beta_{k}}(\tilde{f}_{\beta_{k}}-\chi_{q}^{s}(\tilde{f}_{\beta_{k}}))v-\sum_{i}S_{s}(\tilde{y}_{i})(\tilde{x}_{i}-\chi_{q}^{s}(\tilde{x}_{i}))v=-G_{\beta_{k}}(\tilde{f}_{\beta_{k}}-\chi_{q}^{s}(\tilde{f}_{\beta_{k}}))v$

since $\tilde{x}_{i}\in\mathbb{C}_{\mathcal{B}}([-\beta_{1},-\beta_{k-1}])$.
The previous identity and the fact that the element
$G_{\beta_{k}}\in\mathbb{C}_{\mathcal{B}}[N_{+}]$ is invertible yield

$(\tilde{f}_{\beta_{k}}-\chi_{q}^{s}(\tilde{f}_{\beta_{k}}))v=0.$

Now observe that by Proposition 3.2.4 any element $x$ of
$\mathbb{C}_{\mathcal{B}}([-\beta_{1},-\beta_{k}])$ can be uniquely
represented in the form $x=\tilde{f}_{\beta_{k}}z+z^{\prime}$, where
$z,z^{\prime}\in\mathbb{C}_{\mathcal{B}}([-\beta_{1},-\beta_{k-1}])$.
Therefore

$xv=(\tilde{f}_{\beta_{k}}z+z^{\prime})v=(\tilde{f}_{\beta_{k}}\chi_{q}^{s}(z)+\chi_{q}^{s}(z^{\prime}))v=(\chi_{q}^{s}(\tilde{f}_{\beta_{k}})\chi_{q}^{s}(z)+\chi_{q}^{s}(z^{\prime}))v=\chi_{q}^{s}(\tilde{f}_{\beta_{k}}z+z^{\prime})v=\chi_{q}^{s}(x)v.$

This establishes the induction step and proves the first claim of this
proposition.

From the second description of the space $W_{\mathcal{B}}^{s}(G)$ it follows
that if $v_{1},v_{2}\in{\mathbb{C}}_{\mathcal{B}}[G^{*}]$ are any
representatives of elements $v_{1}+I_{q},v_{2}+I_{q}\in
W_{\mathcal{B}}^{s}(G)$ then the formula

$(v_{1}+I_{q})(v_{2}+I_{q})=v_{1}v_{2}+I_{q}$

defines a multiplication in $W_{\mathcal{B}}^{s}(G)$. This completes the
proof.

∎

The space $W_{\mathcal{B}}^{s}(G)$ equipped with the multiplication opposite
to the one defined in the previous proposition is called the q-W–algebra
associated to (the conjugacy class of) the Weyl group element $s\in W$.

In conclusion we obtain some results on the structure of $Q_{\mathcal{B}}$.
Consider the Lie algebra $\mathfrak{L}_{\mathcal{B}}$ associated to the
associative algebra ${\mathbb{C}}_{\mathcal{B}}[M_{+}]$, i.e.
$\mathfrak{L}_{\mathcal{B}}$ is the Lie algebra which is isomorphic to
${\mathbb{C}}_{\mathcal{B}}[M_{+}]$ as a linear space, and the Lie bracket in
$\mathfrak{L}_{\mathcal{B}}$ is given by the usual commutator of elements in
${\mathbb{C}}_{\mathcal{B}}[M_{+}]$.

Define an action of the Lie algebra $\mathfrak{L}_{\mathcal{B}}$ on the space
${\mathbb{C}}_{\mathcal{B}}[G^{*}]/I_{\mathcal{B}}$:

$m\cdot(x+I_{\mathcal{B}})=\rho_{\chi^{s}_{q}}([m,x]).$ (3.2.23)

where $x\in{\mathbb{C}}_{\mathcal{B}}[G^{*}]$ is any representative of
$x+I_{\mathcal{B}}\in{\mathbb{C}}_{\mathcal{B}}[G^{*}]/I_{\mathcal{B}}$ and
$m\in{\mathbb{C}}_{\mathcal{B}}[M_{+}]$. The algebra $W_{\mathcal{B}}^{s}(G)$
can be regarded as the intersection of the space of invariants with respect to
action (3.2.23) with the subspace
$Q_{\mathcal{B}}\subset{\mathbb{C}}_{\mathcal{B}}[G^{*}]/I_{\mathcal{B}}$.

Note also that since ${\chi_{q}^{s}}$ is a character of
${\mathbb{C}}_{\mathcal{B}}[M_{+}]$ the ideal $I_{\mathcal{B}}$ is stable
under the action of ${\mathbb{C}}_{\mathcal{B}}[M_{+}]$ on
${\mathbb{C}}_{\mathcal{B}}[G^{*}]$ by commutators.

Denote by $\mathcal{B}_{\chi_{q}^{s}}$ the rank one representation of the
algebra ${\mathbb{C}}_{\mathcal{B}}[M_{+}]$ defined by the character
$\chi_{q}^{s}$. Using the description of the algebra $W_{\mathcal{B}}^{s}(G)$
in terms of action (3.2.23) and the isomorphism
${\mathbb{C}}_{\mathcal{B}}[G^{*}]/I_{\mathcal{B}}={\mathbb{C}}_{\mathcal{B}}[G^{*}]\otimes_{{\mathbb{C}}_{\mathcal{B}}[M_{+}]}\mathcal{B}_{\chi_{q}^{s}}$
one can also define the algebra $W_{\mathcal{B}}^{s}(G)$ as the intersection

$W_{\mathcal{B}}^{s}(G)={\rm
Hom}_{{\mathbb{C}}_{\mathcal{B}}[M_{+}]}(\mathcal{B}_{\chi_{q}^{s}},{\mathbb{C}}_{\mathcal{B}}[G^{*}]\otimes_{{\mathbb{C}}_{\mathcal{B}}[M_{+}]}\mathcal{B}_{\chi_{q}^{s}})\cap
Q_{\mathcal{B}}.$

Using Frobenius reciprocity we also have

${\rm
Hom}_{{\mathbb{C}}_{\mathcal{B}}[M_{+}]}(\mathcal{B}_{\chi_{q}^{s}},{\mathbb{C}}_{\mathcal{B}}[G^{*}]\otimes_{{\mathbb{C}}_{\mathcal{B}}[M_{+}]}\mathcal{B}_{\chi_{q}^{s}})={\rm
End}_{{\mathbb{C}}_{\mathcal{B}}[G^{*}]}({\mathbb{C}}_{\mathcal{B}}[G^{*}]\otimes_{{\mathbb{C}}_{\mathcal{B}}[M_{+}]}\mathcal{B}_{\chi_{q}^{s}}).$

Hence the algebra $W_{\mathcal{B}}^{s}(G)$ acts on the space
${\mathbb{C}}_{\mathcal{B}}[G^{*}]\otimes_{{\mathbb{C}}_{\mathcal{B}}[M_{+}]}\mathcal{B}_{\chi_{q}^{s}}$
from the right by operators commuting with the natural left
${\mathbb{C}}_{\mathcal{B}}[G^{*}]$–action on
${\mathbb{C}}_{\mathcal{B}}[G^{*}]\otimes_{{\mathbb{C}}_{\mathcal{B}}[M_{+}]}\mathcal{B}_{\chi_{q}^{s}}$.
By the definition of $W_{\mathcal{B}}^{s}(G)$ this action preserves
$Q_{\mathcal{B}}$ and by the above presented arguments it commutes with the
natural left ${\mathbb{C}}_{\mathcal{B}}[G_{*}]$–action on $Q_{\mathcal{B}}$.

Thus $Q_{\mathcal{B}}$ is a
${\mathbb{C}}_{\mathcal{B}}[G_{*}]$–$W_{\mathcal{B}}^{s}(G)$ bimodule equipped
also with the adjoint action of ${\mathbb{C}}_{\mathcal{B}}[M_{+}]$. By
(3.2.18) the adjoint action satisfies

${\rm Ad}x(yv)={\rm Ad}x^{1}(y){\rm
Ad}x^{2}(v),~{}x\in{\mathbb{C}}_{\mathcal{B}}[M_{+}],~{}y\in{\mathbb{C}}_{\mathcal{B}}[G_{*}],v\in
Q_{\mathcal{B}},$ (3.2.24)

and $\Delta_{s}(x)=x^{1}\otimes x^{2}$.

Denote by $1\in Q_{\mathcal{B}}$ the image of the element
$1\in{\mathbb{C}}_{\mathcal{B}}[G_{*}]$ in the quotient $Q_{\mathcal{B}}$
under the canonical projection ${\mathbb{C}}_{\mathcal{B}}[G_{*}]\rightarrow
Q_{\mathcal{B}}$. Obviously $1$ is the generating vector for $Q_{\mathcal{B}}$
as a module over ${\mathbb{C}}_{\mathcal{B}}[G_{*}]$. Using formula (3.2.24)
and recalling that $Q_{\mathcal{B}}$ is a
${\mathbb{C}}_{\mathcal{B}}[G_{*}]$–$W_{\mathcal{B}}^{s}(G)$ bimodule, for
$x\in{\mathbb{C}}_{\mathcal{B}}[M_{+}],y\in{\mathbb{C}}_{\mathcal{B}}[G_{*}]$,
and for a representative $w\in{\mathbb{C}}_{\varepsilon}[G_{*}]$ of an element
$w+I_{\mathcal{B}}\in W_{\mathcal{B}}^{s}(G)$ we have

${\rm Ad}x(wy1)={\rm Ad}x(yw1)={\rm Ad}x_{1}(y){\rm Ad}x_{2}(w1)=$ $={\rm
Ad}x_{1}(y)\varepsilon_{s}(x_{2})w1={\rm Ad}x(y)w1=w{\rm Ad}x(y1).$

Since $Q_{\mathcal{B}}$ is generated by the vector $1$ over
${\mathbb{C}}_{\mathcal{B}}[G_{*}]$ the last relation implies that the action
of $W_{\mathcal{B}}^{s}(G)$ on $Q_{\mathcal{B}}$ commutes with the adjoint
action.

We can summarize the results of the discussion above in the following
proposition.

######  Proposition 3.2.9.

The space $Q_{\mathcal{B}}$ is naturally equipped with the structure of a left
${\mathbb{C}}_{\mathcal{B}}[G_{*}]$–module, a right
${\mathbb{C}}_{\mathcal{B}}[M_{+}]$–module via the adjoint action and a right
$W_{\mathcal{B}}^{s}(G)$–module in such a way that the left
${\mathbb{C}}_{\mathcal{B}}[G_{*}]$–action and the right
${\mathbb{C}}_{\mathcal{B}}[M_{+}]$–action commute with the right
$W_{\mathcal{B}}^{s}(G)$–action and compatibility condition (3.2.24) is
satisfied.

Finally we remark that by specializing $q$ to a particular value
$\varepsilon\in\mathbb{C}$, $\varepsilon\neq 0$, one can define a complex
associative algebra
${\mathbb{C}}_{\varepsilon}[G_{*}]={\mathbb{C}}_{\mathcal{B}}[G_{*}]/(q^{\frac{1}{dr^{2}}}-\varepsilon^{\frac{1}{dr^{2}}}){\mathbb{C}}_{\mathcal{B}}[G_{*}]$,
its subalgebra ${\mathbb{C}}_{\varepsilon}[M_{+}]$ with a nontrivial character
$\chi_{\varepsilon}^{s}$ and the corresponding W–algebra

$W_{\varepsilon}^{s}(G)={\rm
Hom}_{{\mathbb{C}}_{\varepsilon}[M_{+}]}(\mathbb{C}_{\varepsilon_{s}},Q_{\varepsilon}),$
(3.2.25)

where $\mathbb{C}_{\varepsilon_{s}}$ is the trivial representation of the
algebra ${\mathbb{C}}_{\varepsilon}[M_{+}]$ induced by the counit,
$Q_{\varepsilon}=Q_{\mathcal{B}}/Q_{\mathcal{B}}(q^{\frac{1}{dr^{2}}}-\varepsilon^{\frac{1}{dr^{2}}})$.

Obviously, for generic $\varepsilon$ we have
$W_{\varepsilon}^{s}(G)=W_{\mathcal{B}}^{s}(G)/(q^{\frac{1}{dr^{2}}}-\varepsilon^{\frac{1}{dr^{2}}})W_{\mathcal{B}}^{s}(G)$.

###  3.3 Poisson reduction

In this section we recall basic facts on Poisson reduction. They will be used
in the next section to describe Poisson q-W–algebras as reduced Poisson
algebras.

Let $M,~{}B,~{}B^{\prime}$ be Poisson manifolds. Two Poisson surjections

$\begin{array}[]{ccccc}&&M&&\\\
&\stackrel{{\scriptstyle\pi^{\prime}}}{{\swarrow}}&&\stackrel{{\scriptstyle\pi}}{{\searrow}}&\\\
B^{\prime}&&&&B\end{array}$

form a dual pair if the pullback $\pi^{{}^{\prime}*}C^{\infty}(B^{\prime})$ is
the centralizer of $\pi^{*}C^{\infty}(B)$ in the Poisson algebra
$C^{\infty}(M)$. In this case the sets
$B^{\prime}_{b}=\pi^{\prime}\left(\pi^{-1}(b)\right),~{}b\in B$ are Poisson
submanifolds in $B^{\prime}$ called reduced Poisson manifolds.

Fix an element $b\in B$. Then the algebra of functions
$C^{\infty}(B^{\prime}_{b})$ may be described as follows. Let $I_{b}$ be the
ideal in $C^{\infty}(M)$ generated by elements ${\pi}^{*}(f),~{}f\in
C^{\infty}(B),~{}f(b)=0$. Denote $M_{b}=\pi^{-1}(b)$. Then the algebra
$C^{\infty}(M_{b})$ is simply the quotient of $C^{\infty}(M)$ by $I_{b}$.
Denote by $P_{b}:C^{\infty}(M)\rightarrow
C^{\infty}(M)/I_{b}=C^{\infty}(M_{b})$ the canonical projection onto the
quotient.

######  Lemma 3.3.1.

Suppose that the map $f\mapsto f(b)$ is a character of the Poisson algebra
$C^{\infty}(B)$. Then one can define an action of the Poisson algebra
$C^{\infty}(B)$ on the space $C^{\infty}(M_{b})$ by

$f\cdot\varphi=P_{b}(\\{{\pi}^{*}(f),\tilde{\varphi}\\}),$ (3.3.1)

where $f\in C^{\infty}(B)$, $\varphi\in C^{\infty}(M_{b})$ and
$\tilde{\varphi}\in C^{\infty}(M)$ is a representative of $\varphi$ in
$C^{\infty}(M)$ such that $P_{b}(\tilde{\varphi})=\varphi$. Moreover,
$C^{\infty}(B^{\prime}_{b})$ is the subspace of invariants in
$C^{\infty}(M_{b})$ with respect to this action.

###### Proof.

Let $\varphi\in C^{\infty}(M_{b})$. Choose a representative
$\tilde{\varphi}\in C^{\infty}(M)$ such that $P_{b}(\tilde{\varphi})=\varphi$.
Since the map $f\mapsto f(b)$ is a character of the Poisson algebra
$C^{\infty}(B)$, Hamiltonian vector fields of functions ${\pi}^{*}(f),~{}f\in
C^{\infty}(B)$ are tangent to the submanifold $M_{b}$. Therefore the right
hand side of (3.3.1) only depends on $\varphi$ but not on the representative
$\tilde{\varphi}$, and hence formula (3.3.1) defines an action of the Poisson
algebra $C^{\infty}(B)$ on the space $C^{\infty}(M_{b})$.

Using the definition of the dual pair we obtain that
$\varphi={\pi^{\prime}}^{*}(\psi)$ for some $\psi\in
C^{\infty}(B^{\prime}_{b})$ if and only if
$P_{b}(\\{{\pi}^{*}(f),\tilde{\varphi}\\})=0$ for every $f\in C^{\infty}(B)$.

Finally we obtain that $C^{\infty}(B^{\prime}_{b})$ is exactly the subspace of
invariants in $C^{\infty}(M_{b})$ with respect to action (3.3.1).

∎

######  Definition 3.3.2.

The algebra $C^{\infty}(B^{\prime}_{b})$ is called a reduced Poisson algebra.
We also denote it by $C^{\infty}(M_{b})^{C^{\infty}(B)}$.

######  Remark 3.3.3.

Note that the description of the algebra $C^{\infty}(M_{b})^{C^{\infty}(B)}$
obtained in Lemma 3.3.1 is independent of both the manifold $B^{\prime}$ and
the projection $\pi^{\prime}$. Observe also that the Hamiltonian vector fields
of functions $\pi^{*}(f),f\in C^{\infty}(B)$ are tangent to $M_{b}$, and hence
the reduced space $B^{\prime}_{b}$ may be identified with across–section of
the action of the Poisson algebra $C^{\infty}(B)$ on $M_{b}$ by Hamiltonian
vector fields in the case when this action is free. In particular, in this
case $B^{\prime}_{b}$ may be regarded as a submanifold in $M_{b}$.

In the case when the map $f\mapsto f(b)$ is a character of the Poisson algebra
$C^{\infty}(B)$ the Poisson structure on the algebra
$C^{\infty}(B^{\prime}_{b})$ can be explicitly described as follows. Let
$\varphi_{1},\varphi_{2}\in C^{\infty}(M_{b})^{C^{\infty}(B)}$. Choose
representatives $\tilde{\varphi}_{1,2}\in C^{\infty}(M)$ such that
$P_{b}(\tilde{\varphi}_{1,2})=\varphi_{1,2}$. Then

$\\{\varphi_{1},\varphi_{2}\\}=\\{\tilde{\varphi}_{1},\tilde{\varphi}_{2}\\}\quad{\rm
mod}~{}I_{b}.$ (3.3.2)

By Lemma 3.3.1 the class in $C^{\infty}(M)/I_{b}=C^{\infty}(M_{b})$ of the
function in right hand side of this formula is ${C^{\infty}(B)}$–invariant and
independent of the choice of the representatives $\tilde{\varphi}_{1,2}\in
C^{\infty}(M)$.

An important example of dual pairs is provided by Poisson group actions.
Recall that a (local) left Poisson group action of a Poisson–Lie group $A$ on
a Poisson manifold $M$ is a (local) left group action $A\times M\rightarrow M$
which is also a Poisson map (as usual, we suppose that $A\times M$ is equipped
with the product Poisson structure).

If the space $M/A$ is a smooth manifold, there exists a unique Poisson
structure on $M/A$ such that the canonical projection $M\rightarrow M/A$ is a
Poisson map.

Let $\mathfrak{a}$ be the Lie algebra of $A$. Denote by
$\langle\cdot,\cdot\rangle$ the canonical paring between ${\mathfrak{a}}^{*}$
and $\mathfrak{a}$. A map $\mu:M\rightarrow A^{*}$ is called a moment map for
a (local) left Poisson group action $M\times A\rightarrow M$ if

$L_{\widehat{X}}\varphi=\langle\mu^{*}(\theta_{A^{*}}),X\rangle(\xi_{\varphi}),$
(3.3.3)

where $\theta_{A^{*}}$ is the universal left–invariant Maurer–Cartan form on
$A^{*}$, $X\in{\mathfrak{a}}$, $\widehat{X}$ is the corresponding vector field
on $M$ and $\xi_{\varphi}$ is the Hamiltonian vector field of $\varphi\in
C^{\infty}(M)$.

######  Proposition 3.3.4.

([72], Theorem 4.9) Let $A\times M\rightarrow M$ be a left (local) Poisson
group action of a Poisson–Lie group $A$ on a Poisson manifold $M$ with moment
map $\mu:M\to A^{*}$. Denote by $\Pi_{A^{*}}$ the Poisson tensor of $A^{*}$.
Then there exists a right invariant bivector field $\Lambda$ on $A^{*}$ such
that $\Pi_{\mu}=\Pi_{A^{*}}+\Lambda$ is a Poisson tensor on $A^{*}$ and the
map $\mu:M\to A^{*}_{\mu}$ is Poisson, where $A^{*}_{\mu}$ the manifold
$A^{*}$ equipped with the Poisson structure associated to $\Pi_{\mu}$.

From the definition of the moment map it follows that if $M/A$ is a smooth
manifold then the canonical projection $M\rightarrow M/A$ and the moment map
$\mu:M\rightarrow A^{*}$ form a dual pair.

The main example of Poisson group actions is the so–called dressing action.
The dressing action may be described as follows.

######  Proposition 3.3.5.

Let $G$ be a connected simply connected Poisson–Lie group with factorizable
tangent Lie bialgebra, $G^{*}$ the dual group. Then there exists a unique left
local Poisson group action

$G\times G^{*}\rightarrow G^{*},~{}~{}((L_{+},L_{-}),g)\mapsto
g\circ(L_{+},L_{-}),$

such that the identity mapping $\mu:G^{*}\rightarrow G^{*}$ is the moment map
for this action.

Moreover, let $q:G^{*}\rightarrow G$ be the map defined by

$q(L_{+},L_{-})=L_{-}^{-1}L_{+}.$

Then

$q(g\circ(L_{+},L_{-}))=gL_{-}L_{+}^{-1}g^{-1}.$ (3.3.4)

The notion of Poisson group actions may be generalized as follows. Let
$A\times M\rightarrow M$ be a Poisson group action of a Poisson–Lie group $A$
on a Poisson manifold $M$. A subgroup $K\subset A$ is called admissible if the
set $C^{\infty}\left(M\right)^{K}$ of $K$-invariants is a Poisson subalgebra
in $C^{\infty}\left(M\right)$. If space $M/K$ is a smooth manifold, we may
identify the algebras $C^{\infty}(M/K)$ and $C^{\infty}\left(M\right)^{K}$.
Hence there exists a Poisson structure on $M/K$ such that the canonical
projection $M\rightarrow M/K$ is a Poisson map.

######  Proposition 3.3.6.

([95], Theorem 6; [72], §2) Let
$\left({\mathfrak{a}},{\mathfrak{a}}^{*}\right)$ be the tangent Lie bialgebra
of a Poisson–Lie group $A$. A connected Lie subgroup $K\subset A$ with Lie
algebra ${\mathfrak{k}}\subset{\mathfrak{a}}$ is admissible if the annihilator
${\mathfrak{k}}^{\perp}$ of $\mathfrak{k}$ in ${\mathfrak{a}}^{*}$ is a Lie
subalgebra ${\mathfrak{k}}^{\perp}\subset{\mathfrak{a}}^{*}$.

We shall need the following particular example of dual pairs arising from
Poisson group actions.

Let $A\times M\rightarrow M$ be a left (local) Poisson group action of a
Poisson–Lie group $A$ on a manifold $M$. Suppose that this action possesses a
moment map $\mu:M\rightarrow A^{*}$. Let $K$ be an admissible subgroup in $A$.
Denote by $\mathfrak{k}$ the Lie algebra of $K$. Assume that
${\mathfrak{k}}^{\perp}\subset{\mathfrak{a}}^{*}$ is a Lie subalgebra in
${\mathfrak{a}}^{*}$. Suppose also that there is a splitting
${\mathfrak{a}}^{*}={\mathfrak{t}}+{\mathfrak{k}}^{\perp}$ (direct sum of
vector spaces), and that $\mathfrak{t}$ is a Lie subalgebra in
${\mathfrak{a}}^{*}$. Then the linear space ${\mathfrak{k}}^{*}$ is naturally
identified with $\mathfrak{t}$.

Assume that $A^{*}=TK^{\perp}$ as a manifold, where $K^{\perp},T$ are the Lie
subgroups of $A^{*}$ corresponding to the Lie subalgebras
${\mathfrak{k}}^{\perp},{\mathfrak{t}}\subset{\mathfrak{a}}^{*}$,
respectively. For any $a^{*}=tk^{\perp}\in A^{*}$ with $k^{\perp}\in
K^{\perp}$, $t\in T$ denote $\pi_{K^{\perp}}(a^{*})=k^{\perp}$,
$\pi_{T}(a^{*})=t$. This defines maps $\pi_{K^{\perp}}:A^{*}\to K^{\perp}$,
$\pi_{T}:A^{*}\to T$.

######  Proposition 3.3.7.

Suppose that for any $k^{\perp}\in K^{\perp}$ the transformation

$\displaystyle{\mathfrak{t}}\rightarrow{\mathfrak{t}},$ (3.3.5) $\displaystyle
t\mapsto({\rm Ad}(k^{\perp})t)_{{\mathfrak{t}}},$

where the subscript ${{\mathfrak{t}}}$ stands for the
${{\mathfrak{t}}}$–component with respect to the decomposition
${\mathfrak{a}}^{*}={\mathfrak{t}}+{\mathfrak{k}}^{\perp}$, is invertible.

Define a map $\overline{\mu}:M\rightarrow T$ by

$\overline{\mu}=\pi_{T}\mu.$

Then

(i) $\overline{\mu}^{*}\left(C^{\infty}\left(T\right)\right)$ is a Poisson
subalgebra in $C^{\infty}\left(M\right)$, and hence one can equip $T$ with a
Poisson structure such that $\overline{\mu}:M\rightarrow T$ is a Poisson map.

(ii) Moreover, the algebra $C^{\infty}\left(M\right)^{K}$ is the centralizer
of $\overline{\mu}^{*}\left(C^{\infty}\left(T\right)\right)$ in the Poisson
algebra $C^{\infty}\left(M\right)$. In particular, if $M/K$ is a smooth
manifold the maps

$\begin{array}[]{ccccc}&&M&&\\\
&\stackrel{{\scriptstyle\pi}}{{\swarrow}}&&\stackrel{{\scriptstyle\overline{\mu}}}{{\searrow}}&,\\\
M/K&&&&T\end{array}$ (3.3.6)

form a dual pair.

###### Proof.

(i) We claim that multiplication in $A^{*}$ gives rise to a right Poisson
group action $A^{*}_{\mu}\times A^{*}\to A^{*}_{\mu}$. Indeed, for $g\in