P}\big{)}}^{j_{\scriptscriptstyle\Theta}}$ of
$\,\mathfrak{g}^{\raise-1.0pt\hbox{$\scriptscriptstyle\mathcal{R}$}}_{\scriptscriptstyle
P}$ by $j_{\scriptscriptstyle\Theta}$ occurs) given by
$\;E_{i}\,\mapsto\,E_{i}\,,\;T\,\mapsto\,T\,$ and $\;F_{i}\,\mapsto\,F_{i}\;$
for all $\,i\in I\,$, $\,T\in\mathfrak{h}\,$. In particular, the class of all
MpLbA’s of any fixed Cartan type and of fixed rank is stable by toral twist
deformations. Moreover, inside it the subclass of all such MpLbA’s associated
with straight, resp. small, realizations is stable as well.

###### Proof.

By §3.2.3, the Lie algebra structure in
${\big{(}\mathfrak{g}^{\raise-1.0pt\hbox{$\scriptscriptstyle\mathcal{R}$}}_{\scriptscriptstyle
P}\big{)}}^{j_{\scriptscriptstyle\Theta}}$ is the same as in
$\mathfrak{g}^{\raise-1.0pt\hbox{$\scriptscriptstyle\mathcal{R}$}}_{\scriptscriptstyle
P}\,$, and the latter only depends on the $\alpha_{j}$’s and the sums
$\,S_{j}:=2^{-1}\big{(}T_{i}^{+}\\!+T_{i}^{-}\big{)}\,$ ($\,j\in I\,$). Now,
both the $\alpha_{j}$’s and the $S_{j}$’s do not change (see above) when we
pass from
$\mathfrak{g}^{\raise-1.0pt\hbox{$\scriptscriptstyle\mathcal{R}$}}_{\scriptscriptstyle
P}$ to $\mathfrak{g}_{P_{\Theta}}^{{\mathcal{R}}_{\scriptscriptstyle\Theta}}$
or viceversa; therefore, the formulas in the claim (mapping each generator of
$\mathfrak{g}_{P_{\Theta}}^{{\mathcal{R}}_{\scriptscriptstyle\Theta}}$ onto
the same name generator of
$\,\mathfrak{g}^{\raise-1.0pt\hbox{$\scriptscriptstyle\mathcal{R}$}}_{\scriptscriptstyle
P}={\big{(}\mathfrak{g}^{\raise-1.0pt\hbox{$\scriptscriptstyle\mathcal{R}$}}_{\scriptscriptstyle
P}\big{)}}^{j_{\scriptscriptstyle\Theta}}\,$) provide an isomorphism of Lie
algebras.

Now consider the toral twist
$\;j_{\scriptscriptstyle\Theta}:={\textstyle\sum_{g,k=1}^{t}}\theta_{gk}\,H_{g}\otimes
H_{k}\;$ given in (3.7). By (3.3)

$\delta^{j_{\scriptscriptstyle\Theta}}(x)\;:=\;\delta(x)-\operatorname{ad}_{x}\\!\big{(}j_{\scriptscriptstyle\Theta}\big{)}\;=\;\delta(x)-{\textstyle\sum_{g,k=1}^{t}}\theta_{gk}\,\big{(}\big{[}x,H_{g}\big{]}\otimes
H_{k}+H_{g}\otimes\big{[}x,H_{k}\big{]}\big{)}$

for all $\,x\in\mathfrak{g}\,$. Now take $\,x:=E_{\ell}\,$ ($\,\ell\in I\,$):
then the previous formula reads

$\halign{\hbox to\displaywidth{$\hfil\displaystyle#\hfil$}\cr
0.0pt{\hfil$\displaystyle\delta^{j_{\scriptscriptstyle\Theta}}(E_{\ell})\;:=\;\delta(E_{\ell})-{\textstyle\sum_{g,k=1}^{t}}\,\theta_{gk}\,\big{(}\big{[}E_{\ell}\,,H_{g}\big{]}\otimes
H_{k}+H_{g}\otimes\big{[}E_{\ell}\,,H_{k}\big{]}\big{)}\;=\hfill\cr
0.0pt{\hfil$\displaystyle\quad=\;\delta(E_{\ell})-{\textstyle\sum_{g,k=1}^{t}}\,\theta_{gk}\,\big{(}\\!-\alpha_{\ell}(H_{g})E_{\ell}\otimes
H_{k}-H_{g}\otimes\alpha_{\ell}(H_{k})E_{\ell}\,\big{)}\;=\hfill\cr
0.0pt{\hfil$\displaystyle=\;T^{+}_{\ell}\otimes E_{\ell}-E_{\ell}\otimes
T^{+}_{\ell}+{\textstyle\sum_{g,k=1}^{t}}\,\theta_{gk}\,\big{(}\,\alpha_{\ell}(H_{g})E_{\ell}\otimes
H_{k}+\alpha_{\ell}(H_{k})H_{g}\otimes E_{\ell}\,\big{)}\;=\quad\cr
0.0pt{\hfil$\displaystyle\hfill=\;\Big{(}\,T^{+}_{\ell}+{\textstyle\sum_{g,k=1}^{t}}\,\theta_{kg}\,\alpha_{\ell}(H_{g})H_{k}\Big{)}\otimes
E_{\ell}\,-\,E_{\ell}\otimes\Big{(}\,T^{+}_{\ell}+{\textstyle\sum_{g,k=1}^{t}}\,\theta_{kg}\,\alpha_{\ell}(H_{g})H_{k}\Big{)}\;=\cr
0.0pt{\hfil$\displaystyle\hfill=\;T^{+}_{{\scriptscriptstyle\Theta},\ell}\otimes
E_{\ell}\,-\,E_{\ell}\otimes
T^{+}_{{\scriptscriptstyle\Theta},\ell}\;=\;2\;T^{+}_{{\scriptscriptstyle\Theta},\ell}\wedge\\!E_{\ell}\crcr}}}}}}$

hence in short we get
$\;\delta^{j_{\scriptscriptstyle\Theta}}(E_{\ell})\,=\,2\;T^{+}_{{\scriptscriptstyle\Theta},\ell}\wedge\\!E_{\ell}\;$.
Similar computations give

$\delta^{j_{\scriptscriptstyle\Theta}}(E_{i})=2\,T^{+}_{{\scriptscriptstyle\Theta},i}\wedge\\!E_{i}\;,\\!\quad\delta^{j_{\scriptscriptstyle\Theta}}(T)=0\;,\\!\quad\delta^{j_{\scriptscriptstyle\Theta}}(F_{i})=2\,T^{-}_{{\scriptscriptstyle\Theta},i}\wedge\\!F_{i}\;,\quad\forall\;i\in
I\,,\,T\\!\in\mathfrak{h}$

This means that, through the Lie algebra isomorphism $f_{\scriptscriptstyle
P}^{\scriptscriptstyle\Theta}\,$, the Lie coalgebra structure of
${\big{(}\mathfrak{g}^{\raise-1.0pt\hbox{$\scriptscriptstyle\mathcal{R}$}}_{\scriptscriptstyle
P}\big{)}}^{j_{\scriptscriptstyle\Theta}}$ is described on generators exactly
like that of
$\mathfrak{g}_{P_{\Theta}}^{{\mathcal{R}}_{\scriptscriptstyle\Theta}}\,$, with
the new “coroots” $T^{\pm}_{{\scriptscriptstyle\Theta},i}$ ($\,i\in I\,$) in
${\big{(}\mathfrak{g}^{\raise-1.0pt\hbox{$\scriptscriptstyle\mathcal{R}$}}_{\scriptscriptstyle
P}\big{)}}^{j_{\scriptscriptstyle\Theta}}$ playing the role of the coroots
$T^{\pm}_{i}$ ($\,i\in I\,$) in
$\mathfrak{g}_{P_{\Theta}}^{{\mathcal{R}}_{\scriptscriptstyle\Theta}}\,$.
Therefore $\;f^{\scriptscriptstyle\Theta}_{\scriptscriptstyle
P}:\mathfrak{g}_{P_{\Theta}}^{{\mathcal{R}}_{\scriptscriptstyle\Theta}}\relbar\joinrel\relbar\joinrel\longrightarrow{\big{(}\mathfrak{g}^{\raise-1.0pt\hbox{$\scriptscriptstyle\mathcal{R}$}}_{\scriptscriptstyle
P}\big{)}}^{j_{\scriptscriptstyle\Theta}}\;$ is an isomorphism of Lie
bialgebras, q.e.d. ∎

In fact, the previous result can be reversed, somehow. Namely, our next result
shows, in particular, that every straight small MpLbA can be realized as a
toral twist deformation of the “standard” MpLbA
$\mathfrak{g}^{\scriptscriptstyle D}_{\scriptscriptstyle DA}$ (as in §3.2.3) —
cf. claim (c) below.

######  Theorem 3.3.4.

With assumptions as above, let $P$ and $P^{\prime}$ be two matrices of Cartan
type with the same associated Cartan matrix $A\,$, i.e. such that
$\,P_{s}=P^{\prime}_{s}\,$. (a) Let $\,{\mathcal{R}}$ be a straight
realization of $P$ and let $\mathfrak{g}_{\scriptscriptstyle
P}^{\scriptscriptstyle{\mathcal{R}}}$ be the associated MpLbA. Then there
exists a matrix $\,\Theta\in\mathfrak{so}_{t}(\Bbbk)\,$ such that
$\,P^{\prime}=P_{\scriptscriptstyle\Theta}\,$, the corresponding
${\mathcal{R}}_{\scriptscriptstyle\Theta}$ is a straight realization of
$\,P^{\prime}=P_{\scriptscriptstyle\Theta}\,$, and for the twist element
$j_{\scriptscriptstyle\Theta}$ as in (3.7) we have

$\mathfrak{g}_{\scriptscriptstyle
P^{\prime}}^{\scriptscriptstyle{\mathcal{R}}_{\scriptscriptstyle\Theta}}\;\cong\;{\big{(}\,\mathfrak{g}_{\scriptscriptstyle
P}^{\scriptscriptstyle{\mathcal{R}}}\big{)}}^{j_{\Theta}}$

In a nutshell, if $\,P^{\prime}_{s}=P_{s}\,$ then from any straight MpLbA over
$P$ we can obtain by toral twist deformation a straight MpLbA (of the same
rank) over $P^{\prime}\,$. (b) Let ${\mathcal{R}}$ and
${\mathcal{R}}^{\prime}$ be straight small realizations of $P$ and
$P^{\prime}$ respectively, with
$\,\operatorname{rk}({\mathcal{R}})=\operatorname{rk}({\mathcal{R}}^{\prime})=:t\,$,
and let
$\mathfrak{g}^{\raise-1.0pt\hbox{$\scriptscriptstyle\mathcal{R}$}}_{\scriptscriptstyle
P}$ and $\mathfrak{g}_{\scriptscriptstyle
P^{\prime}}^{\scriptscriptstyle{\mathcal{R}}^{\prime}}$ be the associated
MpLbA’s. Then there exists a matrix $\,\Theta\in\mathfrak{so}_{t}(\Bbbk)\,$
such that for the twist element $j_{\scriptscriptstyle\Theta}$ as in (3.7) we
have

$\mathfrak{g}_{\scriptscriptstyle
P^{\prime}}^{\scriptscriptstyle{\mathcal{R}}^{\prime}}\;\cong\;{\big{(}\,\mathfrak{g}^{\raise-1.0pt\hbox{$\scriptscriptstyle\mathcal{R}$}}_{\scriptscriptstyle
P}\big{)}}^{j_{\Theta}}$

In a nutshell, if $\,P^{\prime}_{s}=P_{s}\,$ then any straight small MpLbA
over $P^{\prime}$ is isomorphic to a toral twist deformation of any straight
small MpLbA over $P$ of the same rank. (c) Every straight small MpLbA is
isomorphic to some toral twist deformation of the standard MpLbA
$\mathfrak{g}^{\scriptscriptstyle D}_{\scriptscriptstyle DA}$ (over
$\,DA=P_{s}\,$, cf. §3.2.3) of the same rank.

###### Proof.

(a) By Theorem 3.3.3 it is enough to find
$\,\Theta\in\mathfrak{so}_{t}(\Bbbk)\,$ such that
$\,P^{\prime}=P_{\scriptscriptstyle\Theta}\,$, that is
$\;P^{\prime}=P-\mathfrak{A}\,\Theta\,\mathfrak{A}^{\,\scriptscriptstyle
T}\,$; but this is guaranteed by Lemma 2.2.5, so we are done. (b) This follows
from claim (a), along with the uniqueness of straight small realizations — cf.
Proposition 2.1.5(b) — and Proposition 3.2.4. (c) This follows from (b),
taking as $\mathfrak{g}_{\scriptscriptstyle
P^{\prime}}^{\scriptscriptstyle{\mathcal{R}}^{\prime}}$ the given straight
small MpLbA and as
$\mathfrak{g}^{\raise-1.0pt\hbox{$\scriptscriptstyle\mathcal{R}$}}_{\scriptscriptstyle
P}$ the “standard” MpLbA $\mathfrak{g}^{\scriptscriptstyle
D}_{\scriptscriptstyle P}$ over $\,P:=DA=P^{\prime}_{s}\,$. ∎

######  Observations 3.3.5.

(a) Theorems 3.3.3 and 3.3.4 have the following interpretation. Our MpLbA’s
$\,\mathfrak{g}^{\raise-1.0pt\hbox{$\scriptscriptstyle\mathcal{R}$}}_{\scriptscriptstyle
P}\,$ depend on the multiparameter $P\,$; at a further level, once we perform
onto
$\,\mathfrak{g}^{\raise-1.0pt\hbox{$\scriptscriptstyle\mathcal{R}$}}_{\scriptscriptstyle
P}\,$ a deformation by toral twist the outcome
$\;\mathfrak{g}^{\scriptscriptstyle{\mathcal{R}}}_{{\scriptscriptstyle
P},{\scriptscriptstyle\Theta}}:={\big{(}\,\mathfrak{g}^{\raise-1.0pt\hbox{$\scriptscriptstyle\mathcal{R}$}}_{\scriptscriptstyle
P}\big{)}}^{j_{\Theta}}\;$ depends on two multiparameters, namely $P$ and
$\Theta\,$. Thus all these
$\mathfrak{g}^{\scriptscriptstyle{\mathcal{R}}}_{{\scriptscriptstyle
P},{\scriptscriptstyle\Theta}}$’s form a seemingly richer family of “twice-
multiparametric” MpLbA’s. Nonetheless, Theorem 3.3.3 above proves that this
coincides with the family of all MpLbA’s, though the latter seems smaller.

In short, Theorems 3.3.3 and 3.3.4 show the following. The dependence of the
Lie bialgebra structure of
$\mathfrak{g}^{\scriptscriptstyle{\mathcal{R}}}_{{\scriptscriptstyle
P},{\scriptscriptstyle\Theta}}$ on the “double parameter” $(P\,,\Theta)$ is
“split” in the algebraic structure — ruled by $P$ — and in the coalgebraic
structure — ruled by $\Theta\,$. Now, Theorems 3.3.3 and 3.3.4 enable us to
“polarize” this dependence so to codify it either entirely within the
algebraic structure (while the coalgebraic one is reduced to a “canonical
form”) — so the single multiparameter $P_{\Theta}$ is enough to describe it —
or entirely within the coalgebraic structure (with the algebraic one being
reduced to the “standard” one) — so the one multiparameter $\Theta_{P}$ is
enough to describe it. (b) As we saw at the end of §2.2.8, the (sub)class of
split realizations is not closed under twist deformation; as a consequence,
the subclass of all MpLbA’s that are “split” is not closed either under twist
deformation.

###  3.4. Deformations of MpLbA’s by toral $2$–cocycles

Let again
$\mathfrak{g}^{\raise-1.0pt\hbox{$\scriptscriptstyle\mathcal{R}$}}_{\scriptscriptstyle
P}$ be a MpLbA as in §3.2.3, and keep notation as above. Dually to what we did
before, we consider now the so-called “toral” 2–cocycles for
$\mathfrak{g}^{\raise-1.0pt\hbox{$\scriptscriptstyle\mathcal{R}$}}_{\scriptscriptstyle
P}\,$.

######  Definition 3.4.1.

Fix an antisymmetric $\Bbbk$–linear map
$\,\chi:\mathfrak{h}\otimes\mathfrak{h}\relbar\joinrel\longrightarrow\Bbbk\,$
such that

$\chi(S_{i}\,,\,-\,)\,=\,0\,=\,\chi(\,-\,,S_{i})\qquad\qquad\forall\;\;i\in I$
(3.8)

where $\,S_{i}:=2^{-1}\big{(}\,T^{+}_{i}+T^{-}_{i}\big{)}\,$ for all $\,i\in
I\,$ (cf. Definition 2.1.2); in other words, we have $\,\chi\in\text{\sl
Alt}_{\,{\Bbbk[[\hbar]]}}^{\,S}(\mathfrak{h})\,$, cf. (2.12). Moreover, let
$\,\;\pi^{\mathfrak{g}^{\raise-1.0pt\hbox{$\scriptscriptstyle\mathcal{R}$}}_{\scriptscriptstyle
P}}_{\mathfrak{h}}:\mathfrak{g}^{\raise-1.0pt\hbox{$\scriptscriptstyle\mathcal{R}$}}_{\scriptscriptstyle
P}\relbar\joinrel\twoheadrightarrow\mathfrak{h}\;\,$ be the canonical
projection induced by any one of the triangular decompositions in (3.6). We
define

$\chi_{\mathfrak{g}}:=\chi\circ\Big{(}\,\pi^{\mathfrak{g}^{\raise-1.0pt\hbox{$\scriptscriptstyle\mathcal{R}$}}_{\scriptscriptstyle
P}}_{\mathfrak{h}}\otimes\pi^{\mathfrak{g}^{\raise-1.0pt\hbox{$\scriptscriptstyle\mathcal{R}$}}_{\scriptscriptstyle
P}}_{\mathfrak{h}}\Big{)}:\mathfrak{g}^{\raise-1.0pt\hbox{$\scriptscriptstyle\mathcal{R}$}}_{\scriptscriptstyle
P}\otimes\mathfrak{g}^{\raise-1.0pt\hbox{$\scriptscriptstyle\mathcal{R}$}}_{\scriptscriptstyle
P}\relbar\joinrel\relbar\joinrel\twoheadrightarrow\mathfrak{h}\otimes\mathfrak{h}\relbar\joinrel\longrightarrow\Bbbk\qquad$

and we call it the toral 2–cocycle (or “2–cocycle of toral type”) associated
with $\,\chi\,$. $\diamondsuit$

Next result follows at once by construction, and explains our use of
terminology:

######  Lemma 3.4.2.

For any antisymmetric $\Bbbk$–linear map
$\,\chi:\mathfrak{h}\otimes\mathfrak{h}\longrightarrow\Bbbk\,$ obeying (3.8),

$\chi_{\mathfrak{g}}:=\chi\circ\Big{(}\pi^{\mathfrak{g}^{\raise-1.0pt\hbox{$\scriptscriptstyle\mathcal{R}$}}_{\scriptscriptstyle
P}}_{\mathfrak{h}}\otimes\pi^{\mathfrak{g}^{\raise-1.0pt\hbox{$\scriptscriptstyle\mathcal{R}$}}_{\scriptscriptstyle
P}}_{\mathfrak{h}}\,\Big{)}:\mathfrak{g}^{\raise-1.0pt\hbox{$\scriptscriptstyle\mathcal{R}$}}_{\scriptscriptstyle
P}\otimes\mathfrak{g}^{\raise-1.0pt\hbox{$\scriptscriptstyle\mathcal{R}$}}_{\scriptscriptstyle
P}\relbar\joinrel\relbar\joinrel\twoheadrightarrow\mathfrak{h}\otimes\mathfrak{h}\relbar\joinrel\longrightarrow\Bbbk$

is a 2–cocycle map for the Lie bialgebra
$\mathfrak{g}^{\raise-1.0pt\hbox{$\scriptscriptstyle\mathcal{R}$}}_{\scriptscriptstyle
P}\,$, in the sense of (3.4).

###### Proof.

We have to check that
$\,\chi_{\mathfrak{g}}\in\operatorname{Hom}_{\Bbbk}\big{(}\,\mathfrak{g}^{\raise-1.0pt\hbox{$\scriptscriptstyle\mathcal{R}$}}_{\scriptscriptstyle
P}\otimes\mathfrak{g}^{\raise-1.0pt\hbox{$\scriptscriptstyle\mathcal{R}$}}_{\scriptscriptstyle
P}\,,\Bbbk\,\big{)}\,$ satisfies (3.4), that is

$\displaystyle\operatorname{ad}_{\psi}\\!\big{(}\,\partial_{*}(\chi_{\mathfrak{g}})-{[\\![\,\chi_{\mathfrak{g}}\,,\chi_{\mathfrak{g}}\,]\\!]}_{*}\,\big{)}\;=\;0\;\;,\;\quad\operatorname{ad}_{\psi}\\!\big{(}\,\chi_{\mathfrak{g}}+{(\chi_{\mathfrak{g}})}_{\\!{}_{2,1}}\big{)}\,=\,0\quad\qquad\forall\;\;\psi\in(\mathfrak{g}^{\raise-1.0pt\hbox{$\scriptscriptstyle\mathcal{R}$}}_{\scriptscriptstyle
P})^{*}$

Since $\,\chi\,$ is antisymmetric, we have that
$\,\chi_{\mathfrak{g}}+{(\chi_{\mathfrak{g}})}_{\\!{}_{2,1}}=0\,$, hence the
second condition is trivially satisfied. On the other hand, the first equality
follows from the fact that actually one has
$\;\partial_{*}(\chi_{\mathfrak{g}})=0\;$ and
$\;{[\\![\,\chi_{\mathfrak{g}}\,,\chi_{\mathfrak{g}}\,]\\!]}_{*}=0\;$. To see
it, let us describe $\,\partial_{*}(\chi_{\mathfrak{g}})\,$,
$\,{[\\![\,\chi_{\mathfrak{g}}\,,\chi_{\mathfrak{g}}\,]\\!]}_{*}\in{\big{(}\mathfrak{g}^{\raise-1.0pt\hbox{$\scriptscriptstyle\mathcal{R}$}}_{\scriptscriptstyle
P}\otimes\mathfrak{g}^{\raise-1.0pt\hbox{$\scriptscriptstyle\mathcal{R}$}}_{\scriptscriptstyle
P}\otimes\mathfrak{g}^{\raise-1.0pt\hbox{$\scriptscriptstyle\mathcal{R}$}}_{\scriptscriptstyle
P})}^{*}\,$ explicitly. For
$\,x,y,z\in\mathfrak{g}^{\raise-1.0pt\hbox{$\scriptscriptstyle\mathcal{R}$}}_{\scriptscriptstyle
P}\,$ we have

$\displaystyle\partial_{*}(\chi_{\mathfrak{g}})(x,y,z)$
$\displaystyle\;=\;\big{(}(\operatorname{id}\otimes\delta_{*})(\chi_{g})+\text{c.p.}\big{)}(x,y,z)\;=\;\chi_{\mathfrak{g}}\big{(}x\otimes[y,z]\big{)}+\text{c.p.}\;=$
$\displaystyle\;=\;\chi_{\mathfrak{g}}\big{(}x,[y,z]\big{)}+\chi_{\mathfrak{g}}\big{(}z,[x,y]\big{)}+\chi_{\mathfrak{g}}\big{(}y,[z,x]\big{)}$
$\displaystyle{[\\![\,\chi_{\mathfrak{g}}\,,\chi_{\mathfrak{g}}\,]\\!]}_{*}(x,y,z)$
$\displaystyle\,=\,\chi_{\mathfrak{g}}(x_{[1]},y)\chi_{\mathfrak{g}}(x_{[2]},z)+\chi_{\mathfrak{g}}(y_{[1]},z)\chi_{\mathfrak{g}}(y_{[2]},x)+\chi_{\mathfrak{g}}(z_{[1]},x)\chi_{\mathfrak{g}}(z_{[2]},y)$

Using (3.6), for
$\,g\in\mathfrak{g}^{\raise-1.0pt\hbox{$\scriptscriptstyle\mathcal{R}$}}_{\scriptscriptstyle
P}\,$ write $\,g=g_{+}+g_{0}+g_{-}\,$ with $\,g_{\pm}\in\mathfrak{n}_{\pm}\,$
and $\,g_{0}\in\mathfrak{h}\,$. As $\chi_{g}$ is defined through the map
$\pi^{\mathfrak{g}^{\raise-1.0pt\hbox{$\scriptscriptstyle\mathcal{R}$}}_{\scriptscriptstyle
P}}_{\mathfrak{h}}\,$, the map $\partial_{*}(\chi_{\mathfrak{g}})$ vanishes
when evaluated at elements in $\mathfrak{n}_{+}$ or $\mathfrak{n}_{-}\,$.
Moreover, since the bracket on $\mathfrak{h}$ is trivial and
$\,[\,\mathfrak{h},\mathfrak{n}_{\pm}\,]\subseteq\mathfrak{n}_{\pm}\,$, we get

$\partial_{*}(\chi_{\mathfrak{g}})(x,y,z)\;=\;\chi\big{(}x_{0}\,,\,[\,y_{+},z_{-}]+[\,y_{-},z_{+}]\big{)}\,+\,\text{c.p.}$

But $\,[\mathfrak{n}_{+},\mathfrak{n}_{-}]\,$ is contained in the Lie
subalgebra spanned by the $S_{i}$’s ($\,i\in I\,$), so we eventually get
$\;\partial_{*}(\chi_{\mathfrak{g}})(x,y,z)=0\;$ — for all
$\,x,y,z\in\mathfrak{g}^{\raise-1.0pt\hbox{$\scriptscriptstyle\mathcal{R}$}}_{\scriptscriptstyle
P}\,$ — by condition (3.8).

To prove that
$\,{[\\![\,\chi_{\mathfrak{g}}\,,\chi_{\mathfrak{g}}\,]\\!]}_{*}(x,y,z)=0\,$,
we use the Lie coalgebra structure. From (3.1) it follows that
$\,\delta(\mathfrak{n}_{\pm})\subseteq\mathfrak{b}^{\scriptscriptstyle
P}_{\pm}\otimes\mathfrak{n}_{\pm}\,$. Since $\,\delta(\mathfrak{h})=0\,$ by
definition and $\,\partial_{*}(\chi_{\mathfrak{g}})\,$ vanishes on
$\mathfrak{n}_{\pm}\,$, we get that each summand of
$\,{[\\![\,\chi_{\mathfrak{g}}\,,\chi_{\mathfrak{g}}\,]\\!]}_{*}(x,y,z)\,$ is
zero, q.e.d. ∎

The second result is the dual analog of Theorem 3.3.3 and of Theorem 3.3.4.

We start with some preliminaries. Let $\,P\in M_{t}(\Bbbk)\,$ be a
multiparameter matrix of Cartan type with associated Cartan matrix $A\,$, let
${\mathcal{R}}$ be a realization of it, and let
$\mathfrak{g}^{\raise-1.0pt\hbox{$\scriptscriptstyle\mathcal{R}$}}_{\scriptscriptstyle
P}$ be the associated multiparameter Lie bialgebra; then, given any
$\,\chi\in\text{\sl Alt}_{\,{\Bbbk[[\hbar]]}}^{\,S}(\mathfrak{h})\,$ as in
(2.12), let
$\,\chi_{\mathfrak{g}}:\mathfrak{g}^{\raise-1.0pt\hbox{$\scriptscriptstyle\mathcal{R}$}}_{\scriptscriptstyle
P}\\!\otimes\mathfrak{g}^{\raise-1.0pt\hbox{$\scriptscriptstyle\mathcal{R}$}}_{\scriptscriptstyle
P}\longrightarrow\Bbbk\,$ be the $2$–cocycle map for
$\mathfrak{g}^{\raise-1.0pt\hbox{$\scriptscriptstyle\mathcal{R}$}}_{\scriptscriptstyle
P}$ as in Lemma 3.4.2.

Consider the antisymmetric matrix
$\,\mathring{X}:={\Big{(}\,\mathring{\chi}_{i{}j}=\chi\big{(}\,T_{i}^{+}\,,T_{j}^{+}\big{)}\Big{)}}_{i,j\in
I}\\!\in\mathfrak{so}_{n}(\Bbbk)\,$. By Proposition 2.3.2 we have a matrix
$P_{(\chi)}$ and a realization ${\mathcal{R}}_{(\chi)}$ of it, given by

$P_{(\chi)}:=\,P\,+\,\mathring{X}\,=\,{\Big{(}\,p^{(\chi)}_{i{}j}:=\,p_{ij}+\mathring{\chi}_{i{}j}\Big{)}}_{\\!i,j\in
I}\;\;,\quad\Pi_{(\chi)}:=\,{\Big{\\{}\,\alpha_{i}^{(\chi)}:=\,\alpha_{i}\pm\chi\big{(}\,\text{--}\,,T_{i}^{\pm}\big{)}\Big{\\}}}_{i\in
I}$

In particular, if $P$ is of Cartan type, then so is $P_{(\chi)}$ as well, and
they are associated with the same Cartan matrix.

######  Theorem 3.4.3.

Keep notation as above. (a) There exists a Lie bialgebra isomorphism
$\;f^{(\chi)}_{\scriptscriptstyle
P}:\mathfrak{g}_{P_{(\chi)}}^{{\mathcal{R}}_{(\chi)}}{\buildrel\cong\over{\lhook\joinrel\relbar\joinrel\relbar\joinrel\relbar\joinrel\twoheadrightarrow}}{\big{(}\mathfrak{g}^{\raise-1.0pt\hbox{$\scriptscriptstyle\mathcal{R}$}}_{\scriptscriptstyle
P}\big{)}}_{\chi_{\mathfrak{g}}}\;$ — where
$\,{\big{(}\mathfrak{g}^{\raise-1.0pt\hbox{$\scriptscriptstyle\mathcal{R}$}}_{\scriptscriptstyle
P}\big{)}}_{\chi_{\mathfrak{g}}}=\big{(}\,\mathfrak{g}^{\raise-1.0pt\hbox{$\scriptscriptstyle\mathcal{R}$}}_{\scriptscriptstyle
P}\,;\,{[\,\ ,\ ]}_{\chi_{\mathfrak{g}}}\,,\,\delta\,\big{)}\,$ is the
2–cocycle deformation of
$\,\mathfrak{g}^{\raise-1.0pt\hbox{$\scriptscriptstyle\mathcal{R}$}}_{\scriptscriptstyle
P}$ as in Definition 3.1.4. In particular, the class of all MpLbA’s of any
fixed Cartan type and of fixed rank is stable by toral 2–cocycle deformations.
Moreover, inside it the subclass of all such MpLbA’s associated with split,
resp. minimal, realizations is stable as well. (b) Let $P$ and $P^{\prime}$ be
two matrices of Cartan type with the same associated Cartan matrix $A\,$, i.e.
such that $\,P_{s}=P^{\prime}_{s}\,$. Then the following holds: (b.1) Let
$\,{\mathcal{R}}$ be a split realization of $P$ and let
$\mathfrak{g}_{\scriptscriptstyle P}^{\scriptscriptstyle{\mathcal{R}}}$ be the
associated MpLbA. Then there is $\,\chi\in\text{\sl
Alt}_{\,\Bbbk}^{\,S}(\mathfrak{h})\,$ such that $\,P^{\prime}=P_{(\chi)}\,$,
the corresponding ${\mathcal{R}}_{(\chi)}$ is a split realization of
$\,P^{\prime}=P_{(\chi)}\,$, and for the 2–cocycle $\chi_{\mathfrak{g}}$ as in
Definition 3.4.1 we have

$\mathfrak{g}_{\scriptscriptstyle
P^{\prime}}^{\scriptscriptstyle{\mathcal{R}}_{(\chi)}}\;\cong\;{\big{(}\,\mathfrak{g}_{\scriptscriptstyle
P}^{\scriptscriptstyle{\mathcal{R}}}\big{)}}_{\chi_{{}_{\mathfrak{g}}}}$

In a nutshell, if $\,P^{\prime}_{s}=P_{s}\,$ then from any split MpLbA over
$P$ we can obtain by toral 2–cocycle deformation a split MpLbA (of the same
rank) over $P^{\prime}\,$. (b.2) Let ${\mathcal{R}}$ and
${\mathcal{R}}^{\prime}$ be split minimal realizations of $P$ and $P^{\prime}$
respectively, and let
$\mathfrak{g}^{\raise-1.0pt\hbox{$\scriptscriptstyle\mathcal{R}$}}_{\scriptscriptstyle
P}$ and $\mathfrak{g}_{\scriptscriptstyle
P^{\prime}}^{\scriptscriptstyle{\mathcal{R}}^{\prime}}$ be the associated
MpLbA’s. Then there exists $\,\chi\in\text{\sl
Alt}_{\,\Bbbk}^{\,S}(\mathfrak{h})\,$ such that for the 2–cocycle
$\chi_{\mathfrak{g}}$ as in Definition 3.4.1 we have

$\mathfrak{g}_{\scriptscriptstyle
P^{\prime}}^{\scriptscriptstyle{\mathcal{R}}^{\prime}}\;\cong\;{\big{(}\,\mathfrak{g}^{\raise-1.0pt\hbox{$\scriptscriptstyle\mathcal{R}$}}_{\scriptscriptstyle
P}\big{)}}_{\chi_{\mathfrak{g}}}$

In a nutshell, if $\,P^{\prime}_{s}=P_{s}\,$ then any split minimal MpLbA over
$P^{\prime}$ is isomorphic to a toral 2–cocycle deformation of any split
minimal MpLbA over $P$. (b.3) Every split minimal MpLbA is isomorphic to some
toral 2–cocycle deformation of the Manin double
$\,\mathfrak{g}^{\scriptscriptstyle D}_{\scriptscriptstyle
DA}:=\mathfrak{b}_{+}\oplus\mathfrak{b}_{-}\,$ associated with $\,DA\,$
endowed with the canonical Lie bialgebra structure given in §3.2.3.

###### Proof.

(a) By §3.1.2, the Lie coalgebra structure in
${\big{(}\mathfrak{g}^{\raise-1.0pt\hbox{$\scriptscriptstyle\mathcal{R}$}}_{\scriptscriptstyle
P}\big{)}}_{\chi_{\mathfrak{g}}}$ is the same as in
$\mathfrak{g}^{\raise-1.0pt\hbox{$\scriptscriptstyle\mathcal{R}$}}_{\scriptscriptstyle
P}\,$, and the latter coincides with the one in
$\mathfrak{g}_{P_{(\chi)}}^{{\mathcal{R}}_{(\chi)}}$. In particular, we have a
Lie coalgebra isomorphism among them. With respect to the Lie algebra
structure, we know that the Lie bracket in
${\big{(}\mathfrak{g}^{\raise-1.0pt\hbox{$\scriptscriptstyle\mathcal{R}$}}_{\scriptscriptstyle
P}\big{)}}_{\chi_{\mathfrak{g}}}$ is a deformation of that of
$\mathfrak{g}^{\raise-1.0pt\hbox{$\scriptscriptstyle\mathcal{R}$}}_{\scriptscriptstyle
P}$ according to (3.5). Let us see that the modified defining relations in
${\big{(}\mathfrak{g}^{\raise-1.0pt\hbox{$\scriptscriptstyle\mathcal{R}$}}_{\scriptscriptstyle
P}\big{)}}_{\chi_{\mathfrak{g}}}$ coincide with the ones in
$\mathfrak{g}_{P_{(\chi)}}^{{\mathcal{R}}_{(\chi)}}\,$, cf. §3.2.3; this will
imply that both objects are isomorphic as Lie bialgebras.

As the Lie cobracket on the Cartan subalgebra $\mathfrak{h}$ of
$\mathfrak{g}_{P_{(\chi)}}^{{\mathcal{R}}_{(\chi)}}$ is trivial, we get that

${[T^{\prime},T^{\prime\prime}]}_{\chi_{{}_{\mathfrak{g}}}}\,=\;[T^{\prime},T^{\prime\prime}]\,-\,T^{\prime}_{[1]}\,\chi_{\mathfrak{g}}\big{(}T^{\prime}_{[2]},T^{\prime\prime}\big{)}\,+\,T^{\prime\prime}_{[1]}\,\chi_{\mathfrak{g}}\big{(}T^{\prime\prime}_{[2]},T^{\prime}\big{)}\,=\,[T^{\prime},T^{\prime\prime}]\,=\,0$

for all $\,T^{\prime},T^{\prime\prime}\in\mathfrak{h}\,$. Take now $\,i\in
I\,$: since $\,\delta(T)=0\,$, $\,\delta(E_{i})=2\,T_{i}^{+}\\!\wedge E_{i}\,$
and $\,\delta(F_{i})=2\,T_{i}^{-}\\!\wedge F_{i}\,$, we get that

$\displaystyle{[T,E_{i}]}_{\chi_{\mathfrak{g}}}$
$\displaystyle\,=\,[T,E_{i}]-E_{i}\,\chi_{\mathfrak{g}}\big{(}T_{i}^{+},T\big{)}\,=\,\alpha_{i}(T)E_{i}+\chi\big{(}T,T_{i}^{+}\big{)}E_{i}\,=\,\alpha_{i}^{(\chi)}(T)\,E_{i}$
$\displaystyle{[T,F_{i}]}_{\chi_{\mathfrak{g}}}$
$\displaystyle\,=\,[T,F_{i}]-F_{i}\,\chi_{\mathfrak{g}}\big{(}T_{i}^{-},T\big{)}\,=\,-\alpha_{i}(T)F_{i}+\chi_{\mathfrak{g}}\big{(}T,T_{i}^{-}\big{)}\,F_{i}\,=$
$\displaystyle=\,-\Big{(}\alpha_{i}(T)F_{i}-\chi\big{(}T,T_{i}^{-}\big{)}\Big{)}\,F_{i}\,=\,-\alpha_{i}^{(\chi)}(T)\,F_{i}$
$\displaystyle{[E_{i},F_{j}]}_{\chi_{\mathfrak{g}}}$
$\displaystyle\,=\,[E_{i},F_{j}]+E_{i}\,\chi_{\mathfrak{g}}\big{(}T_{i}^{+},F_{j}\big{)}\,-\,F_{j}\,\chi_{\mathfrak{g}}\big{(}T_{j}^{-},E_{i}\big{)}\,=\,[E_{i},F_{j}]\,=\,\delta_{ij}\,{{\;T_{i}^{+}\\!+T_{i}^{-}\;}\over{\;2\,d_{i}\;}}$

Note that only the relations involving the roots are changed by the cocycle.
Now, with respect to the Serre relations, as the Lie subalgebras
$\mathfrak{n}_{\pm}$ are contained in the right and left radical of
$\chi_{\mathfrak{g}}\,$, analogous calculations as above yield that

$\displaystyle{\big{(}\textsl{ad}_{\chi_{\mathfrak{g}}}\,(E_{i})\big{)}}^{1-a_{ij}}(E_{j})$
$\displaystyle\,=\,{\big{(}\textsl{ad}\,(E_{i})\big{)}}^{1-a_{ij}}(E_{j})\,=\,0$
$\displaystyle{\big{(}\textsl{ad}_{\chi_{\mathfrak{g}}}\,(F_{i})\big{)}}^{1-a_{ij}}(F_{j})$
$\displaystyle\,=\,{\big{(}\textsl{ad}\,(F_{i})\big{)}}^{1-a_{ij}}(F_{j})\,=\,0$

Here $\textsl{ad}_{\chi_{\mathfrak{g}}}$ denotes the adjoint action with
respect to the deformed bracket $\,{[-,-]}_{\chi_{\mathfrak{g}}}\,$. (b.1) By
claim (a), it is enough to find an antisymmetric $\Bbbk$–linear
$\,\chi:\mathfrak{h}\otimes\mathfrak{h}\longrightarrow\Bbbk\,$ obeying (3.8)
such that $\,P^{\prime}=P_{(\chi)}\,$; this is guaranteed by Proposition
2.3.5. (b.2) This follows from claim (b.1), along with the uniqueness of split
realizations — cf. Proposition 2.1.5(b). (b.3) This follows as an application
of claim (b.2), taking as $\mathfrak{g}_{\scriptscriptstyle
P^{\prime}}^{\scriptscriptstyle{\mathcal{R}}^{\prime}}$ the given split
minimal MpLbA and for
$\mathfrak{g}^{\raise-1.0pt\hbox{$\scriptscriptstyle\mathcal{R}$}}_{\scriptscriptstyle
P}$ the “standard” MpLbA $\mathfrak{g}^{\scriptscriptstyle
D}_{\scriptscriptstyle P}$ over $\,P:=DA=P^{\prime}_{s}\,$ (cf. §3.2.3), which
by definition is straight and split minimal. ∎

##  4\. Formal multiparameter QUEAs (=FoMpQUEAs)

This section is devoted to introduce formal multiparameter quantized universal
enveloping algebras (in short, FoMpQUEAs) and to study their deformations.

###  4.1. The Hopf algebra setup

Our main references for the theory of Hopf algebras are [Mo] and [Ra]. For
topological or $\hbar$–adically complete Hopf algebras see for instance [Ks],
[CP], [KS].

######  4.1.1.

Hopf notation. Let us fix our notation for Hopf algebra theory (mainly
standard, indeed). The comultiplication is denoted $\Delta$, the counit
$\epsilon\,$ and the antipode $\SS\,$; for the first, we use the Heyneman-
Sweedler notation, namely $\,\Delta(x)=x_{(1)}\otimes x_{(2)}\,$.

Hereafter by $\Bbbk$ we denote the ground ring of our algebras, coalgebras,
etc. In any coalgebra $C\,$, the set of group-like elements is denoted by
$G(C)\,$; also, we denote by $\,C^{+}:=\operatorname{Ker}(\epsilon)\,$ the
augmentation ideal. If $\,g,h\in G(C)\,$, the set of $(g,h)$–primitive
elements is defined to be $\;P_{g,h}(C)\,:=\,\big{\\{}\,x\in
C\,|\,\Delta(x)=x\otimes g+h\otimes x\,\big{\\}}\;$. In case $C$ is a
bialgebra, we write $\operatorname{Prim}(C)=P_{1,1}(C)$ for the space of
primitive elements.

For a Hopf algebra $H$ (or just bialgebra), we write $H^{\text{op}}\,$, resp.
$H^{\text{cop}}\,$, for the Hopf algebra (or bialgebra) given by taking in $H$
the opposite product, resp. coproduct.

Given a Hopf algebra map $\,\pi:H\longrightarrow K\,$, then $H$ is a left and
right $K$–comodule, with structure maps
$\;\lambda:=(\pi\otimes\operatorname{id})\Delta:H\longrightarrow K\otimes
H\;$, $\;\rho:=(\operatorname{id}\otimes\pi)\Delta:H\longrightarrow H\otimes
K\;$. The space of left and right coinvariants then is defined, respectively,
by

$\displaystyle{}^{\operatorname{co}\\!K}\\!H$
$\displaystyle\;:=\;{}^{\operatorname{co}\pi}H\,=\,\big{\\{}\,h\in
H\,\big{|}\,(\pi\otimes\operatorname{id})\big{(}\Delta(h)\big{)}=1\otimes
h\,\big{\\}}$ $\displaystyle H^{\,\operatorname{co}\\!K}$
$\displaystyle\;:=\;H^{\operatorname{co}\pi}\,=\,\big{\\{}\,h\in
H\,\big{|}\,(\operatorname{id}\otimes\pi)\big{(}\Delta(h)\big{)}=h\otimes
1\,\big{\\}}$

We recall the (essentially standard) notion of skew-Hopf pairing between two
Hopf algebras and the construction of the Drinfeld’s double.

######  Definition 4.1.2.

[AY, §2.1] Given two Hopf algebras $H$ and $K$ with bijective antipode over
the ring $\Bbbk\,$, a $\Bbbk$–linear map
$\,\eta:H\otimes_{\Bbbk}K\longrightarrow\Bbbk\,$ is called a skew-Hopf pairing
(between $H$ and $K\,$) if, for all $\,h\in H\,$, $\,k\in K\,$, one has

$\displaystyle\eta\big{(}h\,,\,k^{\prime}\,k^{\prime\prime}\big{)}\,=\;\eta\big{(}h_{(1)}\,,\,k^{\prime}\big{)}\,\eta\big{(}h_{(2)}\,,\,k^{\prime\prime}\big{)}$
(4.1)
$\displaystyle\eta\big{(}h^{\prime}\,h^{\prime\prime}\,,\,k\big{)}\,=\;\eta\big{(}h^{\prime},\,k_{(2)}\big{)}\,\eta\big{(}h^{\prime\prime},\,k_{(1)}\big{)}$
(4.2)
$\displaystyle\eta\big{(}h\,,1\big{)}\,=\,\epsilon(h)\;\;,\qquad\eta\big{(}1\,,k\big{)}\,=\,\epsilon(k)$
(4.3) $\displaystyle\eta\big{(}\SS^{\pm
1}(h)\,,k\,\big{)}\,=\;\eta\big{(}h\,,\,\SS^{\mp 1}(k)\big{)}$ (4.4)

Note that the map $\eta$ turns out to be convolution invertible: its inverse
is given by $\;\eta^{-1}(h,k)=\eta(h,\SS(k))=\eta(\SS^{-1}(h),k)\;$ for all
$\,h\in H\,$ and $\,k\in K\,$.

In this setup, the Drinfeld double, or “quantum double”, $D(H,K,\eta)\,$ is
the quotient algebra $\;T(H\oplus K)\Big{/}\mathcal{I}\;$, where $\mathcal{I}$
is the (two-sided) ideal generated by the relations

$1_{H}=\,1\,=\,1_{K}\,\;,\;\quad a\otimes b\,=\,a\,b\,\;,\;\quad
x_{(1)}\otimes y_{(1)}\ \eta(y_{(2)},x_{(2)})\,=\,\eta(y_{(1)},x_{(1)})\
y_{(2)}\otimes x_{(2)}$

for all $\,a\,,b\in H\,$ or $\,a\,,b\in K\,$ and $\,x\in K\,,y\in H\,$. This
is also endowed with a standard Hopf algebra structure, for which $H$ and $K$
are Hopf $\Bbbk$–subalgebras. $\diamondsuit$

######  4.1.3.

Topological issues. We will often deal with topological Hopf algebras, namely
Hopf algebras over the ring ${\Bbbk[[\hbar]]}$ of formal power series over a
field $\Bbbk$ in a formal variable $\hbar\,$. This ring carries a natural
topology, called the $\hbar$–adic topology, coming from the so-called
$\hbar$–adic norm with respect to which it is complete, namely

$\big{\|}\,a_{n}\hbar^{n}+a_{n+1}\hbar^{n+1}+\cdots\big{\|}\;:=\;C^{-n}\qquad\big{(}\,a_{n}\not=0\,\big{)}$

where $\,C>1\,$ is any fixed constant in $\mathbb{R}\,$. In this sense, we
shall consider topological ${\Bbbk[[\hbar]]}$–modules and the completed tensor
products among them, which we denote by
$\,\widehat{\otimes}_{\Bbbk[[\hbar]]}\,$ or simply by $\,\widehat{\otimes}\,$.
For any $\Bbbk$–vector space $V\,$, set

$V[[\hbar]]\;:=\;\Big{\\{}\,{\textstyle\sum_{n\geq
0}}v_{n}\hbar^{n}\,\Big{|}\,v_{n}\in V\,,\;\forall\,n\geq 0\,\Big{\\}}$

then $V[[\hbar]]$ is a complete ${\Bbbk[[\hbar]]}$–module. We call a
topological ${\Bbbk[[\hbar]]}$–module topologically free if it is isomorphic
to $V[[\hbar]]$ for some $\Bbbk$–vector space $V\,$. For two topologically
free modules $V[[\hbar]]$ and $W[[\hbar]]$ one has that
$\;V[[\hbar]]\,\widehat{\otimes}\,W[[\hbar]]\,\cong\,\big{(}V\otimes
W\big{)}[[\hbar]]\,$, see [Ks, Proposition XVI.3.2]. All completed tensor
products between ${\Bbbk[[\hbar]]}$–modules then will be denoted simply by
$\otimes\,$, unless we intend to stress the topological aspect. In particular,
we make no distinction on the notation between Hopf algebras and topological
Hopf algebras; we assume it is well-understood from the context.

######  4.1.4.

Hopf algebra deformations. There exist two standard methods to deform Hopf
algebras, leading to so-called “$2$–cocycle deformations” and to “twist
deformations”: hereafter we recall both procedures, adapting them to the setup
of topological Hopf algebras, then later on we apply them to formal quantum
groups.

$\underline{\text{Twist deformations}}$: Let $H$ be a Hopf algebra (over a
commutative ring), and let $\,{\mathcal{F}}\in H\otimes H\,$ be an invertible
element in $H^{\otimes 2}$ (later called a “twist”) such that

${\mathcal{F}}_{1{}2}\,\big{(}\Delta\otimes\text{id}\big{)}({\mathcal{F}})\,=\,{\mathcal{F}}_{2{}3}\,\big{(}\text{id}\otimes\Delta\big{)}({\mathcal{F}})\quad,\qquad\big{(}\epsilon\otimes\text{id}\big{)}({\mathcal{F}})=1=\big{(}\text{id}\otimes\epsilon\big{)}({\mathcal{F}})$

Then $H$ bears a second Hopf algebra structure, denoted $H^{\mathcal{F}}$ and
called twist deformation of the old one, with the old product, unit and
counit, but with new “twisted” coproduct $\Delta^{\mathcal{F}}$ and antipode
$\SS^{\mathcal{F}}$ given by

$\qquad\Delta^{\\!{\mathcal{F}}}(x)\,:=\,{\mathcal{F}}\,\Delta(x)\,{\mathcal{F}}^{-1}\quad,\qquad\SS^{\mathcal{F}}(x)\,:=\,v\,\SS(x)\,v^{-1}\quad\qquad\forall\;\;x\in
H$

where $\,v:=\sum_{\mathcal{F}}\SS(f^{\prime}_{1})\,f^{\prime}_{2}\,$ — with
$\,\sum_{\mathcal{F}}f^{\prime}_{1}\otimes
f^{\prime}_{2}={\mathcal{F}}^{-1}\,$ — is invertible in $H$ (see, [CP,
§4.2.E], for further details). When $H$ is in fact a topological Hopf algebra
— meaning that, in particular, its coproduct $\Delta$ takes values into
$\,H\otimes H\,$ where now “$\,\otimes\,$” stands for a suitable topological
tensor product — then the same notions still make sense, and the related
results apply again, up to properly reading them.

$\underline{\text{Cocycle deformations}}$: Let
$\,\big{(}H,m\,,1\,,\Delta\,,\epsilon\big{)}\,$ be a bialgebra over a ring
$\Bbbk\,$. A convolution invertible linear map $\sigma$ in
$\,\operatorname{Hom}_{\Bbbk}(H\otimes H,\Bbbk\,)\,$ is called a (normalized)
Hopf 2-cocycle (or just a “$2$–cocycle” if no confusion arises) if

$\sigma(b_{(1)},c_{(1)})\,\sigma(a,b_{(2)}c_{(2)})\;=\;\sigma(a_{(1)},b_{(1)})\,\sigma(a_{(2)}b_{(2)},c)$

and $\,\sigma(a,1)=\epsilon(a)=\sigma(1,a)\,$ for all $\,a,b,c\in H\,$, see
[Mo, Sec. 7.1].

Using a $2$–cocycle $\sigma$ it is possible to define a new algebra structure
on $H$ by deforming the multiplication. Indeed, define
$\,m_{\sigma}=\sigma*m*\sigma^{-1}:H\otimes H\longrightarrow H\,$ by

$m_{\sigma}(a,b)\,=\,a\cdot_{\sigma}b\,=\,\sigma(a_{(1)},b_{(1)})\,a_{(2)}\,b_{(2)}\,\sigma^{-1}(a_{(3)},b_{(3)})$
$None$

If in addition $H$ is a Hopf algebra with antipode $\SS\,$, then define also
$\,\SS_{\sigma}:H\longrightarrow H\,$ as $\,\SS_{\sigma}:H\longrightarrow H\,$
where

$\SS_{\sigma}(a)\,=\,\sigma(a_{(1)},\SS(a_{(2)}))\,\SS(a_{(3)})\,\sigma^{-1}(\SS(a_{(4)}),a_{(5)})$
$None$

It is known that $\,\big{(}H,m_{\sigma},1,\Delta,\epsilon\big{)}\,$ is in turn
a bialgebra, and
$\,\big{(}H,m_{\sigma},1,\Delta,\epsilon,\SS_{\sigma}\big{)}\,$ is a Hopf
algebra: we shall call such a new structure on $H$ a cocycle deformation of
the old one, and we shall graphically denote it by $H_{\sigma}\,$; see [Doi]
for more details.

######  4.1.5.

Deformations and duality. The two notions of “$2$–cocycle” and of “twist” are
so devised as to be dual to each other with respect to Hopf duality. The proof
of the following result (an exercise in Hopf theory) is left to the reader:

######  Proposition 4.1.6.

Let $H$ be a Hopf algebra over a field, and $H^{*}$ its linear dual. (a) Let
${\mathcal{F}}$ be a twist for $H\,$, and $\sigma_{{}_{\mathcal{F}}}$ the
image of ${\mathcal{F}}$ in ${(H\otimes H)}^{*}$ for the natural composed
embedding $\,H\otimes H\lhook\joinrel\relbar\joinrel\longrightarrow
H^{**}\otimes
H^{**}\lhook\joinrel\relbar\joinrel\longrightarrow{\big{(}H^{*}\otimes
H^{*}\big{)}}^{*}\,$. Then $\sigma_{{}_{\mathcal{F}}}$ is a 2–cocycle for
$H^{*}\,$, and there exists a canonical isomorphism
$\,{\big{(}H^{*}\big{)}}_{\sigma_{{}_{\mathcal{F}}}}\cong{\big{(}H^{\mathcal{F}}\,\big{)}}^{*}\,$.
(b) Let $\sigma$ be a 2–cocycle for $H\,$; assume that $H$ is finite-
dimensional, and let ${\mathcal{F}}_{\sigma}$ be the image of $\sigma$ in the
natural identification $\,{(H\otimes H)}^{*}=H^{*}\otimes H^{*}\,$. Then
${\mathcal{F}}_{\sigma}$ is a twist for $H^{*}\,$, and there exists a
canonical isomorphism
$\,{\big{(}H^{*}\big{)}}^{{\mathcal{F}}_{\sigma}}\cong{\big{(}H_{\sigma}\big{)}}^{*}\,$.
∎

######  4.1.7.

Some $q$–numbers. Let ${\mathbb{Z}\big{[}\hskip 1.0ptq,q^{-1}\big{]}}$ be the
ring of Laurent polynomials with integral coefficients in the indeterminate
$q\,$. For every $\,n\in\mathbb{N}_{+}\,$ we define

$\halign{\hbox to\displaywidth{$\hfil\displaystyle#\hfil$}\cr
0.0pt{\hfil$\displaystyle{(0)}_{q}\,:=\;1\;\;,\quad{(n)}_{q}\,:=\;\frac{\,q^{n}-1\,}{\,q-1\,}\;=\;1+q+\cdots+q^{n-1}\;=\;{\textstyle\sum\limits_{s=0}^{n-1}}\,q^{s}\qquad\big{(}\,\in\,\mathbb{Z}[q]\,\big{)}\cr
0.0pt{\hfil$\displaystyle{(n)}_{q}!\,:=\;{(0)}_{q}{(1)}_{q}\cdots{(n)}_{q}:=\,{\textstyle\prod\limits_{s=0}^{n}}{(s)}_{q}\;\;,\quad{\binom{n}{k}}_{\\!q}\,:=\;\frac{\,{(n)}_{q}!\,}{\;{(k)}_{q}!{(n-k)}_{q}!\,}\qquad\big{(}\,\in\,\mathbb{Z}[q]\,\big{)}\cr
0.0pt{\hfil$\displaystyle{[0]}_{q}:=\,1\;,\;\;{[n]}_{q}:=\,\frac{\,q^{n}-q^{-n}\,}{\,q-q^{-1}\,}=\,q^{-(n-1)}+\cdots+q^{n-1}={\textstyle\sum\limits_{s=0}^{n-1}}\,q^{2\,s-n+1}\\!\quad\big{(}\in{\mathbb{Z}\big{[}\hskip
1.0ptq,q^{-1}\big{]}}\,\big{)}\cr
0.0pt{\hfil$\displaystyle{[n]}_{q}!\,:=\;{[0]}_{q}{[1]}_{q}\cdots{[n]}_{q}=\,{\textstyle\prod\limits_{s=0}^{n}}{[s]}_{q}\;\;,\quad{\bigg{[}\,{n\atop
k}\,\bigg{]}}_{q}\,:=\;\frac{\,{[n]}_{q}!\,}{\;{[k]}_{q}!{[n-k]}_{q}!\,}\qquad\big{(}\,\in\,{\mathbb{Z}\big{[}\hskip
1.0ptq,q^{-1}\big{]}}\,\big{)}\crcr}}}}}$

In particular, we have the identities

${(n)}_{q^{2}}=q^{n-1}{[n]}_{q}\;\;,\qquad{(n)}_{q^{2}}!=q^{\frac{n(n-1)}{2}}{[n]}_{q}\;\;,\qquad{\binom{n}{k}}_{\\!\\!q^{2}}=q^{k(n-k)}{\bigg{[}\,{n\atop
k}\,\bigg{]}}_{q}\;\;.$

Moreover, for any field $\mathbb{F}$ we can think of Laurent polynomials as
functions on $\mathbb{F}^{\times}$, hence for any
$\,q\in\mathbb{F}^{\times}\,$ we shall read every symbol above as a suitable
element in $\,\mathbb{F}\,$.

###  4.2. Formal multiparameter QUEAs

We introduce now the notion of formal multiparameter quantum universal
enveloping algebra — or just “FoMpQUEA”, in short.

Hereafter, $\Bbbk$ is a field of characteristic zero, ${\Bbbk[[\hbar]]}$ the
ring of formal power series in $\hbar$ with coefficients in $\Bbbk\,$. In any
topological, ${\Bbbk[[\hbar]]}$–adically complete ${\Bbbk[[\hbar]]}$–algebra
$\mathcal{A}\,$, if $\,X\in\mathcal{A}\,$ we use the standard notation
$\;e^{\hbar\,X}:=\exp(\hbar\,X)\,=\,{\textstyle\sum\limits_{n=0}^{+\infty}}\,\hbar^{n}X^{n}\big{/}n!\;\in\,\mathcal{A}\;$.

######  Definition 4.2.1.

Let $\,A:={\big{(}\,a_{i,j}\big{)}}_{i,j\in I}\,$ be some fixed generalized
symmetrizable Cartan matrix, and let $\,P:={\big{(}\,p_{i,j}\big{)}}_{i,j\in
I}\in M_{n}\big{(}{\Bbbk[[\hbar]]}\big{)}\,$ be a matrix of Cartan type
associated with $A$ as in the sense of Definition 2.1.2(d), that is
$\,P+P^{\,\scriptscriptstyle T}=2\,D\,A\,$, i.e.
$\,p_{ij}+p_{ji}=2\,d_{i}\,a_{ij}\,$ for all $\,i,j\in I\,$, which implies
$\,p_{ii}=2\,d_{i}\not=0\,$ for all $\,i\in I\,$.

We define in ${\Bbbk[[\hbar]]}$ the following elements:
$\,q:=e^{\hbar}=\exp(\hbar)\in{\Bbbk[[\hbar]]}\,$,
$\,q_{i}:=e^{\hbar\,d_{i}}\,\big{(}\\!=q^{d_{i}}\big{)}\,$,
$\,q_{ij}:=e^{\hbar\,p_{ij}}\,\big{(}\\!=q^{p_{ij}}\big{)}\;$ for all
$\,i,j\in I\,$, and also $\;q_{ij}^{1/2}:=e^{\hbar\,p_{ij}/2}\;$ for all
$\,i,j\in I\,$. In particular we have
$\,q_{ii}^{1/2}=e^{\hbar{}d_{i}}=q_{i}\,$ and
$\;q_{ij}\,q_{ji}=q_{ii}^{\,a_{ij}}\;$ for all $\,i,j\in I\,$. $\diamondsuit$

We can now define our FoMpQUEAs, using notation as in Definition 4.2.1 above.

######  Definition 4.2.2.

Let $\,A:={\big{(}\,a_{i,j}\big{)}}_{i,j\in I}\,$ be a generalized
symmetrizable Cartan matrix, and $\,P:={\big{(}\,p_{i,j}\big{)}}_{i,j\in I}\in
M_{n}\big{(}{\Bbbk[[\hbar]]}\big{)}\,$ a matrix of Cartan type associated with
$A$. We fix a realization
$\,{\mathcal{R}}:=\big{(}\,\mathfrak{h}\,,\Pi\,,\Pi^{\vee}\,\big{)}\,$ of $P$
as in Definition 2.1.2. (a) We define the formal multiparameter quantum
universal enveloping algebra — in short formal MpQUEA, or simply FoMpQUEA —
with multiparameter $P$ and realization ${\mathcal{R}}$ as follows. It is the
unital, associative, topological, $\hbar$–adically complete
${\Bbbk[[\hbar]]}$–algebra $U^{\,{\mathcal{R}}}_{\\!P,\hskip
0.7pt\hbar}(\hskip 0.5pt\mathfrak{g})$ generated by the
${\Bbbk[[\hbar]]}$–submodule $\mathfrak{h}$ together with elements
$\,E_{i}\,$, $F_{i}\,$ (for all $\,i\in I\,$), with relations (for all
$\,T,T^{\prime},T^{\prime\prime}\in\mathfrak{h}\,$, $\,i\,,j\in I\,$)

$\displaystyle T\,E_{j}\,-\,E_{j}\,T\,=\,+\alpha_{j}(T)\,E_{j}\;\;,\qquad
T\,F_{j}\,-\,F_{j}\,T\;=\;-\alpha_{j}(T)\,F_{j}$ (4.5) $\displaystyle
T^{\prime}\,T^{\prime\prime}\;=\;T^{\prime\prime}\,T^{\prime}\;\;,\qquad
E_{i}\,F_{j}\,-\,F_{j}\,E_{i}\;=\;\delta_{i,j}\,{{\;e^{+\hbar\,T_{i}^{+}}-\,e^{-\hbar\,T_{i}^{-}}\;}\over{\;q_{i}^{+1}-\,q_{i}^{-1}\;}}$
$\displaystyle\sum\limits_{k=0}^{1-a_{ij}}(-1)^{k}{\left[{1-a_{ij}\atop
k}\right]}_{\\!q_{i}}q_{ij}^{+k/2\,}q_{ji}^{-k/2}\,E_{i}^{1-a_{ij}-k}E_{j}E_{i}^{k}\;=\;0\qquad(\,i\neq
j\,)$ $\displaystyle\sum\limits_{k=0}^{1-a_{ij}}(-1)^{k}{\left[{1-a_{ij}\atop
k}\right]}_{\\!q_{i}}q_{ij}^{+k/2\,}q_{ji}^{-k/2}\,F_{i}^{1-a_{ij}-k}F_{j}F_{i}^{k}\;=\;0\qquad(\,i\neq
j\,)$

We say that the FoMpQUEA $U^{\,{\mathcal{R}}}_{\\!P,\hskip 0.7pt\hbar}(\hskip
0.5pt\mathfrak{g})$ is straight, or small, or minimal, or split, if such is
${\mathcal{R}}\,$; also, we define the rank of
$U^{\,{\mathcal{R}}}_{\\!P,\hskip 0.7pt\hbar}(\hskip 0.5pt\mathfrak{g})$ as
$\;\operatorname{rk}\\!\big{(}U^{\,{\mathcal{R}}}_{\\!P,\hskip
0.7pt\hbar}(\hskip
0.5pt\mathfrak{g})\big{)}:=\operatorname{rk}({\mathcal{R}})=\operatorname{rk}_{\Bbbk[[\hbar]]}(\mathfrak{h})\,$.
(b) We define the Cartan subalgebra
$U_{P,\hbar}^{\,{\mathcal{R}}}(\mathfrak{h})\,$, or just
$U_{\hbar}(\mathfrak{h})\,$, of a FoMpQUEA $U^{\,{\mathcal{R}}}_{\\!P,\hskip
0.7pt\hbar}(\hskip 0.5pt\mathfrak{g})$ as being the unital, $\hbar$–adically
complete topological ${\Bbbk[[\hbar]]}$–subalgebra of
$U^{\,{\mathcal{R}}}_{\\!P,\hskip 0.7pt\hbar}(\hskip 0.5pt\mathfrak{g})$
generated by the ${\Bbbk[[\hbar]]}$–submodule $\mathfrak{h}\,$. (c) We define
the positive, resp. the negative, Borel subalgebra
$U^{\,{\mathcal{R}}}_{\\!P,\hskip 0.7pt\hbar}(\hskip 0.5pt\mathfrak{b}_{+})$,
resp. $U^{\,{\mathcal{R}}}_{\\!P,\hskip 0.7pt\hbar}(\hskip
0.5pt\mathfrak{b}_{-})$, of $\,U^{\,{\mathcal{R}}}_{\\!P,\hskip
0.7pt\hbar}(\hskip 0.5pt\mathfrak{g})$ to be the unital, $\hbar$–adically
complete topological ${\Bbbk[[\hbar]]}$–subalgebra of
$U^{\,{\mathcal{R}}}_{\\!P,\hskip 0.7pt\hbar}(\hskip 0.5pt\mathfrak{g})$
generated by $\mathfrak{h}$ and all the $E_{i}$’s, resp. by $\mathfrak{h}$ and
all the $F_{i}$’s ($\,i\in I\,$). (d) We define the positive, resp. negative,
nilpotent subalgebra $U^{\,{\mathcal{R}}}_{\\!P,\hskip 0.7pt\hbar}(\hskip
0.5pt\mathfrak{n}_{+})$, resp. $U^{\,{\mathcal{R}}}_{\\!P,\hskip
0.7pt\hbar}(\hskip 0.5pt\mathfrak{n}_{-})$, of a FoMpQUEA
$U^{\,{\mathcal{R}}}_{\\!P,\hskip 0.7pt\hbar}(\hskip 0.5pt\mathfrak{g})$ to be
the unital, $\hbar$–adically complete topological
${\Bbbk[[\hbar]]}$–subalgebra of $U^{\,{\mathcal{R}}}_{\\!P,\hskip
0.7pt\hbar}(\hskip 0.5pt\mathfrak{g})$ generated by the $E_{i}$’s, resp. the
$F_{i}$’s, with $\,i\in I\,$. $\diamondsuit$

The following two results underscore that the dependence of FoMpQUEAs on
realizations (which includes that on the multiparameter matrix) is functorial:

######  Proposition 4.2.3.

Let $\,P\in M_{n}\big{(}{\Bbbk[[\hbar]]}\big{)}\,$. If both
${\mathcal{R}}^{\prime}$ and $\,{\mathcal{R}}^{\prime\prime}$ are realizations
of $P$ and
$\,\underline{\phi}:{\mathcal{R}}^{\prime}\relbar\joinrel\relbar\joinrel\longrightarrow{\mathcal{R}}^{\prime\prime}\,$
is a morphism between them, then there exists a unique morphism of unital
topological $\,{\Bbbk[[\hbar]]}$–algebras
$\;U_{\underline{\phi}}:U^{\,{\mathcal{R}}^{\prime}}_{\\!P,\hbar}(\mathfrak{g})\relbar\joinrel\relbar\joinrel\longrightarrow
U^{\,{\mathcal{R}}^{\prime\prime}}_{\\!P,\hbar}(\mathfrak{g})\;$ that extends
the morphism
$\,\phi:\mathfrak{h}^{\prime}\relbar\joinrel\relbar\joinrel\longrightarrow\mathfrak{h}^{\prime\prime}\,$
of $\,{\Bbbk[[\hbar]]}$–modules given by $\underline{\phi}\,$; moreover,
$\,U_{\underline{\operatorname{id}}_{\mathcal{R}}}=\operatorname{id}_{U^{\,{\mathcal{R}}}_{\\!P,\hskip
0.7pt\hbar}(\hskip 0.5pt\mathfrak{g})}\,$ and
$\;U_{\underline{\phi}^{\prime}\circ\,\underline{\phi}}=U_{\underline{\phi}^{\prime}}\circ
U_{\underline{\phi}}\;$ (whenever
$\,\underline{\phi}^{\prime}\circ\,\underline{\phi}\,$ is defined).

Thus, the construction $\,{\mathcal{R}}\mapsto
U^{\,{\mathcal{R}}}_{\\!P,\hskip 0.7pt\hbar}(\hskip 0.5pt\mathfrak{g})\,$ —
for any fixed $P$ — is functorial in ${\mathcal{R}}\,$.

Moreover, if $\underline{\phi}$ is an epimorphism, resp. a monomorphism, then
$U_{\underline{\phi}}$ is an epimorphism, resp. a monomorphism, as well.

Finally, for any morphism
$\,\underline{\phi}:{\mathcal{R}}^{\prime}\relbar\joinrel\relbar\joinrel\relbar\joinrel\longrightarrow{\mathcal{R}}^{\prime\prime}\,$,
the kernel $\,\operatorname{Ker}\big{(}U_{\underline{\phi}}\big{)}$ of
$\,U_{\underline{\phi}}$ is the two-sided ideal in
$\,U^{\,{\mathcal{R}}^{\prime}}_{\\!P,\hbar}(\mathfrak{g})$ generated by
$\,\operatorname{Ker}(\phi)\,$, and the latter is central in
$\,U^{\,{\mathcal{R}}^{\prime}}_{\\!P,\hbar}(\mathfrak{g})\,$.

###### Proof.

Everything is obvious, we only spend some words on the centrality of
$\operatorname{Ker}(\phi)\,$. Lemma 2.1.9 gives
$\;\operatorname{Ker}(\phi)\,\subseteq\,\bigcap\limits_{j\in
I}\operatorname{Ker}(\alpha^{\prime}_{j})\;$; then (4.5) implies that each
element in $\operatorname{Ker}(\phi)$ commutes with all generators of
$\,U^{\,{\mathcal{R}}^{\prime}}_{\\!P,\hbar}(\mathfrak{g})\,$, so
$\operatorname{Ker}(\phi)$ is central in
$\,U^{\,{\mathcal{R}}^{\prime}}_{\\!P,\hbar}(\mathfrak{g})\,$. ∎

######  Corollary 4.2.4.

With notation as above, if
$\;{\mathcal{R}}^{\prime}\cong{\mathcal{R}}^{\prime\prime}\,$ then
$\;U^{\,{\mathcal{R}}^{\prime}}_{P,\hbar}(\mathfrak{g})\cong
U^{\,{\mathcal{R}}^{\prime\prime}}_{P,\hbar}(\mathfrak{g})\;$.

In particular, all FoMpQUEAs built upon split realizations, respectively small
realizations, of the same matrix $P$ and sharing the same rank of
$\,\mathfrak{h}$ are isomorphic to each other, hence they are independent (up
to isomorphisms) of the specific realization, but only depend on $P$ and on
the rank of $\,\mathfrak{h}\,$.

###### Proof.

This follows at once from Proposition 4.2.3 together with the uniqueness
result in Proposition 2.1.5 and Proposition 2.1.7. ∎

We conclude this subsection with an important structure result, namely the
“triangular decomposition” for FoMpQUEAs. We begin with some preliminaries.

######  Definition 4.2.5.

Let $\,A:={\big{(}\,a_{i,j}\big{)}}_{i,j\in I}\,$ be a generalized
symmetrizable Cartan matrix, and $\,P:={\big{(}\,p_{i,j}\big{)}}_{i,j\in I}\in
M_{n}\big{(}{\Bbbk[[\hbar]]}\big{)}\,$ a matrix of Cartan type associated with
$A$. We fix a realization
$\,{\mathcal{R}}:=\big{(}\,\mathfrak{h}\,,\Pi\,,\Pi^{\vee}\,\big{)}\,$ of $P$
as in Definition 2.1.2. (a) We define $\widehat{U}^{+}$, resp.
$\widehat{U}^{-}$, to be the unital, associative, topological,
$\hbar$–adically complete ${\Bbbk[[\hbar]]}$–algebra with generators
$E_{i}\,(i\in I\,)\,$, resp. $F_{i}\,(i\in I\,)\,$, and relations

$u^{X}_{ij}\,:=\,\sum\limits_{k=0}^{1-a_{ij}}(-1)^{k}{\left[{1-a_{ij}\atop
k}\right]}_{\\!q_{i}}q_{ij}^{+k/2\,}q_{ji}^{-k/2}\,X_{i}^{1-a_{ij}-k}X_{j}X_{i}^{k}\,\;=\;\,0\qquad(\,\forall\;i\neq
j\,)$

with $\,X:=E\,$, resp. $\,X:=F\,$. (b) We define $\,\widehat{U}^{0}\,$ to be
the unital, associative, commutative, topological, $\hbar$–adically complete
${\Bbbk[[\hbar]]}$–algebra generated by $\mathfrak{h}\,$. In other words, it
is $\,\widehat{U}^{0}:=\widehat{S}_{\hbar}(\mathfrak{h})\,$, the $\hbar$–adic
completion of the symmetric ${\Bbbk[[\hbar]]}$–algebra over the
${\Bbbk[[\hbar]]}$–module $\mathfrak{h}\,$. (c) We define
$\;\overrightarrow{U}^{{\mathcal{R}},\otimes}_{\\!P,\,\hbar}(\mathfrak{g}):=\widehat{U}^{-}\,\widehat{\otimes}_{\Bbbk[[\hbar]]}\,\widehat{U}^{0}\,\widehat{\otimes}_{\Bbbk[[\hbar]]}\,\widehat{U}^{+}\,$,
and we introduce notation

$\halign{\hbox to\displaywidth{$\hfil\displaystyle#\hfil$}\cr
0.0pt{\hfil$\displaystyle\widehat{U}^{-}_{\otimes}:=\widehat{U}^{-}\otimes{\Bbbk[[\hbar]]}\otimes{\Bbbk[[\hbar]]}\;\;,\\!\\!\quad\widehat{U}^{0}_{\otimes}(\mathfrak{h}):={\Bbbk[[\hbar]]}\otimes\widehat{U}^{0}\otimes{\Bbbk[[\hbar]]}\;\;,\\!\\!\quad\widehat{U}^{+}_{\otimes}:={\Bbbk[[\hbar]]}\otimes{\Bbbk[[\hbar]]}\otimes\widehat{U}^{+}\cr
0.0pt{\hfil$\displaystyle F^{\otimes}:=F\otimes 1\otimes
1\;,\,\;H^{\otimes}:=1\otimes H\otimes 1\;,\,\;E^{\otimes}:=1\otimes 1\otimes
E\quad\;\forall\;F\in\widehat{U}^{-},H\in\widehat{U}^{0},E\in\widehat{U}^{+}\crcr}}}$

Similarly, we set
$\;\overleftarrow{U}^{{\mathcal{R}},\otimes}_{\\!P,\,\hbar}(\mathfrak{g}):=\widehat{U}^{+}\,\widehat{\otimes}_{\Bbbk[[\hbar]]}\,\widehat{U}^{0}\,\widehat{\otimes}_{\Bbbk[[\hbar]]}\,\widehat{U}^{-}\,$.

The following, key technical result is also interesting in itself:

######  Lemma 4.2.6.

There exists on
$\,\overrightarrow{U}^{{\mathcal{R}},\otimes}_{\\!P,\,\hbar}(\mathfrak{g})$ a
unique structure of unital, associative, topological, $\hbar$–adically
complete ${\Bbbk[[\hbar]]}$–algebra such that $\,\widehat{U}^{-}_{\otimes}\,$,
$\widehat{U}^{0}_{\otimes}$ and $\,\widehat{U}^{+}_{\otimes}$ are all
${\Bbbk[[\hbar]]}$–subalgebras in
$\,U^{{\mathcal{R}},\otimes}_{\\!P,\,\hbar}(\mathfrak{g})\,$, and moreover

$\halign{\hbox to\displaywidth{$\hfil\displaystyle#\hfil$}\cr
0.0pt{\hfil$\displaystyle F_{i}^{\otimes}\cdot T^{\otimes}\;=\;F_{i}\otimes
T\otimes 1\;\;,\quad T^{\otimes}\cdot E_{j}^{\otimes}\;=\;1\otimes T\otimes
E_{j}\;\;,\quad F_{i}^{\otimes}\cdot E_{j}^{\otimes}\;=\;F_{i}\otimes 1\otimes
E_{j}\cr 0.0pt{\hfil$\displaystyle T^{\otimes}\cdot
F_{i}^{\otimes}\;=\;F_{i}\otimes T\otimes
1\,-\,\alpha_{i}(T)\,F_{i}^{\otimes}\quad,\qquad E_{j}^{\otimes}\cdot
T^{\otimes}\;=\;1\otimes T\otimes E_{j}\,-\,\alpha_{j}(T)\,E_{j}^{\otimes}\cr
0.0pt{\hfil$\displaystyle E_{j}^{\otimes}\cdot
F_{i}^{\otimes}\;=\;F_{i}\otimes 1\otimes
E_{j}\,+\,\delta_{i{}j}\,1\otimes{{\;\;e^{+\hbar\,T_{i}^{+}}-e^{-\hbar\,T_{i}^{-}}\;}\over{e^{+\hbar\,d_{i}}-e^{-\hbar\,d_{i}}}}\otimes
1\crcr}}}}$

An entirely similar claim holds true for
$\,\overleftarrow{U}^{{\mathcal{R}},\otimes}_{\\!P,\,\hbar}(\mathfrak{g}):=\widehat{U}^{+}\,\widehat{\otimes}_{\Bbbk[[\hbar]]}\,\widehat{U}^{0}\,\widehat{\otimes}_{\Bbbk[[\hbar]]}\,\widehat{U}^{-}\,$.

###### Proof.

It is enough to prove the statement about
$\overrightarrow{U}^{{\mathcal{R}},\otimes}_{\\!P,\,\hbar}(\mathfrak{g})\,$.

We introduce a structure of an $\hbar$–adically complete, topological
${\Bbbk[[\hbar]]}$–algebra
$\overrightarrow{U}^{{\mathcal{R}},\otimes}_{\\!P,\,\hbar}(\mathfrak{g})$ as
required by hands, somehow. First, we assume that in this algebra the
submodules $\widehat{U}^{-}_{\otimes}\,$, $\widehat{U}^{0}_{\otimes}\,$ and
$\widehat{U}^{+}_{\otimes}\,$ sit as ${\Bbbk[[\hbar]]}$–subalgebras — there is
no obstruction to such a requirement. After this, the structure will be
uniquely determined once we fix the products among elements in any two
(different) of these subalgebras. Moreover, as the subalgebra
$\,\widehat{U}^{-}_{\otimes}\,$, resp.
$\,\widehat{U}^{0}_{\otimes}(\mathfrak{h})\,$, resp.
$\,\widehat{U}^{+}_{\otimes}\,$, is (topologically) generated by the
$F_{i}^{\otimes}$’s ($\,i\in I\,$), resp. the $T^{\otimes}$’s
($\,T\in\mathfrak{h}\,$), resp. the $E_{j}^{\otimes}$’s ($\,j\in I\,$), it is
enough to fix the products among any two such generators (from different
sets). Eventually, recall that the $F_{i}$’s, resp. the $E_{j}$’s, are indeed
generators for $\widehat{U}^{-}$, resp. for $\widehat{U}^{+}$, which are only
subject to the “quantum Serre relations” in Definition 4.2.5(b), while the
$T$’s are “commutative free” — but for the fact that they are related by
obvious, built-in relations such as
$\,T=c^{\prime}\,T^{\prime}+c^{\prime\prime}T^{\prime\prime}\,$ inside
$\mathfrak{h}$ implies
$\,T=c^{\prime}\,T^{\prime}+c^{\prime\prime}T^{\prime\prime}\,$ in
$\widehat{U}^{0}_{\otimes}\,$, as $\mathfrak{h}$ is naturally mapped
(linearly) into $\widehat{U}^{0}_{\otimes}\,$. Thus, one can define the values
of the product among $F_{i}^{\otimes}$, $E_{j}^{\otimes}$’ and $T^{\otimes}$
in any possible way as soon as all “quantum Serre relations” among the
$F_{i}^{\otimes}$’s and among the $E_{j}^{\otimes}$’s, as well as the “obvious
relations” among the $T$’s from $\mathfrak{h}$ — namely the “linear relations”
(such as $\,T=c^{\prime}\,T^{\prime}+c^{\prime\prime}T^{\prime\prime}\,$) and
the commutation relations (of the form
$\,T^{\prime}T^{\prime\prime}=T^{\prime\prime}T^{\prime}\,$) are respected.

By the above discussion, the following choices

$\halign{\hbox to\displaywidth{$\hfil\displaystyle#\hfil$}\cr
0.0pt{\hfil$\displaystyle F_{i}^{\otimes}\cdot T^{\otimes}\;:=\;F_{i}\otimes
T\otimes 1\;\;,\quad T^{\otimes}\cdot E_{j}^{\otimes}\;:=\;1\otimes T\otimes
E_{j}\;\;,\quad F_{i}^{\otimes}\cdot E_{j}^{\otimes}\;:=\;F_{i}\otimes
1\otimes E_{j}\cr 0.0pt{\hfil$\displaystyle T^{\otimes}\cdot
F_{i}^{\otimes}:=\,F_{i}\otimes T\otimes 1\,-\,\alpha_{i}(T)\,F_{i}\otimes
1\otimes 1\;,\\!\\!\\!\quad E_{j}^{\otimes}\cdot T^{\otimes}:=\,1\otimes
T\otimes E_{j}\,-\,\alpha_{j}(T)\,1\otimes 1\otimes E_{j}\cr
0.0pt{\hfil$\displaystyle E_{j}^{\otimes}\cdot
F_{i}^{\otimes}\;:=\;F_{i}\otimes 1\otimes E_{j}\,+\,\delta_{i\,j}\cdot
1\otimes{{\;\;e^{+\hbar\,T_{i}^{+}}\\!-e^{-\hbar\,T_{i}^{-}}\;}\over{e^{+\hbar\,d_{i}}-e^{-\hbar\,d_{i}}}}\otimes
1\crcr}}}}$

for the values of the product among two generators — from different
subalgebras $\,\widehat{U}^{-}_{\otimes}\,$, $\,\widehat{U}^{0}_{\otimes}\,$
or $\,\widehat{U}^{+}_{\otimes}\,$ — are enough to determine a unique algebra
structure as required: we only have still to check that, using these defining
formulas for the product, the relations mentioned above among generators are
respected. First of all, we consider all linear relations and commutation
relations among the $T$’s: in this case, the check is entirely trivial.
Second, we consider the case of quantum Serre relations. Concerning them, in
order to have more readable formulas, we simplify notation (with a slight
abuse) by writing, instead of “$\,F^{\otimes}\,$”
($\,\forall\;F\in\widehat{U}^{-}\,$) just “$F\,$” again, and similarly “$H\,$”
instead of ”$\,H^{\otimes}\,$” ($\,\forall\;H\in\widehat{U}^{0}\,$) and
“$E\,$” instead of ”$\,E^{\otimes}\,$” ($\,\forall\;E\in\widehat{U}^{+}\,$).

Our goal now is to check that the multiplication defined by the previous
formulas “respects” the quantum Serre relations, which boils down to verify
the following: all products between a factor chosen in
$\,\big{\\{}\,F_{i}\,,T,E_{j}\,|\,i\,,j\in I,\,T\in\mathfrak{h}\,\big{\\}}\,$
and another (in either order) chosen in
$\,\big{\\{}\,u^{E}_{ij}\,,u^{F}_{ij}\,\big{|}\,i\,,j\in
I,\,i\not=j\,\big{\\}}\,$ is zero. Clearly all products of type
$\,u^{F}_{ij}\cdot F_{\ell}\,$ and $\,F_{\ell}\cdot u^{F}_{ij}\,$, resp.
$\,u^{E}_{ij}\cdot E_{t}\,$ and $\,E_{t}\cdot u^{E}_{ij}\,$, are zero because
so they are in the subalgebra $U^{-}$, resp. $U^{+}$. The non-trivial cases
are

$T\cdot u^{E}_{ij}\;\;,\\!\quad u^{E}_{ij}\cdot T\;,\\!\quad T\cdot
u^{F}_{ij}\;\;,\\!\quad u^{F}_{ij}\cdot T\;,\\!\quad u^{E}_{ij}\cdot
F_{\ell}\;\;,\\!\quad F_{\ell}\cdot u^{E}_{ij}\;\;,\\!\quad u^{F}_{ij}\cdot
E_{t}\;\;,\\!\quad E_{t}\cdot u^{F}_{ij}$

but among these, four cases are indeed almost trivial, as definitions give

$\halign{\hbox to\displaywidth{$\hfil\displaystyle#\hfil$}\cr
0.0pt{\hfil$\displaystyle T\cdot u^{E}_{ij}\,=\,1\otimes T\otimes
u^{E}_{ij}\,=\,1\otimes T\otimes 0\,=\,0\,\;,\\!\\!\quad u^{F}_{ij}\cdot
T\,=\,u^{F}_{ij}\otimes T\otimes 1\,=\,0\,\otimes T\otimes 1\,=\,0\cr
0.0pt{\hfil$\displaystyle F_{\ell}\cdot u^{E}_{ij}\,=\,F_{\ell}\otimes
1\otimes u^{E}_{ij}\,=\,F_{\ell}\otimes 1\otimes 0\,=\,0\,\;,\\!\\!\quad
u^{F}_{ij}\cdot E_{t}\,=\,u^{F}_{ij}\otimes 1\otimes E_{t}\,=\,0\,\otimes
1\otimes E_{t}\,=\,0\crcr}}}$

Eventually, the remaining, really non-trivial cases are the following four

$u^{E}_{ij}\cdot T\;\;,\qquad T\cdot u^{F}_{ij}\;\;,\qquad u^{E}_{ij}\cdot
F_{\ell}\;\;,\qquad E_{t}\cdot u^{F}_{ij}$

that we now go and analyze in detail. Let us consider the product $\,T\cdot
u^{F}_{ij}\,$: straightforward calculations give

$\halign{\hbox to\displaywidth{$\hfil\displaystyle#\hfil$}\cr
0.0pt{\hfil$\displaystyle T\cdot
u^{F}_{ij}\;=\;T\,\bigg{(}\,{\textstyle\sum\limits_{k=0}^{1-a_{ij}}}{(-1)}^{k}{\displaystyle{\left[{1-a_{ij}\atop
k}\right]}_{\\!q_{i}}}q_{ij}^{+k/2\,}q_{ji}^{-k/2}\,F_{i}^{1-a_{ij}-k}\,F_{j}\,F_{i}^{k}\bigg{)}\;=\hfill\cr
0.0pt{\hfil$\displaystyle=\;{\textstyle\sum\limits_{k=0}^{1-a_{ij}}}{(-1)}^{k}{\displaystyle{\left[{1-a_{ij}\atop
k}\right]}_{\\!q_{i}}}q_{ij}^{+k/2\,}q_{ji}^{-k/2}\,T\,F_{i}^{1-a_{ij}-k}\,F_{j}\,F_{i}^{k}\;=\hfill\cr
0.0pt{\hfil$\displaystyle\hfill={\textstyle\sum\limits_{k=0}^{1-a_{ij}}}{(-1)}^{k}{\displaystyle{\left[{1-a_{ij}\atop
k}\right]}_{\\!q_{i}}}\\!q_{ij}^{+k/2\,}q_{ji}^{-k/2}\,F_{i}^{1-a_{ij}-k}\,F_{j}\,F_{i}^{k}\otimes\Big{(}T-\big{(}\big{(}1-a_{ij}\big{)}\alpha_{i}+\alpha_{j}\big{)}(T)\Big{)}\otimes
1\,=\cr
0.0pt{\hfil$\displaystyle\hfill=\,u^{F}_{ij}\otimes\Big{(}T-\big{(}\big{(}1-a_{ij}\big{)}\alpha_{i}+\alpha_{j}\big{)}(T)\Big{)}\otimes
1\,=\,0\otimes\Big{(}T-\big{(}\big{(}1-a_{ij}\big{)}\alpha_{i}+\alpha_{j}\big{)}(T)\Big{)}\otimes
1\,=\,0\crcr}}}}}$

which is good. The product $\,u^{E}_{ij}\cdot T\,$ is dealt with in a similar
way. Let us now consider the product $\,u^{E}_{ij}\cdot F_{\ell}\,$: again,
direct calculations yield different results, depending on whether
$\,\ell\in\\{i\,,j\\}\,$ or not. The first possible case is

$\halign{\hbox to\displaywidth{$\hfil\displaystyle#\hfil$}\cr
0.0pt{\hfil$\displaystyle\framebox{$\,\ell\not\in\\{i\,,j\\}$}\;\;\Longrightarrow\;\;u^{E}_{ij}\cdot
F_{\ell}\;=\bigg{(}\,{\textstyle\sum\limits_{k=0}^{1-a_{ij}}}{(-1)}^{k}{\displaystyle{\left[{1-a_{ij}\atop
k}\right]}_{\\!q_{i}}}\\!q_{ij}^{+k/2\,}q_{ji}^{-k/2}\,E_{i}^{1-a_{ij}-k}\,E_{j}\,E_{i}^{k}\bigg{)}F_{\ell}\;=\hfill\cr
0.0pt{\hfil$\displaystyle=\;F_{\ell}\otimes
1\otimes\bigg{(}\,{\textstyle\sum\limits_{k=0}^{1-a_{ij}}}{(-1)}^{k}{\displaystyle{\left[{1-a_{ij}\atop
k}\right]}_{\\!q_{i}}}\\!q_{ij}^{+k/2\,}q_{ji}^{-k/2}\,E_{i}^{1-a_{ij}-k}\,E_{j}\,E_{i}^{k}\bigg{)}\;=\cr
0.0pt{\hfil$\displaystyle\hfill=\;F_{\ell}\otimes 1\otimes
u^{E}_{ij}\;=\;F_{\ell}\otimes 1\otimes 0\;=\;0\crcr}}}}$

which stands good! The second case is

$\halign{\hbox to\displaywidth{$\hfil\displaystyle#\hfil$}\cr
0.0pt{\hfil$\displaystyle\framebox{$\,\ell=j\,$}\;\Longrightarrow\;u^{E}_{ij}\cdot
F_{\ell}\;=\bigg{(}\,{\textstyle\sum\limits_{k=0}^{1-a_{ij}}}{(-1)}^{k}{\displaystyle{\left[{1-a_{ij}\atop
k}\right]}_{\\!q_{i}}}\\!q_{ij}^{+k/2\,}q_{ji}^{-k/2}\,E_{i}^{1-a_{ij}-k}\,E_{j}\,E_{i}^{k}\bigg{)}F_{j}\;=\hfill\cr
0.0pt{\hfil$\displaystyle=\;{\textstyle\sum\limits_{k=0}^{1-a_{ij}}}{(-1)}^{k}{\displaystyle{\left[{1-a_{ij}\atop
k}\right]}_{\\!q_{i}}}\\!q_{ij}^{+k/2\,}q_{ji}^{-k/2}\,E_{i}^{1-a_{ij}-k}\cdot
E_{j}\cdot F_{j}\cdot E_{i}^{k}\;=\hfill\cr
0.0pt{\hfil$\displaystyle\hfill=\;{\textstyle\sum\limits_{k=0}^{1-a_{ij}}}{(-1)}^{k}{\displaystyle{\left[{1-a_{ij}\atop
k}\right]}_{\\!q_{i}}}\\!q_{ij}^{+k/2\,}q_{ji}^{-k/2}\,E_{i}^{1-a_{ij}-k}\cdot\bigg{(}F_{j}\cdot
E_{j}+{{\;e^{+\hbar\,T_{j}^{+}}-e^{-\hbar\,T_{j}^{-}}\;}\over{e^{+\hbar\,d_{j}}-e^{-\hbar\,d_{j}}}}\bigg{)}\cdot
E_{i}^{k}\;=\cr 0.0pt{\hfil$\displaystyle\hfill=\;F_{j}\cdot
u^{E}_{ij}\,+\,{\textstyle\sum\limits_{k=0}^{1-a_{ij}}}{(-1)}^{k}{\displaystyle{\left[{1-a_{ij}\atop
k}\right]}_{\\!q_{i}}}\\!q_{ij}^{+k/2\,}q_{ji}^{-k/2}\,E_{i}^{1-a_{ij}-k}\cdot{{\;e^{+\hbar\,T_{j}^{+}}-e^{-\hbar\,T_{j}^{-}}\;}\over{e^{+\hbar\,d_{j}}-e^{-\hbar\,d_{j}}}}\cdot
E_{i}^{k}\;=\cr 0.0pt{\hfil$\displaystyle\qquad=\;F_{j}\otimes 1\otimes
u^{E}_{ij}\;+\hfill\cr
0.0pt{\hfil$\displaystyle\qquad\qquad+\,{\textstyle\sum\limits_{k=0}^{1-a_{ij}}}{(-1)}^{k}{\displaystyle{\left[{1-a_{ij}\atop
k}\right]}_{\\!q_{i}}}\\!q_{ij}^{+k/2\,}q_{ji}^{-k/2}\,q_{ji}^{k-1+a_{ij}}\cdot{{e^{+\hbar\,T_{j}^{+}}}\over{\;e^{+\hbar\,d_{j}}-e^{-\hbar\,d_{j}}\;}}\cdot
E_{i}^{1-a_{ij}}\;-\hfill\cr
0.0pt{\hfil$\displaystyle\qquad\qquad\qquad-\,{\textstyle\sum\limits_{k=0}^{1-a_{ij}}}{(-1)}^{k}{\displaystyle{\left[{1-a_{ij}\atop
k}\right]}_{\\!q_{i}}}\\!q_{ij}^{+k/2\,}q_{ji}^{-k/2}\,q_{ij}^{1-a_{ij}-k}\cdot{{e^{-\hbar\,T_{j}^{-}}}\over{\;e^{+\hbar\,d_{j}}-e^{-\hbar\,d_{j}}\;}}\cdot
E_{i}^{1-a_{ij}}\;=\hfill\cr 0.0pt{\hfil$\displaystyle\hfill=\;F_{j}\otimes
1\otimes
u^{E}_{ij}\,+\,1\otimes{{\;C^{+}_{ij}(q)\,e^{+\hbar\,T_{j}^{+}}\\!-C^{-}_{ij}(q)\,e^{-\hbar\,T_{j}^{-}}\;}\over{e^{+\hbar\,d_{j}}-e^{-\hbar\,d_{j}}}}\otimes
E_{i}^{1-a_{ij}}\crcr}}}}}}}}}$

where in the last line we have $\,F_{j}\otimes 1\otimes
u^{E}_{ij}=F_{j}\otimes 1\otimes 0=0\,$, and

$\halign{\hbox to\displaywidth{$\hfil\displaystyle#\hfil$}\cr
0.0pt{\hfil$\displaystyle
C^{+}_{ij}(q)\;:=\;{\textstyle\sum\limits_{k=0}^{1-a_{ij}}}{(-1)}^{k}{\displaystyle{\left[{1-a_{ij}\atop
k}\right]}_{\\!q_{i}}}\\!q_{ij}^{+k/2\,}q_{ji}^{-k/2}\,q_{ji}^{k-1+a_{ij}}\;=\hfill\cr
0.0pt{\hfil$\displaystyle\hfill=\,{\textstyle\sum\limits_{k=0}^{1-a_{ij}}}{(-1)}^{k}{\displaystyle{\left[{1-a_{ij}\atop
k}\right]}_{\\!q_{i}}}\\!\\!{\big{(}q_{ij}\,q_{ji}\big{)}}^{+k/2}\,q_{ji}^{-1+a_{ij}}\,=\;q_{ji}^{a_{ij}-1}{\textstyle\sum\limits_{k=0}^{1-a_{ij}}}{(-1)}^{k}{\displaystyle{\left[{1-a_{ij}\atop
k}\right]}_{\\!q_{i}}}q_{i}^{\,ka_{ij}}\;=\;0\crcr}}}$

where the very last identity follows from the general, combinatorial
$q$–identity (see for example [Ja, Chapter 0], or [Lu, §1.3.4])

${\textstyle\sum\limits_{k=0}^{N}}{(-1)}^{k}{\displaystyle{\left[{N\atop
k}\right]}_{\\!q_{i}}}q_{i}^{\,k(1-N)}\;=\;0$ $None$

In a parallel way we get $\,C^{-}_{ij}(q)=0\,$, hence from the above analysis
we conclude that $\,u^{E}_{ij}\cdot F_{\ell}=u^{E}_{ij}\cdot F_{j}=0\,$
whenever $\,\ell=j\,$. The third and last case is when $\,\ell=i\,$. To deal
with that, let us notice that standard computations give us, for all
$\,n\in\mathbb{N}\,$,

$\halign{\hbox to\displaywidth{$\hfil\displaystyle#\hfil$}\cr
0.0pt{\hfil$\displaystyle E_{i}^{n}\cdot
F_{i}\;=\;F_{i}\,E_{i}^{n}\,+\,\big{[}E_{i}^{n},F_{i}\big{]}\;=\;F_{i}\,E_{i}^{n}\,+\,{\textstyle\sum\limits_{\ell=0}^{n}}\,E_{i}^{\ell}\,\big{[}E_{i}\,,F_{i}\big{]}\,E_{i}^{n-1-\ell}\;=\hfill\cr
0.0pt{\hfil$\displaystyle\qquad=\;F_{i}\,E_{i}^{n}\,+\,{\textstyle\sum\limits_{\ell=0}^{n-1}}\,E_{i}^{\ell}\,{{\;e^{+\hbar\,T_{i}^{+}}-e^{-\hbar\,T_{i}^{-}}\;}\over{e^{+\hbar\,d_{i}}-e^{-\hbar\,d_{i}}}}\,E_{i}^{n-1-\ell}\;=\hfill\cr
0.0pt{\hfil$\displaystyle\qquad\qquad=\;F_{i}\cdot
E_{i}^{n}\,+\,{\;{\textstyle\sum_{\ell=0}^{n-1}}\,q_{i{}i}^{-\ell}\,{e^{+\hbar\,T_{i}^{+}}\,-\,{\textstyle\sum_{\ell=0}^{n-1}}\,q_{i{}i}^{+\ell}\,{e^{-\hbar\,T_{i}^{-}}}\;}\over{q_{i}^{+1}-\,q_{i}^{-1}}}\cdot
E_{i}^{n-1}\;=\hfill\cr
0.0pt{\hfil$\displaystyle\qquad\qquad\qquad=\;F_{i}\cdot
E_{i}^{n}\,+\,{[n]}_{q_{i}}\,{\;\,q_{i}^{-n+1}\,{e^{+\hbar\,T_{i}^{+}}-\,q_{i}^{+n-1}\,{e^{-\hbar\,T_{i}^{-}}}\;}\over{q_{i}^{+1}-\,q_{i}^{-1}}}\cdot
E_{i}^{n-1}\hfill\crcr}}}}}$

Using this, we compute still more, and get, for all $\,r,s\in\mathbb{N}\,$,

$\halign{\hbox to\displaywidth{$\hfil\displaystyle#\hfil$}\cr
0.0pt{\hfil$\displaystyle E_{i}^{r}\,E_{j}\,E_{i}^{s}\cdot
F_{i}\;=\;E_{i}^{r}\,E_{j}\,\Big{(}F_{i}\cdot
E_{i}^{s}\,+\,{[s]}_{q_{i}}\,{\;\,q_{i}^{-s+1}\,{e^{+\hbar\,T_{i}^{+}}-\,q_{i}^{+s-1}\,{e^{-\hbar\,T_{i}^{-}}}\;}\over{q_{i}^{+1}-\,q_{i}^{-1}}}\cdot
E_{i}^{s-1}\,\Big{)}\;=\hfill\cr 0.0pt{\hfil$\displaystyle\hskip
37.0pt=\;E_{i}^{r}\cdot F_{i}\cdot
E_{j}\,E_{i}^{s}\,+\,{[s]}_{q_{i}}\,E_{i}^{r}\,E_{j}\cdot{{\;\,q_{i}^{-s+1}\,e^{+\hbar\,T_{i}^{+}}-\,q_{i}^{+s-1}\,e^{-\hbar\,T_{i}^{-}}\;}\over{q_{i}^{+1}-\,q_{i}^{-1}}}\cdot
E_{i}^{s-1}\;=\hfill\cr 0.0pt{\hfil$\displaystyle\hskip 74.0pt=\;F_{i}\cdot
E_{i}^{r}\,E_{j}\,E_{i}^{s}\,+\,{[r]}_{q_{i}}\,{{\;\,q_{i}^{-r+1}\,e^{+\hbar\,T_{i}^{+}}-\,q_{i}^{+r-1}\,e^{-\hbar\,T_{i}^{-}}\;}\over{q_{i}^{+1}-\,q_{i}^{-1}}}\cdot
E_{i}^{r-1}\,E_{j}\,E_{i}^{s}\;+\hfill\cr 0.0pt{\hfil$\displaystyle\hskip
111.0pt+\;\,{[s]}_{q_{i}}\,{{\;\,q_{i}^{-s+1-2r}q_{ij}^{-1}\,e^{+\hbar\,T_{i}^{+}}-\,q_{i}^{+s-1+2r}q_{ji}^{+1}\,e^{-\hbar\,T_{i}^{-}}\;}\over{q_{i}^{+1}-\,q_{i}^{-1}}}\cdot
E_{i}^{r}\,E_{j}\,E_{i}^{s-1}\hfill\crcr}}}}}$

Applying this result, we get the following:

$\halign{\hbox to\displaywidth{$\hfil\displaystyle#\hfil$}\cr
0.0pt{\hfil$\displaystyle u^{E}_{ij}\cdot
F_{i}\;={\textstyle\sum\limits_{r+s=1-a_{ij}}}\hskip-5.0pt{(-1)}^{s}{\displaystyle{\left[{1-a_{ij}\atop
s}\right]}_{\\!q_{i}}}\\!q_{ij}^{+s/2\,}q_{ji}^{-s/2}\,E_{i}^{r}\,E_{j}\,E_{i}^{s}\cdot
F_{i}\;=\hfill\cr 0.0pt{\hfil$\displaystyle\hskip
15.0pt={\textstyle\sum\limits_{r+s=1-a_{ij}}}\hskip-5.0pt{(-1)}^{s}{\displaystyle{\left[{1-a_{ij}\atop
s}\right]}_{\\!q_{i}}}\\!q_{ij}^{+s/2\,}q_{ji}^{-s/2}\cdot\bigg{(}F_{i}\cdot
E_{i}^{r}\,E_{j}\,E_{i}^{s}\;\;+\hfill\cr 0.0pt{\hfil$\displaystyle\hskip
30.0pt+\;\;{[r]}_{q_{i}}\,{{\;\,q_{i}^{-r+1}\,e^{+\hbar\,T_{i}^{+}}-\,q_{i}^{+r-1}\,e^{-\hbar\,T_{i}^{-}}\;}\over{q_{i}^{+1}\\!-q_{i}^{-1}}}\cdot
E_{i}^{r-1}\,E_{j}\,E_{i}^{s}\;\;+\hfill\cr 0.0pt{\hfil$\displaystyle\hskip
45.0pt+\;\;{[s]}_{q_{i}}\,{{\;\,q_{i}^{-s+1-2r}q_{ij}^{-1}\,e^{+\hbar\,T_{i}^{+}}-\,q_{i}^{+s-1+2r}q_{ji}^{+1}\,e^{-\hbar\,T_{i}^{-}}\;}\over{q_{i}^{+1}\\!-q_{i}^{-1}}}\cdot
E_{i}^{r}\,E_{j}\,E_{i}^{s-1}\bigg{)}\;\;=\hfill\cr
0.0pt{\hfil$\displaystyle\hskip
5.0pt=\;\;F_{i}\cdot\hskip-5.0pt{\textstyle\sum\limits_{r+s=1-a_{ij}}}\hskip-5.0pt{(-1)}^{s}{\displaystyle{\left[{1-a_{ij}\atop
s}\right]}_{\\!q_{i}}}\\!q_{ij}^{+s/2\,}q_{ji}^{-s/2}\,E_{i}^{r}\,E_{j}\,E_{i}^{s}\;\;+\hfill\cr
0.0pt{\hfil$\displaystyle\hskip
10.0pt+{\textstyle\sum\limits_{h+k=-a_{ij}}}\hskip-1.0pt{{{\;(-1)}^{k}\,\varGamma_{h,k}^{+}\;}\over{\;q_{i}^{+1}\\!-q_{i}^{-1}\;}}\;e^{+\hbar\,T_{i}^{+}}\\!\cdot
E_{i}^{h}\,E_{j}\,E_{i}^{k}\;+{\textstyle\sum\limits_{h+k=-a_{ij}}}\hskip-1.0pt{{{\;(-1)}^{k}\,\varGamma_{h,k}^{-}\;}\over{\;q_{i}^{+1}\\!-q_{i}^{-1}\;}}\;e^{-\hbar\,T_{i}^{-}}\\!\cdot
E_{i}^{h}\,E_{j}\,E_{i}^{k}\crcr}}}}}}}$

where the coefficients $\,\varGamma_{h,k}^{+}\,$ and $\,\varGamma_{h,k}^{-}\,$
are given by

$\hskip-7.0pt\varGamma_{h,k}^{\pm}\;=\;q_{ij}^{+k/2}\,q_{ji}^{-k/2}\,q_{i}^{\mp
h}\left({\left[{{h+k+\\!1}\atop
k}\right]}_{\\!q_{i}}\\!\\!{\big{[}h+\\!1\big{]}}_{\\!q_{i}}\,-\,{\left[{{h+k+\\!1}\atop{k+\\!1}}\right]}_{\\!q_{i}}\\!\\!{\big{[}k+\\!1\big{]}}_{\\!q_{i}}\right)\;=\;\,0$

Plugging this result in the previous formulas, we get

$\halign{\hbox to\displaywidth{$\hfil\displaystyle#\hfil$}\cr
0.0pt{\hfil$\displaystyle\framebox{$\,\ell=i\,$}\,\;\Longrightarrow\;\,u^{E}_{ij}\cdot
F_{\ell}\,\;=\;\,F_{i}\,\cdot\hskip-5.0pt{\textstyle\sum\limits_{r+s=1-a_{ij}}}\hskip-7.0pt{(-1)}^{s}{\displaystyle{\left[{1-a_{ij}\atop
s}\right]}_{\\!q_{i}}}\\!q_{ij}^{+s/2\,}q_{ji}^{-s/2}\,E_{i}^{r}\,E_{j}\,E_{i}^{s}\;+\;0\;+\;0\;\;=\hfill\cr
0.0pt{\hfil$\displaystyle\hfill=\;\;F_{i}\,u^{E}_{ij}\;+\;0\;+\;0\;\;=\;\;F_{i}\otimes
1\otimes u^{E}_{ij}\;\;=\;\;F_{i}\otimes 1\otimes 0\;\;=\;\;0\crcr}}}$

which makes the job. The case of $\,E_{t}\cdot u^{F}_{ij}\,$ is entirely
similar, hence we are done. ∎

The next result shows that
$\overrightarrow{U}^{{\mathcal{R}},\otimes}_{\\!P,\,\hbar}(\mathfrak{g})$ and
$\overleftarrow{U}^{{\mathcal{R}},\otimes}_{\\!P,\,\hbar}(\mathfrak{g})$ are
nothing but different, explicit realizations of our FoMpQUEA
$U^{\,{\mathcal{R}}}_{\\!P,\hskip 0.7pt\hbar}(\hskip 0.5pt\mathfrak{g})\,$;
moreover, from this we deduce an explicit description of the nilpotent, Borel
and Cartan quantum subalgebras in $U^{\,{\mathcal{R}}}_{\\!P,\hskip
0.7pt\hbar}(\hskip 0.5pt\mathfrak{g})\,$.

######  Theorem 4.2.7.

(a) There exist natural isomorphisms of topological
${\Bbbk[[\hbar]]}$–algebras

$\halign{\hbox to\displaywidth{$\hfil\displaystyle#\hfil$}\cr
0.0pt{\hfil$\displaystyle\overrightarrow{U}^{{\mathcal{R}},\otimes}_{\\!P,\,\hbar}(\mathfrak{g})\,:=\,\widehat{U}^{-}\,\widehat{\otimes}_{\Bbbk[[\hbar]]}\,\widehat{U}^{0}\,\widehat{\otimes}_{\Bbbk[[\hbar]]}\,\widehat{U}^{+}\,\;\cong\;\,U^{\,{\mathcal{R}}}_{\\!P,\hskip
0.7pt\hbar}(\hskip 0.5pt\mathfrak{g})\cr
0.0pt{\hfil$\displaystyle\overleftarrow{U}^{{\mathcal{R}},\otimes}_{\\!P,\,\hbar}(\mathfrak{g})\,:=\,\widehat{U}^{+}\,\widehat{\otimes}_{\Bbbk[[\hbar]]}\,\widehat{U}^{0}\,\widehat{\otimes}_{\Bbbk[[\hbar]]}\,\widehat{U}^{-}\,\;\cong\;\,U^{\,{\mathcal{R}}}_{\\!P,\hskip
0.7pt\hbar}(\hskip 0.5pt\mathfrak{g})\crcr}}}$

induced by multiplication in $U^{\,{\mathcal{R}}}_{\\!P,\hskip
0.7pt\hbar}(\hskip 0.5pt\mathfrak{g})\,$. (b) With notation as in Definition
4.2.2, the isomorphisms in claims (a) above induce by restriction similar
isomorphisms

$\halign{\hbox to\displaywidth{$\hfil\displaystyle#\hfil$}\cr
0.0pt{\hfil$\displaystyle U^{\,{\mathcal{R}}}_{\\!P,\hskip 0.7pt\hbar}(\hskip
0.5pt\mathfrak{n}_{-})\,\;\cong\;\,\widehat{U}^{-}\quad,\qquad
U_{\hbar}(\mathfrak{h})\,\;\cong\;\,\widehat{U}^{0}\quad,\qquad
U^{\,{\mathcal{R}}}_{\\!P,\hskip 0.7pt\hbar}(\hskip
0.5pt\mathfrak{n}_{+})\,\;\cong\;\,\widehat{U}^{+}\cr
0.0pt{\hfil$\displaystyle\widehat{U}^{\pm}\,\widehat{\otimes}_{\Bbbk[[\hbar]]}\,\widehat{U}^{0}\;\cong\;U^{\,{\mathcal{R}}}_{\\!P,\hskip
0.7pt\hbar}(\hskip
0.5pt\mathfrak{b}_{\pm})\quad,\qquad\widehat{U}^{0}\,\widehat{\otimes}_{\Bbbk[[\hbar]]}\,\widehat{U}^{\pm}\;\cong\;U^{\,{\mathcal{R}}}_{\\!P,\hskip
0.7pt\hbar}(\hskip 0.5pt\mathfrak{b}_{\pm})\crcr}}}$

It follows then that $U^{\,{\mathcal{R}}}_{\\!P,\hskip 0.7pt\hbar}(\hskip
0.5pt\mathfrak{n}_{\pm})\,$, $U_{\hbar}(\mathfrak{h})\,$, and
$U^{\,{\mathcal{R}}}_{\\!P,\hskip 0.7pt\hbar}(\hskip 0.5pt\mathfrak{b}_{\pm})$
admit the obvious descriptions (in particular, by generators and relations)
inherited from Definition 4.2.5.

###### Proof.

(a) The two cases are similar, so we prove the claim only for
$\overrightarrow{U}^{{\mathcal{R}},\otimes}_{\\!P,\,\hbar}(\mathfrak{g})\,$.
Consider the ${\Bbbk[[\hbar]]}$–algebra
$\;\overrightarrow{U}^{{\mathcal{R}},\otimes}_{\\!P,\,\hbar}(\mathfrak{g}):=\widehat{U}^{-}\,\widehat{\otimes}\,\widehat{U}^{0}\,\widehat{\otimes}\,\widehat{U}^{+}\;$
of Lemma 4.2.6. By construction, it is (topologically) generated by the
elements $\,F_{i}^{\otimes}$, $T^{\otimes}$, $E_{j}^{\otimes}$ ($\,i\,,j\in
I\,,T\in\mathfrak{h}\,$), and these generators obey the same relations (up to
“inserting/removing” any super/sub-script “$\,\otimes$”) as the analogous
generators of $U^{\,{\mathcal{R}}}_{\\!P,\hskip 0.7pt\hbar}(\hskip
0.5pt\mathfrak{g})\,$. Therefore, there exists an epimorphism of topological
${\Bbbk[[\hbar]]}$–algebras

$\pi\,:\,U^{\,{\mathcal{R}}}_{\\!P,\hskip 0.7pt\hbar}(\hskip
0.5pt\mathfrak{g})\relbar\joinrel\relbar\joinrel\relbar\joinrel\relbar\joinrel\twoheadrightarrow
U^{{\mathcal{R}},\otimes}_{\\!P,\,\hbar}(\mathfrak{g})\;,\qquad F_{i}\mapsto
F_{i}^{\otimes}\;,\;\;T\mapsto T^{\otimes}\;,\;\;E_{j}\mapsto
E_{j}^{\otimes}\;\quad\big{(}\,\forall\;i\,,j\,,T\,\big{)}$

On the other hand, for each $\,\bullet\in\\{-\,,0\,,+\\}\,$ there is a
morphism
$\,\widehat{U}^{\bullet}\,{\buildrel{\eta_{\bullet}}\over{\relbar\joinrel\relbar\joinrel\longrightarrow}}\,U^{\,{\mathcal{R}}}_{\\!P,\hskip
0.7pt\hbar}(\hskip 0.5pt\mathfrak{g})\,$ mapping every generator of
$\widehat{U}^{\bullet}$ onto the corresponding generator in
$U^{\,{\mathcal{R}}}_{\\!P,\hskip 0.7pt\hbar}(\hskip 0.5pt\mathfrak{g})\,$.
Composing these with “threefold-multiplication” $m_{3}$ in
$U^{\,{\mathcal{R}}}_{\\!P,\hskip 0.7pt\hbar}(\hskip 0.5pt\mathfrak{g})$ we
obtain a morphism

$\mu_{3}:=m_{3}\circ\big{(}\eta_{-}\otimes\eta_{0}\otimes\eta_{+}\big{)}:U^{{\mathcal{R}},\otimes}_{\\!P,\,\hbar}(\mathfrak{g}):=\widehat{U}^{-}\,\widehat{\otimes}\,\widehat{U}^{0}\,\widehat{\otimes}\,\widehat{U}^{+}\\!\relbar\joinrel\relbar\joinrel\relbar\joinrel\longrightarrow
U^{\,{\mathcal{R}}}_{\\!P,\hskip 0.7pt\hbar}(\hskip 0.5pt\mathfrak{g})$

of topological ${\Bbbk[[\hbar]]}$–modules. Now, by construction, the map
$\mu_{3}$ is inverse to $\pi\,$, so the latter is bijective, hence it is a
${\Bbbk[[\hbar]]}$–algebra isomorphism. As on both sides we consider
$\hbar$–adic topology, this $\pi$ is then an homeomorphism of topological
spaces too, so it is an isomorphism of topological
${\Bbbk[[\hbar]]}$–algebras. Therefore $\,\mu_{3}\,$, being the inverse of
$\pi\,$, is an isomorphism of topological ${\Bbbk[[\hbar]]}$–algebras too,
q.e.d. (b) This follows quite easily from claim (a). ∎

######  Observation 4.2.8.

The proof above relies on an ad-hoc argument which has roots on well-known
Hopf theoretic constructions: bosonization and lifting. Indeed, the algebras
$\widehat{U}^{\pm}$ admit a braided Hopf algebra structure whose
comultiplication is defined by setting the generators $E_{i}\,$, resp.
$F_{i}\,$, to be primitive elements, for all $\,i\in I\,$. Hereafter, by
braided Hopf algebra we mean a Hopf algebra in a braided tensor category; in
the present case, the category is the category
${}_{\widehat{U}^{0}}^{\widehat{U}^{0}}\mathcal{Y\,D}$ of Yetter-Drinfeld
modules over the topological Hopf algebra $\widehat{U}^{0}\,$. Given a Hopf
algebra $B$ in a category of Yetter-Drinfeld modules
${}_{H}^{H}\mathcal{Y\,D}$ over a Hopf algebra $H$, there is a process to
construct a usual Hopf algebra, called the “Radford biproduct” or
bosonization: it is kind of a semidirect product and coproduct, discovered by
Radford and interpreted categorically by Majid. As a vector space, the
bosonization $\,B\\#H\,$ is just the vector space $\,B\otimes H\,$. In our
case, the (completed) tensor product
$\,\widehat{U}^{-}\otimes\widehat{U}^{+}\,$ of the braided Hopf algebras is
again a braided Hopf algebra, and the bosonization
$\,\big{(}\widehat{U}^{-}\otimes\widehat{U}^{+}\big{)}\,\\#\,\widehat{U}^{0}\,$
is a topological, complete Hopf ${\Bbbk[[\hbar]]}$–algebra. By construction,
it can be presented by the generators $\,T\in\mathfrak{h}\,$ and $E_{i}\,$,
$F_{i}$ with $\,i\in I\,$, satisfying all the relations in (4.5) except the
commuting relation between $E_{i}$ and $F_{j}\,$; in fact, the latter now is
replaced simply by $\,\big{[}E_{i}\,,F_{j}\big{]}=0\,$. Here enters into the
picture the process of lifting or deformation: through this process, one
deforms the relations in a specific way, in our case, the element
$\,\big{[}E_{i}\,,F_{j}\big{]}\in\widehat{U}^{-}\otimes\widehat{U}^{+}\,$
becomes skew-primitive in the bosonization
$\,\big{(}\widehat{U}^{-}\otimes\widehat{U}^{+}\big{)}\,\\#\,\widehat{U}^{0}\,$
and one change the relation by setting it equal to the difference between the
group-like elements appearing in the comultiplication, that is
$\;E_{i}\,F_{j}\,-\,F_{j}\,E_{i}\;=\;\delta_{i,j}\,{{\;e^{+\hbar\,T_{i}^{+}}-\,e^{-\hbar\,T_{i}^{-}}\;}\over{\;q_{i}^{+1}-\,q_{i}^{-1}\;}}\;$.

In general, the lifting process can also be described via cocycle
deformations.

All these constructions are described explicitly in [Gar] in the case of
polynomial multiparameter quantum groups. It is worth noting that through
bosonizations and quantum doubles one can generalize triangular decompositions
to more general families of Hopf algebras or quantum groups; this implies
special features in representation theory, see for example [PV].

Eventually, we can prove the “triangular decomposition” for our FoMpQUEAs:

######  Theorem 4.2.9.

(“Triangular Decompositions” in $U^{\,{\mathcal{R}}}_{\\!P,\hskip
0.7pt\hbar}(\hskip 0.5pt\mathfrak{g})$) There exist natural isomorphisms of
topological ${\Bbbk[[\hbar]]}$–algebras

$\halign{\hbox to\displaywidth{$\hfil\displaystyle#\hfil$}\cr
0.0pt{\hfil$\displaystyle U^{\,{\mathcal{R}}}_{\\!P,\hskip 0.7pt\hbar}(\hskip
0.5pt\mathfrak{n}_{\mp})\,\widehat{\otimes}_{\Bbbk[[\hbar]]}\,U_{\hbar}(\mathfrak{h})\,\;\cong\;\,U^{\,{\mathcal{R}}}_{\\!P,\hskip
0.7pt\hbar}(\hskip 0.5pt\mathfrak{b}_{\mp})\quad,\qquad
U_{\hbar}(\mathfrak{h})\,\widehat{\otimes}_{\Bbbk[[\hbar]]}\,U^{\,{\mathcal{R}}}_{\\!P,\hskip
0.7pt\hbar}(\hskip
0.5pt\mathfrak{n}_{\mp})\,\;\cong\;\,U^{\,{\mathcal{R}}}_{\\!P,\hskip
0.7pt\hbar}(\hskip 0.5pt\mathfrak{b}_{\mp})\cr 0.0pt{\hfil$\displaystyle
U^{\,{\mathcal{R}}}_{\\!P,\hskip 0.7pt\hbar}(\hskip
0.5pt\mathfrak{n}_{-})\,\widehat{\otimes}_{\Bbbk[[\hbar]]}\,U_{\hbar}(\mathfrak{h})\,\widehat{\otimes}_{\Bbbk[[\hbar]]}\,U^{\,{\mathcal{R}}}_{\\!P,\hskip
0.7pt\hbar}(\hskip
0.5pt\mathfrak{n}_{+})\,\;\cong\;\,U^{\,{\mathcal{R}}}_{\\!P,\hskip
0.7pt\hbar}(\hskip 0.5pt\mathfrak{g})\cr 0.0pt{\hfil$\displaystyle
U^{\,{\mathcal{R}}}_{\\!P,\hskip 0.7pt\hbar}(\hskip
0.5pt\mathfrak{n}_{+})\,\widehat{\otimes}_{\Bbbk[[\hbar]]}\,U_{\hbar}(\mathfrak{h})\,\widehat{\otimes}_{\Bbbk[[\hbar]]}\,U^{\,{\mathcal{R}}}_{\\!P,\hskip
0.7pt\hbar}(\hskip
0.5pt\mathfrak{n}_{-})\,\;\cong\;\,U^{\,{\mathcal{R}}}_{\\!P,\hskip
0.7pt\hbar}(\hskip 0.5pt\mathfrak{g})\crcr}}}}$

(notation as in Definition 4.2.2) induced by multiplication in
$U^{\,{\mathcal{R}}}_{\\!P,\hskip 0.7pt\hbar}(\hskip 0.5pt\mathfrak{g})\,$.

###### Proof.

This is a direct consequence of Theorem 4.2.7 above. ∎

###  4.3. Hopf structure on FoMpQUEAs

We introduce now on our FoMpQUEAs a structure of topological Hopf algebra.
Another proof of its existence will follow from an alternative construction
(cf. §4.5). To begin with, we explain the link between our “formal” and the
“polynomial” one.

######  Observation 4.3.1.

The usual, formal QUEA $U_{\hbar}(\mathfrak{g})$ by Drinfeld has a
“polynomial” sibling $U_{q}(\mathfrak{g})$ introduced by Jimbo and Lusztig:
the latter is a (Hopf) subalgebra of the former, over the subring
$\,\Bbbk\big{[}\,q\,,q^{-1}\,\big{]}\,$ of ${\Bbbk[[\hbar]]}\,$, with
$\,q^{\pm 1}:=e^{\,\pm\hbar}\,$.

Our notion of formal multiparameter QUEA is explicitly tailored so that this
parallelism extend to the multiparameter setting, linking our formal
multiparameter $U^{\,{\mathcal{R}}}_{\\!P,\hskip 0.7pt\hbar}(\hskip
0.5pt\mathfrak{g})$ with a “polynomial” multiparameter QUEA
$U_{\mathbf{q}}(\mathfrak{g})$ as in [HPR] or [GG1].

To see this, we consider a matrix $P$ of Cartan type, whose associated Cartan
matrix is $A$, and a split realization
$\,{\mathcal{R}}=\big{(}\,\mathfrak{h}\,,\Pi\,,\Pi^{\vee}\,\big{)}\,$ of it:
for this the algebra $U^{\,{\mathcal{R}}}_{\\!P,\hskip 0.7pt\hbar}(\hskip
0.5pt\mathfrak{g})$ is defined and we begin by modifying the presentation of
the latter. First, replace each $F_{i}$ by $\,\dot{F}_{i}:=q_{i}F_{i}\,$ for
all $\,i\in I\,$. Then the fourth relation in (4.5) reads

$E_{i}\,\dot{F}_{j}\,-\,\dot{F}_{j}\,E_{i}\,\;=\;\,\delta_{i,j}\,q_{ii}\,{{\;e^{+\hbar\,T_{i}^{+}}-\,e^{-\hbar\,T_{i}^{-}}\;}\over{\;q_{ii}-\,1\;}}\quad\qquad\forall\;\;i\,,j\in
I\quad$

Second, using the relation $\;{\bigg{(}{\displaystyle{n\atop
k}}\bigg{)}}_{\\!\\!q_{ii}}\\!=\,q_{i}^{\,k\,(n-k)}{\bigg{[}\,{\displaystyle{n\atop
k}}\,\bigg{]}}_{q_{i}}\;$ (cf. §4.1.7) along with the identity
$\;q_{ij}\,q_{ji}=q_{ii}^{\,a_{ij}}\;$ — which follows from the assumption
$\,P+P^{\scriptscriptstyle t}=2\,D\,A\,$ and definitions, see Definition
4.2.1, we can re-write the two last relations in (4.5) as

$\displaystyle\sum\limits_{k=0}^{1-a_{ij}}(-1)^{k}{\left({1-a_{ij}\atop
k}\right)}_{\\!\\!q_{ii}}q_{ii}^{k\choose
2}q_{ij}^{\,k}\,E_{i}^{1-a_{ij}-k}E_{j}E_{i}^{k}\;=\;0\quad\qquad(\,i\neq
j\,)$ $\displaystyle\sum\limits_{k=0}^{1-a_{ij}}(-1)^{k}{\left({1-a_{ij}\atop
k}\right)}_{\\!\\!q_{ii}}q_{ii}^{k\choose
2}q_{ij}^{\,k}\,\dot{F}_{i}^{1-a_{ij}-k}\dot{F}_{j}\dot{F}_{i}^{k}\;=\;0\quad\qquad(\,i\neq
j\,)$

With this reformulation, $U^{\,{\mathcal{R}}}_{\\!P,\hskip 0.7pt\hbar}(\hskip
0.5pt\mathfrak{g})$ has the following, alternative presentation: it is the
unital, associative, topological, $\hbar$–adically complete
${\Bbbk[[\hbar]]}$–algebra generated by the ${\Bbbk[[\hbar]]}$–submodule
$\mathfrak{h}$ and the elements $\,E_{i}\,$, $\dot{F}_{i}\,$ (for all $\,i\in
I\,$), with relations

$\displaystyle T\,E_{j}\,-\,E_{j}\,T\,=\,+\alpha_{j}(T)\,E_{j}\;\;,\qquad
T\,\dot{F}_{j}\,-\,\dot{F}_{j}\,T\;=\;-\alpha_{j}(T)\,\dot{F}_{j}$ (4.6)
$\displaystyle\hskip-13.0ptT^{\prime}\,T^{\prime\prime}\;=\;T^{\prime\prime}\,T^{\prime}\;\;,\;\quad
E_{i}\,\dot{F}_{j}\,-\,\dot{F}_{j}\,E_{i}\;=\;\delta_{i,j}\,q_{ii}\,{{\;e^{+\hbar\,T_{i}^{+}}-\,e^{-\hbar\,T_{i}^{-}}\;}\over{\;q_{ii}-\,1\;}}\quad(\,i,j\in
I\,)$ $\displaystyle\sum\limits_{k=0}^{1-a_{ij}}(-1)^{k}{\left[{1-a_{ij}\atop
k}\right]}_{\\!q_{i}}q_{ij}^{+k/2\,}q_{ji}^{-k/2}\,E_{i}^{1-a_{ij}-k}E_{j}E_{i}^{k}\;=\;0\qquad(\,i\neq
j\,)$ $\displaystyle\sum\limits_{k=0}^{1-a_{ij}}(-1)^{k}{\left[{1-a_{ij}\atop
k}\right]}_{\\!q_{i}}q_{ij}^{+k/2\,}q_{ji}^{-k/2}\,\dot{F}_{i}^{1-a_{ij}-k}\dot{F}_{j}\dot{F}_{i}^{k}\;=\;0\qquad(\,i\neq
j\,)$

for all $\,T,T^{\prime},T^{\prime\prime}\in\mathfrak{h}\,$, $\,i,j\in I\,$.

Now we set $\,q^{\pm 1}:=e^{\,\pm\hbar}\in{\Bbbk[[\hbar]]}\,$ and consider the
$\Bbbk$–subalgebra $\,\Bbbk\big{[}\,q\,,q^{-1}\,\big{]}\,$ of
${\Bbbk[[\hbar]]}\,$, and the elements $\;K_{i}^{\pm
1}:=e^{\pm\hbar\,T_{i}^{+}}\,$, $\;L_{i}^{\pm 1}:=e^{\mp\hbar\,T_{i}^{-}}\;$
($\,i\in I\,$) in $\,U^{\,{\mathcal{R}}}_{\\!P,\hskip 0.7pt\hbar}(\hskip
0.5pt\mathfrak{g})\,$. Then in the scalar extension
$\;\mathbb{U}^{\,{\mathcal{R}}}_{\\!P,\hskip 0.7pt\hbar}(\hskip
0.5pt\mathfrak{g})\,:=\,{\Bbbk(q)}\operatorname{\otimes}\limits_{\Bbbk[q\,,\,q^{\pm
1}]}U^{\,{\mathcal{R}}}_{\\!P,\hskip 0.7pt\hbar}(\hskip 0.5pt\mathfrak{g})\,$,
we slightly abuse notation by writing $\,X:=1\otimes X\,$ for any $\,X\in
U^{\,{\mathcal{R}}}_{\\!P,\hskip 0.7pt\hbar}(\hskip 0.5pt\mathfrak{g})\,$, and
we define $U_{\mathbf{q}}(\mathfrak{g})$ to be the unital
$\Bbbk(q)$–subalgebra of $\mathbb{U}^{\,{\mathcal{R}}}_{\\!P,\hskip
0.7pt\hbar}(\hskip 0.5pt\mathfrak{g})$ generated by $\,{\big{\\{}K_{i}^{\pm
1},L_{i}^{\pm 1},E_{i}\,,\dot{F}_{i}\,\big{\\}}}_{i\in I}\,$. By construction
and by (4.6), we can describe $U_{\mathbf{q}}(\mathfrak{g})$ as being the
unital, associative algebra over ${\Bbbk(q)}$ with generators $\,K_{i}^{\pm
1}\,$, $L_{i}^{\pm 1}\,$, $\,E_{i}\,$, $\dot{F}_{i}\,$ (for all $\,i\in I\,$)
and relations

$\displaystyle K_{i}^{\pm 1}K_{j}^{\pm 1}\,=\,K_{j}^{\pm 1}K_{i}^{\pm
1}\;,\qquad K_{i}^{\pm 1}L_{j}^{\pm 1}\,=\,L_{j}^{\pm 1}K_{i}^{\pm 1}\;,\qquad
L_{i}^{\pm 1}L_{j}^{\pm 1}\,=\,L_{j}^{\pm 1}L_{i}^{\pm 1}$ (4.7)
$\displaystyle K_{i}^{+1}K_{i}^{-1}=\,1\,=\,K_{i}^{-1}K_{i}^{+1}\quad,\qquad
L_{i}^{+1}L_{i}^{-1}=\,1\,=\,L_{i}^{-1}L_{i}^{+1}$ $\displaystyle K_{i}^{\pm
1}\,E_{j}\,K_{i}^{\mp 1}\,=\;q_{i,j}^{\,\pm 1}\,E_{j}\quad,\qquad L_{i}^{\pm
1}\,E_{j}\,L_{i}^{\mp 1}\,=\;q_{j,i}^{\,\mp 1}\,E_{j}$ $\displaystyle
K_{i}^{\pm 1}\,\dot{F}_{j}\,K_{i}^{\mp 1}\,=\;q_{i,j}^{\,\mp
1}\,\dot{F}_{j}\quad,\qquad L_{i}^{\pm 1}\,\dot{F}_{j}\,L_{i}^{\mp
1}\,=\;q_{j,i}^{\,\pm 1}\,\dot{F}_{j}$ $\displaystyle
E_{i}\,\dot{F}_{j}\,-\,\dot{F}_{j}\,E_{i}\;=\;\delta_{i,j}\,q_{ii}\,{{\;K_{i}-\,L_{i}\;}\over{\;q_{ii}-\,1\;}}$
$\displaystyle\sum\limits_{k=0}^{1-a_{ij}}(-1)^{k}{\left({1-a_{ij}\atop
k}\right)}_{\\!\\!q_{ii}}q_{ii}^{k\choose
2}q_{ij}^{k}\,E_{i}^{1-a_{ij}-k}E_{j}E_{i}^{k}\;=\;0\quad\qquad(\,i\neq j\,)$
$\displaystyle\sum\limits_{k=0}^{1-a_{ij}}(-1)^{k}{\left({1-a_{ij}\atop
k}\right)}_{\\!\\!q_{ii}}q_{ii}^{k\choose
2}q_{ij}^{k}\,\dot{F}_{i}^{1-a_{ij}-k}\dot{F}_{j}\dot{F}_{i}^{k}\;=\;0\quad\qquad(\,i\neq
j\,)$

Next we consider the scalar extension
$\;\mathbb{U}_{\hbar}(\mathfrak{h}):={\Bbbk(q)}\operatorname{\otimes}\limits_{\Bbbk[q\,,\,q^{\pm
1}]}U_{\hbar}(\hskip 0.0pt\mathfrak{h})\;$ of $\,U_{\hbar}(\hskip
0.0pt\mathfrak{h})\,$ — cf. Definition 4.2.2(b) — which clearly embeds as a
“Cartan subalgebra” into $\;\mathbb{U}^{\,{\mathcal{R}}}_{\\!P,\hskip
0.7pt\hbar}(\hskip
0.5pt\mathfrak{g})\,:=\,{\Bbbk(q)}\operatorname{\otimes}\limits_{\Bbbk[q\,,\,q^{\pm
1}]}U^{\,{\mathcal{R}}}_{\\!P,\hskip 0.7pt\hbar}(\hskip 0.5pt\mathfrak{g})\;$;
let also $U_{\mathbf{q}}(\mathfrak{h})$ be the ${\Bbbk(q)}$–subalgebra —
inside $U_{\mathbf{q}}(\mathfrak{g})\,$, $U_{\hbar}(\mathfrak{h})$ and
$\mathbb{U}_{\hbar}(\mathfrak{h})$ — generated by all the $K_{i}^{\pm 1}$’s
and all the $L_{i}^{\pm 1}$’s. Note that inside
$\mathbb{U}_{\hbar}(\mathfrak{h})$ we have $\;\pm
T_{i}^{+}=\hbar^{-1}\log\big{(}K_{i}^{\pm 1}\big{)}\;$ and $\;\pm
T_{i}^{-}=\hbar^{-1}\log\big{(}L_{i}^{\pm 1}\big{)}\;$. By construction, both
$\mathbb{U}_{\hbar}(\mathfrak{h})$ and $U_{\mathbf{q}}(\mathfrak{g})$ are
modules (on the right and on the left, respectively) for the commutative
algebra $U_{\mathbf{q}}(\mathfrak{h})\,$; therefore, the
$U_{\mathbf{q}}(\mathfrak{h})$–module
$\,\mathbb{U}_{\hbar}(\mathfrak{h})\hskip-3.0pt\mathop{\otimes}\limits_{U_{\mathbf{q}}(\mathfrak{h})}\hskip-3.0ptU_{\mathbf{q}}(\mathfrak{g})\,$
is well-defined. Finally, the $\hbar$–adic completion of the latter actually
identifies with its closure inside
$\,\mathbb{U}^{\,{\mathcal{R}}}_{\\!P,\hskip 0.7pt\hbar}(\hskip
0.5pt\mathfrak{g})\,$, which is exactly all of
$\,\mathbb{U}^{\,{\mathcal{R}}}_{\\!P,\hskip 0.7pt\hbar}(\hskip
0.5pt\mathfrak{g})\,$: in a nutshell, we have a (complete) tensor product
factorization $\;\mathbb{U}^{\,{\mathcal{R}}}_{\\!P,\hskip 0.7pt\hbar}(\hskip
0.5pt\mathfrak{g})=\mathbb{U}_{\hbar}(\mathfrak{h})\hskip-2.0pt\mathop{\widehat{\otimes}}\limits_{U_{\mathbf{q}}(\mathfrak{h})}\hskip-2.0ptU_{\mathbf{q}}(\mathfrak{g})\;$.

The previous observation is our bridge to achieve the key point about the
notion of formal multiparameter QUEA, that is the following:

######  Theorem 4.3.2.

Every FoMpQUEA $U^{\,{\mathcal{R}}}_{\\!P,\hskip 0.7pt\hbar}(\hskip
0.5pt\mathfrak{g})$ as in Definition 4.2.2 bears a structure of topological
Hopf algebra over ${\Bbbk[[\hbar]]}$ — with coproduct taking values into the
$\hbar$–adically completed tensor product $\;U^{\,{\mathcal{R}}}_{\\!P,\hskip
0.7pt\hbar}(\hskip
0.5pt\mathfrak{g})\\!\mathop{\widehat{\otimes}}\limits_{\Bbbk[[\hbar]]}\\!U^{\,{\mathcal{R}}}_{\\!P,\hskip
0.7pt\hbar}(\hskip 0.5pt\mathfrak{g})\;$ — given by
($\;\forall\;T\in\mathfrak{h}\,$, $\,\ell\in I$)

$\begin{aligned} \Delta\big{(}E_{\ell}\big{)}\,=\,E_{\ell}\otimes
1\,+\,e^{\hbar\,T_{\ell}^{+}}\otimes E_{\ell}\\\
\Delta\big{(}T\big{)}\,=\,T\otimes 1\,+\,1\otimes T\hskip 25.0pt\\\
\Delta\big{(}F_{\ell}\big{)}\,=\,F_{\ell}\otimes
e^{-\hbar\,T_{\ell}^{-}}\,+\,1\otimes F_{\ell}\end{aligned}\qquad$ (4.8)
$\displaystyle\epsilon\big{(}E_{\ell}\big{)}\,=\,0\;\;\;,\qquad$
$\displaystyle\;\quad\epsilon\big{(}T\big{)}\,=\,0\;\;\;,$
$\displaystyle\;\qquad\epsilon\big{(}F_{\ell}\big{)}\,=\,0\;\;\;\;$ (4.9)
$\displaystyle\SS\big{(}E_{\ell}\big{)}\,=\,-e^{-\hbar\,T_{\ell}^{+}}E_{\ell}\;\;,$
$\displaystyle\;\;\;\SS\big{(}T\big{)}\,=\,-T\;\;,$
$\displaystyle\;\;\;\SS\big{(}F_{\ell}\big{)}\,=\,-F_{\ell}\,e^{+\hbar\,T_{\ell}^{-}}\qquad$
(4.10)

###### Proof.

We provide hereafter two proofs; a third, independent one will follow from
another approach, that is detailed in §4.5 later on. $\underline{\text{First
Proof}}$: Via a direct approach, the proof is a matter of computation. First,
in the free, topological, $\hbar$–adically complete, unital, associative
${\Bbbk[[\hbar]]}$–algebra $\mathcal{F}_{\mathcal{R}}$ generated by the
${\Bbbk[[\hbar]]}$–submodule $\mathfrak{h}$ together with the $E_{i}$’s and
the $F_{i}$’s, the formulas (4.8) and (4.9) define a structure of
(topological) bialgebra. Then one has to check that the two-sided ideal
$\mathcal{I}_{\mathcal{R}}$ in $\mathcal{F}_{\mathcal{R}}$ generated by
relations (4.5) is a bi-ideal for that bialgebra. Second, one has to check
that the map $\;\SS:U^{\,{\mathcal{R}}}_{\\!P,\hskip 0.7pt\hbar}(\hskip
0.5pt\mathfrak{g})\longrightarrow U^{\,{\mathcal{R}}}_{\\!P,\hskip
0.7pt\hbar}(\hskip 0.5pt\mathfrak{g})^{\operatorname{op}}\;$ defined on
$\;\mathcal{F}_{\mathcal{R}}\Big{/}\mathcal{I}_{\mathcal{R}}=:U^{\,{\mathcal{R}}}_{\\!P,\hskip
0.7pt\hbar}(\hskip 0.5pt\mathfrak{g})\;$ by the second line in (4.10) is an
algebra anti-morphism with the “right” properties for the antipode map on the
generators. In all cases, computations are the same as for [CP, Definition-
Proposition 6.5.1] which treats Drinfeld’s special case with $\mathfrak{h}$ of
minimal rank. It is worth stressing, though, a key feature of this generalized
result: namely, the assumption that $P$ be of Cartan type is what one uses to
prove that the quantum Serre’s relations — i.e., the last two relations in
(4.5) — are mapped into
$\,\mathcal{F}_{\\!{\mathcal{R}}}\,\widehat{\otimes}\,\mathcal{I}_{\mathcal{R}}+\mathcal{I}_{\mathcal{R}}\,\widehat{\otimes}\,\mathcal{F}_{\mathcal{R}}\,$
— where $\,\widehat{\otimes}\,$ denotes the $\hbar$–adic completion of the
algebraic tensor product — by the given coproduct on
$\mathcal{F}_{\mathcal{R}}\,$. Actually, one shows that the elements $E_{i,j}$
and $F_{i,j}$ represented by the left hand side of these equalities are skew-
primitives, namely $\;\Delta(E_{i,j})=E_{i,j}\otimes
1+e^{+(1-a_{ij})\,\hbar\,T_{i}^{+}+\hbar\,T_{j}^{+}}\otimes E_{i,j}\;$ and
similarly $\;\Delta(F_{i,j})=F_{i,j}\otimes
e^{-(1-a_{ij})\,\hbar\,T_{i}^{-}-\hbar\,T_{j}^{-}}+1\otimes F_{i,j}\;$.
$\underline{\text{Second Proof}}$: This alternative method goes through an
indirect approach. First of all, we assume the realization ${\mathcal{R}}$ to
be split. In this case, Observation 4.3.1 provides a factorization
$\;\mathbb{U}^{\,{\mathcal{R}}}_{\\!P,\hskip 0.7pt\hbar}(\hskip
0.5pt\mathfrak{g})\,:=\,{\Bbbk(q)}\operatorname{\otimes}\limits_{\Bbbk[q\,,\,q^{\pm
1}]}U^{\,{\mathcal{R}}}_{\\!P,\hskip 0.7pt\hbar}(\hskip
0.5pt\mathfrak{g})\,=\,\mathbb{U}_{\hbar}(\mathfrak{h})\hskip-2.0pt\mathop{\widehat{\otimes}}\limits_{U_{\mathbf{q}}(\mathfrak{h})}\hskip-2.0ptU_{\mathbf{q}}(\mathfrak{g})\;$.

Due to its presentation in Observation 4.3.1, our
$\,U_{\mathbf{q}}(\mathfrak{g})\,$ is a “multiparameter quantum group” in the
sense of [HPR] — cf. also [GG1], where such an example of multiparameter
quantum group is referred to as being “rational”, in that each $q_{ij}$ is a
power of a single, common parameter $q\,$. The key point then is that any such
(“polynomial”) multiparameter quantum group $\,U_{\mathbf{q}}(\mathfrak{g})\,$
has a specific Hopf algebra structure — cf. [HPR] and [Gar] — given (for all
$\,\ell\in I\,$) by

$\quad\begin{matrix}\Delta\big{(}E_{\ell}\big{)}\,=\,E_{\ell}\otimes
1\,+\,K_{\ell}\otimes
E_{\ell}\;\;,&\quad\epsilon\big{(}E_{\ell}\big{)}\,=\,0\;\;,\quad&\SS\big{(}E_{\ell}\big{)}\,=\,-K_{\ell}^{-1}E_{\ell}\\\
\Delta\big{(}K_{\ell}^{\pm 1}\big{)}\,=\,K_{\ell}^{\pm 1}\otimes K_{\ell}^{\pm
1}\;\;,&\quad\epsilon\big{(}K_{\ell}^{\pm
1}\big{)}\,=\,1\;\;,\quad&\SS\big{(}K_{\ell}^{\pm 1}\big{)}\,=\,K_{\ell}^{\mp
1}\\\ \Delta\big{(}L_{\ell}^{\pm 1}\big{)}\,=\,L_{\ell}^{\pm 1}\otimes
L_{\ell}^{\pm 1}\;\;,&\quad\epsilon\big{(}L_{\ell}^{\pm
1}\big{)}\,=\,1\;\;,\quad&\SS\big{(}L_{\ell}^{\pm 1}\big{)}\,=\,L_{\ell}^{\mp
1}\\\ \Delta\big{(}\dot{F}_{\ell}\big{)}\,=\,\dot{F}_{\ell}\otimes
L_{\ell}\,+\,1\otimes\dot{F}_{\ell}\;\;,&\quad\epsilon\big{(}\dot{F}_{\ell}\big{)}\,=\,0\;\;,\quad&\SS\big{(}\dot{F}_{\ell}\big{)}\,=\,-\dot{F}_{\ell}\,L_{\ell}^{-1}\end{matrix}$
(4.11)

Let now $\,\widetilde{U}_{\mathbf{q}}(\mathfrak{g})\,$ be the
$\Bbbk\big{[}q\,,q^{-1}\big{]}$–subalgebra of
$\,U_{\mathbf{q}}(\mathfrak{g})\,$ — hence of
$U^{\,{\mathcal{R}}}_{\\!P,\hskip 0.7pt\hbar}(\hskip 0.5pt\mathfrak{g})$ —
generated by $\,{\big{\\{}E_{i}\,,K_{i}^{\pm 1},L_{i}^{\pm
1},\dot{F}_{i}\big{\\}}}_{i\in I}\,$: it is $\hbar$–adic dense in
$\,U_{\mathbf{q}}(\mathfrak{g})\,$, and restricting the coproduct $\Delta$ to
$\,\widetilde{U}_{\mathbf{q}}(\mathfrak{g})\,$ yields
$\;\Delta\big{(}\widetilde{U}_{\mathbf{q}}(\mathfrak{g})\big{)}\,\subseteq\,\widetilde{U}_{\mathbf{q}}(\mathfrak{g})\hskip-3.0pt\mathop{\otimes}\limits_{\Bbbk[q,q^{-1}]}\hskip-3.0pt\widetilde{U}_{\mathbf{q}}(\mathfrak{g})\;$.
Therefore, since $\,K_{i}=\exp\big{(}\\!+\hbar\,T_{i}^{+}\big{)}\,$ and
$\,L_{i}=\exp\big{(}\\!-\hbar\,T_{i}^{-}\big{)}\,$, there exists one and only
one way to extend (continuously)
$\Delta{\Big{|}}_{\widetilde{U}_{\mathbf{q}}(\mathfrak{g})}$ to
$U^{\,{\mathcal{R}}}_{\\!P,\hskip 0.7pt\hbar}(\hskip 0.5pt\mathfrak{g})$,
which gives a map

$\Delta\,:\,U^{\,{\mathcal{R}}}_{\\!P,\hskip 0.7pt\hbar}(\hskip
0.5pt\mathfrak{g})\relbar\joinrel\relbar\joinrel\longrightarrow
U^{\,{\mathcal{R}}}_{\\!P,\hskip 0.7pt\hbar}(\hskip 0.5pt\mathfrak{g})\hskip
3.0pt\widehat{\otimes}\hskip 3.0ptU^{\,{\mathcal{R}}}_{\\!P,\hskip
0.7pt\hbar}(\hskip 0.5pt\mathfrak{g})$ (4.12)

described by (4.11) plus the additional constraint that all the
$T_{i}^{\pm}$’s be primitive.

Since the original map $\Delta$ on $U_{\mathbf{q}}(\mathfrak{g})$ obey the
axioms of a coproduct, the same holds true for the map in (4.12) as well —
though in a topological framework. With similar arguments, we deal with counit
and antipode map, so that we end up with a (topological) Hopf structure for
$\,U^{\,{\mathcal{R}}}_{\\!P,\hskip 0.7pt\hbar}(\hskip 0.5pt\mathfrak{g})\,$,
uniquely induced from the one on $U_{\mathbf{q}}(\mathfrak{g})\,$. Tracking
the whole construction, we eventually see that such a structure is described
on generators by (4.8), (4.9) and (4.10), q.e.d. Now we consider the case of a
realization
$\;{\mathcal{R}}\,:=\,\big{(}\,\mathfrak{h}\,,\Pi\,,\Pi^{\vee}\,\big{)}\;$ of
any kind. By Lemma 2.1.8, we can then pick a split realization (of the same
matrix $P$ as ${\mathcal{R}}\,$), say
$\;\dot{{\mathcal{R}}}\,:=\,\big{(}\,\dot{\mathfrak{h}}\,,\dot{\Pi}\,,{\dot{\Pi}}^{\vee}\,\big{)}\;$
and an epimorphism of realizations
$\;\underline{\pi}:\dot{{\mathcal{R}}}\relbar\joinrel\twoheadrightarrow{\mathcal{R}}\;$.
By functoriality (cf. Proposition 4.2.3), such a $\underline{\pi}$ induces an
epimorphism of FoMpQUEAs
$\;U_{\underline{\pi}}:U^{\,\dot{{\mathcal{R}}}}_{\\!P,\hbar}(\mathfrak{g})\relbar\joinrel\twoheadrightarrow
U^{\,{\mathcal{R}}}_{\\!P,\hskip 0.7pt\hbar}(\hskip 0.5pt\mathfrak{g})\;$,
whose kernel $\operatorname{Ker}\big{(}U_{\underline{\pi}}\,\big{)}$ is
generated by $\operatorname{Ker}(\pi)\,$, and the latter is central in
$U^{\,\dot{{\mathcal{R}}}}_{\\!P,\hbar}(\mathfrak{g})\,$; moreover, every
element in $\operatorname{Ker}(\pi)$ is primitive. Therefore
$\operatorname{Ker}\big{(}U_{\underline{\pi}}\,\big{)}$ is indeed a Hopf ideal
in the Hopf algebra $U^{\,\dot{{\mathcal{R}}}}_{\\!P,\hbar}(\mathfrak{g})\,$,
hence $U^{\,{\mathcal{R}}}_{\\!P,\hbar}(\mathfrak{g})$ automatically inherits
via $U_{\underline{\pi}}$ a quotient Hopf algebra structure from
$U^{\,\dot{{\mathcal{R}}}}_{\\!P,\hbar}(\mathfrak{g})\,$, which is again
described by the formulas in the statement, q.e.d. ∎

The following is now immediate:

######  Corollary 4.3.3.

The Cartan subalgebra $U_{\hbar}(\mathfrak{h})$ and the Borel subalgebras
$U^{\,{\mathcal{R}}}_{\\!P,\hskip 0.7pt\hbar}(\hskip 0.5pt\mathfrak{b}_{+})$
and $U^{\,{\mathcal{R}}}_{\\!P,\hskip 0.7pt\hbar}(\hskip
0.5pt\mathfrak{b}_{-})$ are actually (topological) Hopf subalgebras of
$\,U^{\,{\mathcal{R}}}_{\\!P,\hskip 0.7pt\hbar}(\hskip 0.5pt\mathfrak{g})\,$,
their Hopf structure being described again via formulas (4.8) and (4.10).

In addition, when ${\mathcal{R}}$ is split we have that
$\;U^{\,{\mathcal{R}}}_{\\!P,\hskip 0.7pt\hbar}(\hskip
0.5pt\mathfrak{b}_{\pm})\,\cong\,U_{\hbar}(\mathfrak{h}_{\pm})\hskip-2.0pt\mathop{\widehat{\;\ltimes\,}}\limits_{U_{\mathbf{q}}(\mathfrak{h}_{\pm})}\hskip-1.0ptU_{\mathbf{q}}(\mathfrak{b}_{\pm})\;$,
where $U_{\mathbf{q}}(\mathfrak{b}_{\pm})\,$ is the multiparameter quantum
group corresponding to the Borel subalgebras.

The next result follows at once from the second proof of Theorem 4.3.2 above.

######  Proposition 4.3.4.

Let
$\,\underline{\phi}:{\mathcal{R}}^{\prime}\\!\relbar\joinrel\relbar\joinrel\longrightarrow\\!{\mathcal{R}}^{\prime\prime}\,$
be a morphism between realizations of a same matrix $\,P\in
M_{n}\big{(}{\Bbbk[[\hbar]]}\big{)}\,$. Then the morphism of unital
topological $\,{\Bbbk[[\hbar]]}$–algebras
$\,U_{\underline{\phi}}:U^{\,{\mathcal{R}}^{\prime}}_{\\!P,\hbar}(\mathfrak{g})\relbar\joinrel\relbar\joinrel\longrightarrow
U^{\,{\mathcal{R}}^{\prime\prime}}_{\\!P,\hbar}(\mathfrak{g})\,$ granted by
Proposition 4.2.3 is indeed a morphism of (unital topological) Hopf
$\,{\Bbbk[[\hbar]]}$–algebras. If we set
$\;\mathfrak{k}:=\operatorname{Ker}(\phi)\,$, then
$\,U_{\hbar}(\mathfrak{k})\,$ is a unital, $\hbar$–adically complete
$\,{\Bbbk[[\hbar]]}$–subalgebra of
$U^{\,{\mathcal{R}}^{\prime}}_{\\!P,\hbar}(\mathfrak{g})$ which is a central
Hopf subalgebra, isomorphic to a quantum Cartan (in the sense of Definition
4.2.2(b)), and
$\;\operatorname{Ker}(U_{\underline{\phi}})=U^{\,{\mathcal{R}}^{\prime}}_{\\!P,\hbar}(\mathfrak{g})\,{U_{\hbar}(\mathfrak{k})}^{+}\,$
— where $\,{U_{\hbar}(\mathfrak{k})}^{+}\,$ is the augmentation ideal of
$\,U_{\hbar}(\mathfrak{k})\,$.

In particular, if $\,U_{\underline{\phi}}$ is an epimorphism, then
$\;U^{\,{\mathcal{R}}^{\prime\prime}}_{\\!P,\hbar}(\mathfrak{g})\,\cong\,U^{\,{\mathcal{R}}^{\prime}}_{\\!P,\hbar}(\mathfrak{g})\Big{/}U^{\,{\mathcal{R}}^{\prime}}_{\\!P,\hbar}(\mathfrak{g})\,{U_{\hbar}(\mathfrak{k})}^{+}\;$.

######  Example 4.3.5.

Fix $\,P:=DA\,$, $\,r:=\operatorname{rk}\\!\big{(}DA\big{)}\,$ and let
$\,\hat{{\mathcal{R}}}:=\big{(}\,\hat{\mathfrak{h}}\,,\hat{\Pi}\,,\hat{\Pi}^{\vee}\big{)}\,$
and $\,{\mathcal{R}}:=\big{(}\,\mathfrak{h}\,,\Pi\,,\Pi^{\vee}\big{)}\,$ be
realizations of $\,DA\,$, where $\hat{{\mathcal{R}}}$ is straight and split
with $\,\operatorname{rk}\\!\big{(}\hat{\mathfrak{h}}\big{)}=2\,(2\,n-r)\,$
while ${\mathcal{R}}$ is straight and small with
$\,\operatorname{rk}(\mathfrak{h})=2\,n-r\,$; more precisely, we assume
$\,T_{i}^{+}=T_{i}^{-}\,$ in ${\mathcal{R}}\,$, for all $\,i\in I\,$. With
this setup, $U^{\,{\mathcal{R}}}_{\\!DA,\hbar}(\mathfrak{g})$ is the usual
Drinfeld’s QUEA $U_{\hbar}\big{(}\mathfrak{g}_{{}_{A}}\big{)}$ for the Kac-
Moody algebra $\mathfrak{g}_{{}_{A}}$ associated with the Cartan matrix $A$ as
in §2.1.1; in particular, its semiclassical limit is $U(\mathfrak{g})\,$.
Instead, $U^{\,\hat{{\mathcal{R}}}}_{\\!DA,\hbar}(\mathfrak{g})$ has
semiclassical limit $U(\mathfrak{g}^{\raise-1.0pt\hbox{${\scriptscriptstyle
D}$}})\,$, with $\mathfrak{g}^{\raise-1.0pt\hbox{${\scriptscriptstyle D}$}}$
the Manin double of $\,\mathfrak{g}=\mathfrak{g}_{{}_{A}}\,$ (cf. §2.1.1).
Now, there exists a (non-unique, if $\,r\lneqq n\,$) epimorphism
$\,\underline{\phi}:\hat{{\mathcal{R}}}\relbar\joinrel\relbar\joinrel\twoheadrightarrow{\mathcal{R}}\,$
such that $\,\phi\big{(}\hat{T}_{i}^{\pm}\big{)}=T_{i}^{\pm}\,$ ($\,i\in
I\,$); then
$\,\mathfrak{z}:=\operatorname{Ker}(\phi)\subseteq\bigcap\limits_{j\in
I}\operatorname{Ker}(\hat{\alpha}_{j})\,$. Since every element of
$\hat{\mathfrak{h}}$ is primitive inside
$U^{\,\hat{{\mathcal{R}}}}_{\\!DA,\hbar}(\mathfrak{g})\,$, the subalgebra
$U^{\,\hat{{\mathcal{R}}}}_{\hbar}(\mathfrak{z})$ of
$U^{\,\hat{{\mathcal{R}}}}_{\\!DA,\hbar}(\mathfrak{g})$ generated by
$\mathfrak{z}$ is indeed a Hopf subalgebra; moreover, it is central in
$U^{\,\hat{{\mathcal{R}}}}_{\\!DA,\hbar}(\mathfrak{g})$ because
$\,\mathfrak{z}\subseteq\bigcap\limits_{j\in
I}\operatorname{Ker}(\hat{\alpha}_{j})\,$. Also,
$U^{\,\hat{{\mathcal{R}}}}_{\hbar}(\mathfrak{z})$ is the $\hbar$–adic
completion of the polynomial ${\Bbbk[[\hbar]]}$–algebra over
$\mathfrak{z}^{*}$, so we can loosely think of it as being a “quantum Cartan
algebra” of “rank” $\,2\,n-r\,$. By Proposition 4.3.4, $\,\underline{\phi}\,$
yields a (Hopf) epimorphism
$\,U_{\underline{\phi}}:U^{\,\hat{{\mathcal{R}}}}_{\\!DA,\hbar}(\mathfrak{g})\relbar\joinrel\twoheadrightarrow
U^{\,{\mathcal{R}}}_{\\!DA,\hbar}(\mathfrak{g})\,$; by construction, the
kernel of latter is the two-sided ideal generated by $\mathfrak{z}\,$, that is

$\operatorname{Ker}\big{(}U_{\underline{\phi}}\big{)}\;=\;U^{\,\hat{{\mathcal{R}}}}_{\\!DA,\hbar}(\mathfrak{g})\,U^{\,\hat{{\mathcal{R}}}}_{\hbar}{(\mathfrak{z})}^{+}\,U^{\,\hat{{\mathcal{R}}}}_{\\!DA,\hbar}(\mathfrak{g})\;=\;U^{\,\hat{{\mathcal{R}}}}_{\\!DA,\hbar}(\mathfrak{g})\,U^{\,\hat{{\mathcal{R}}}}_{\hbar}{(\mathfrak{z})}^{+}\,=\;U^{\,\hat{{\mathcal{R}}}}_{\hbar}{(\mathfrak{z})}^{+}\,U^{\,\hat{{\mathcal{R}}}}_{\\!DA,\hbar}(\mathfrak{g}),$

and we have that
$\;U^{\,{\mathcal{R}}}_{\\!DA,\hbar}(\mathfrak{g})\,\cong\,U^{\,\hat{{\mathcal{R}}}}_{\\!DA,\hbar}(\mathfrak{g})\Big{/}U^{\,\hat{{\mathcal{R}}}}_{\\!DA,\hbar}(\mathfrak{g})\,{U^{\,\hat{{\mathcal{R}}}}_{\hbar}(\mathfrak{z})}^{+}\;$.

Finally, if we deal instead with ${\mathcal{R}}$ small and
$\hat{{\mathcal{R}}}$ split such that
$\,\operatorname{rk}\big{(}\hat{\mathfrak{h}}\big{)}=2\,n\,$ and
$\,\operatorname{rk}(\mathfrak{h})=n\,$, then
$\,\mathfrak{g}=\mathfrak{g}_{{}_{A}}\,$ has to be replaced by the derived
algebra $\mathfrak{g}^{\prime}$ associated with $A\,$, and
$\mathfrak{g}^{\raise-1.0pt\hbox{${\scriptscriptstyle D}$}}$ by the Manin
double of $\mathfrak{g}^{\prime}$. The previous analysis then works again.

######  Remark 4.3.6.

Let us now take any matrix $P$ (of Cartan type), a realization ${\mathcal{R}}$
of it that is minimal and small with
$\,\operatorname{rk}(\mathfrak{h})=2\,n-r\,$ — with
$\,r:=\operatorname{rk}\\!\big{(}P+P^{\,\scriptscriptstyle T}\big{)}\,$ — and
the associated FoMpQUEA $U^{\,{\mathcal{R}}}_{\\!P,\hskip 0.7pt\hbar}(\hskip
0.5pt\mathfrak{g})\,$; then we can still find another realization
$\dot{{\mathcal{R}}}$ of $P$ that is split with
$\,\operatorname{rk}(\mathfrak{h})=2\,(2\,n-r)\,$ and an epimorphism of
realizations
$\,\underline{\pi}:\dot{{\mathcal{R}}}\relbar\joinrel\relbar\joinrel\twoheadrightarrow{\mathcal{R}}\,$
so that $\,\mathfrak{z}_{\pi}:=\operatorname{Ker}(\pi)\,$ is again free of
rank $\,2\,n-r\,$ — see Lemma 2.1.8 and its proof. Then the previous analysis
— that was based upon
$\,\phi:\hat{{\mathcal{R}}}\relbar\joinrel\relbar\joinrel\twoheadrightarrow{\mathcal{R}}\,$
and
$\,U_{\underline{\phi}}:U^{\,\hat{{\mathcal{R}}}}_{\\!DA,\hbar}(\mathfrak{g})\relbar\joinrel\relbar\joinrel\twoheadrightarrow
U^{\,{\mathcal{R}}}_{\\!DA,\hbar}(\mathfrak{g})\,$ — can be repeated now, step
by step, basing instead upon
$\,\underline{\pi}:\dot{{\mathcal{R}}}\relbar\joinrel\relbar\joinrel\twoheadrightarrow{\mathcal{R}}\,$
and the associated epimorphism
$\,U_{\underline{\pi}}:U^{\,\dot{{\mathcal{R}}}}_{\\!P,\hbar}(\mathfrak{g})\relbar\joinrel\relbar\joinrel\twoheadrightarrow
U^{\,{\mathcal{R}}}_{\\!P,\hbar}(\mathfrak{g})\,$ of FoMpQUEAs: this leads to
the sequence of Hopf algebra maps

$U^{\,\dot{{\mathcal{R}}}}_{\hbar}(\mathfrak{z}_{\pi})\;\relbar\joinrel\relbar\joinrel\longrightarrow\;U^{\,\dot{{\mathcal{R}}}}_{\\!P,\hbar}(\mathfrak{g})\;{\buildrel
U_{\underline{\pi}}\over{\relbar\joinrel\relbar\joinrel\relbar\joinrel\longrightarrow}}\;U^{\,{\mathcal{R}}}_{\\!P,\hbar}(\mathfrak{g})$
(4.13)

where $U^{\,\dot{{\mathcal{R}}}}_{\hbar}(\mathfrak{z}_{\pi})$ is the (central)
subalgebra of $\,U^{\,\dot{{\mathcal{R}}}}_{\\!P,\hbar}(\mathfrak{g})\,$
generated by $\mathfrak{z}_{\pi}\,$, which is again a “quantum Cartan algebra”
of “rank” $\,2\,n-r\,$. Again, we obtain that

$U^{\,{\mathcal{R}}}_{\\!P,\hbar}(\mathfrak{g})\,\cong\,U^{\,\dot{{\mathcal{R}}}}_{\\!P,\hbar}(\mathfrak{g})\Big{/}U^{\,\dot{{\mathcal{R}}}}_{\\!P,\hbar}(\mathfrak{g})\,{U^{\,\dot{{\mathcal{R}}}}_{\hbar}(\mathfrak{z}_{\pi})}^{+}$

Therefore, the situation in general is much similar to what happens in the
special, “standard” case of $\,P=DA\,$; what does actually change, indeed, is
the explicit description of $\mathfrak{z}_{\pi}$ — with respect to that of
$\mathfrak{z}\,$, that was quite clear — hence of the “quantum Cartan algebra
of rank $\,2\,n-r\,$”
$U^{\,\dot{{\mathcal{R}}}}_{\hbar}(\mathfrak{z}_{\pi})\,$.

######  4.3.7.

The case of split (and) minimal FoMpQUEAs. We consider now the special case of
a FoMpQUEA $U^{\,{\mathcal{R}}}_{\\!P,\hskip 0.7pt\hbar}(\hskip
0.5pt\mathfrak{g})$ — as defined in Definition 4.2.2 — for which the
realization ${\mathcal{R}}$ is split and minimal — in short, split minimal. In
this case, it follows by definition that $U^{\,{\mathcal{R}}}_{\\!P,\hskip
0.7pt\hbar}(\hskip 0.5pt\mathfrak{g})$ can be described as follows: it is the
unital, associative, topological, $\hbar$–adically complete algebra over
${\Bbbk[[\hbar]]}$ with generators $\,E_{i}\,$, $T_{i}^{+}$, $T_{i}^{-}$,
$F_{i}\,$ (for all $\,i\in I\,$) and relations

$\displaystyle
T_{i}^{+}\,E_{j}\,-\,E_{j}\,T_{i}^{+}\;=\;+p_{i,j}\,E_{j}\quad,\qquad
T_{i}^{-}\,E_{j}\,-\,E_{j}\,T_{i}^{-}\;=\;+p_{j,i}\,E_{j}$ $\displaystyle
T_{i}^{+}\,F_{j}\,-\,F_{j}\,T_{i}^{+}\;=\;-p_{i,j}\,F_{j}\quad,\qquad
T_{i}^{-}\,F_{j}\,-\,F_{j}\,T_{i}^{-}\;=\;-p_{j,i}\,F_{j}$ $\displaystyle
T_{i}^{\pm}\,T_{j}^{\pm}\,=\,T_{j}^{\pm}\,T_{i}^{\pm}\;\,,\\!\quad
E_{i}\,F_{j}\,-\,F_{j}\,E_{i}\;=\;\delta_{i,j}\,{{\;e^{+\hbar\,T_{i}^{+}}-\,e^{-\hbar\,T_{i}^{-}}\;}\over{\;q_{i}^{+1}-\,q_{i}^{-1}\;}}\;\;,\\!\quad
T_{i}^{\pm}\,T_{j}^{\mp}\,=\,T_{j}^{\mp}\,T_{i}^{\pm}$
$\displaystyle\sum\limits_{k=0}^{1-a_{ij}}(-1)^{k}{\left[{1-a_{ij}\atop
k}\right]}_{\\!q_{i}}q_{ij}^{+k/2\,}q_{ji}^{-k/2}\,X_{i}^{1-a_{ij}-k}X_{j}X_{i}^{k}\;=\;0\qquad\forall\;\;X\in\\{E\,,F\\}\quad(\,i\neq
j\,)$

and bearing the (topological) Hopf ${\Bbbk[[\hbar]]}$–algebra structure given
(for all $\,\ell\in I\,$) by

$\halign{\hbox to\displaywidth{$\hfil\displaystyle#\hfil$}\cr
0.0pt{\hfil$\displaystyle\Delta\big{(}E_{\ell}\big{)}\,=\,E_{\ell}\otimes
1\,+\,e^{+\hbar\,T_{\ell}^{+}}\otimes
E_{\ell}\;\;,\qquad\epsilon\big{(}E_{\ell}\big{)}\,=\,0\;\;,\qquad\SS\big{(}E_{\ell}\big{)}\,=\,-e^{-\hbar\,T_{\ell}^{+}}E_{\ell}\cr
0.0pt{\hfil$\displaystyle\Delta\big{(}T^{\pm}_{\ell}\big{)}\,=\,T^{\pm}_{\ell}\otimes
1\,+\,1\otimes
T^{\pm}_{\ell}\;\;,\;\;\qquad\epsilon\big{(}T^{\pm}_{\ell}\big{)}\,=\,0\;\;,\;\;\qquad\SS\big{(}T^{\pm}_{\ell}\big{)}\,=\,-T^{\pm}_{\ell}\;\;\;\cr
0.0pt{\hfil$\displaystyle\Delta\big{(}F_{\ell}\big{)}\,=\,F_{\ell}\otimes
e^{-\hbar\,T_{\ell}^{-}}\,+\,1\otimes
F_{\ell}\;\;,\qquad\epsilon\big{(}F_{\ell}\big{)}\,=\,0\;\;,\qquad\SS\big{(}F_{\ell}\big{)}\,=\,-F_{\ell}\,e^{+\hbar\,T_{\ell}^{-}}\crcr}}}}$

Note then that in this case $U^{\,{\mathcal{R}}}_{\\!P,\hskip
0.7pt\hbar}(\hskip 0.5pt\mathfrak{g})$ depends only on the matrix $P$. Indeed,
in the spirit of Observation 4.3.1, in this special case the formal MpQUEA
$U^{\,{\mathcal{R}}}_{\\!P,\hskip 0.7pt\hbar}(\hskip 0.5pt\mathfrak{g})$ is
just a “logarithmic version” of the polynomial MpQUEA
$U_{\mathbf{q}}(\mathfrak{g})$ in Observation 4.3.1. In addition, in this case
the FoMpQUEA $U^{\,{\mathcal{R}}}_{\\!P,\hskip 0.7pt\hbar}(\hskip
0.5pt\mathfrak{g})$ admits an alternative, somewhat significant presentation,
as follows. Consider inside $U^{\,{\mathcal{R}}}_{\\!P,\hskip
0.7pt\hbar}(\hskip 0.5pt\mathfrak{g})$ the vectors
$\,S_{i}:=2^{-1}\big{(}\,T_{i}^{+}\\!+T_{i}^{-}\big{)}\,$ and
$\,\varLambda_{i}:=2^{-1}\big{(}\,T_{i}^{+}\\!-T_{i}^{-}\big{)}\,$ — for all
$\,i\in I\,$; these clearly form yet another ${\Bbbk[[\hbar]]}$–basis of
$\mathfrak{h}\,$. Then $U^{\,{\mathcal{R}}}_{\\!P,\hskip 0.7pt\hbar}(\hskip
0.5pt\mathfrak{g})$ admits the obvious presentation given by construction and
taking into account that $\,{\big{\\{}T_{i}^{+},T_{i}^{-}\big{\\}}}_{i\in
I}\,$ is a ${\Bbbk[[\hbar]]}$–basis of $\mathfrak{h}\,$, but also the
following, alternative one: it is the $\hbar$–adically complete, unital
${\Bbbk[[\hbar]]}$–algebra with generators $E_{i}$, $F_{i}$, $S_{i}$,
$\varLambda_{i}$ ($\,i\in I\,$) and relations

$\halign{\hbox to\displaywidth{$\hfil\displaystyle#\hfil$}\cr
0.0pt{\hfil$\displaystyle[S_{i}\,,E_{j}]\,=\,\frac{+(\,p_{ij}+p_{ji})}{2}\,E_{j}\,=\,+d_{i}\,a_{ij}\,E_{j}\;,\;\;[S_{i}\,,F_{j}]\,=\,\frac{-(\,p_{ij}+p_{ji})}{2}\,F_{j}\,=\,-d_{i}\,a_{ij}\,F_{j}\cr
0.0pt{\hfil$\displaystyle[\,\varLambda_{i}\,,E_{j}\,]\,=\,+(\,p_{ij}-p_{ji})\,E_{j}\;\;,\qquad[\,\varLambda_{i}\,,F_{j}\,]\,=\,-(\,p_{ij}-p_{ji})\,F_{j}\cr
0.0pt{\hfil$\displaystyle[\,\varLambda_{i}\,,S_{j}\,]\,=\,0\;\;,\qquad
E_{i}\,F_{j}\,-\,F_{j}\,E_{i}\,\;=\;\,\delta_{i,j}\,e^{+\hbar\,\varLambda_{i}^{+}}\,{{\;e^{+\hbar\,S_{i}}-\,e^{-\hbar\,S_{i}}\;}\over{\;q_{i}^{+1}-\,q_{i}^{-1}\;}}\cr
0.0pt{\hfil$\displaystyle\sum\limits_{k=0}^{1-a_{ij}}(-1)^{k}{\left[{1-a_{ij}\atop
k}\right]}_{\\!q_{i}}q_{ij}^{+k/2\,}q_{ji}^{-k/2}\,X_{i}^{1-a_{ij}-k}X_{j}X_{i}^{k}\;=\;0\qquad\forall\;\;X\in\\{E\,,F\\}\quad(\,i\neq
j\,)\crcr}}}}}$

Moreover, the Hopf structure of $U^{\,{\mathcal{R}}}_{\\!P,\hskip
0.7pt\hbar}(\hskip 0.5pt\mathfrak{g})$ is then described (for all $\,i\in
I\,$) by

$\displaystyle\Delta\big{(}E_{\ell}\big{)}$ $\displaystyle\,=\,E_{\ell}\otimes
1\,+\,e^{+\hbar\,\varLambda_{\ell}}e^{+\hbar\,S_{\ell}}\\!\otimes E_{\ell}$
$\displaystyle\Delta\big{(}S_{\ell}\big{)}\,=\,S_{\ell}\otimes 1$
$\displaystyle+\,1\otimes
S_{\ell}\quad,\qquad\Delta\big{(}\varLambda_{\ell}\big{)}\,=\,\varLambda_{\ell}\otimes
1\,+\,1\otimes\varLambda_{\ell}$ $\displaystyle\Delta\big{(}F_{\ell}\big{)}$
$\displaystyle\,=\,F_{\ell}\otimes
e^{-\hbar\,S_{\ell}}e^{+\hbar\,\varLambda_{\ell}}+\,1\otimes F_{\ell}$

In particular, this implies that the $\hbar$–adically complete, unital
subalgebra $U_{\hbar}(\mathfrak{b}_{+})\,$, rep.
$U_{\hbar}(\mathfrak{b}_{-})\,$, of $U^{\,{\mathcal{R}}}_{\\!P,\hskip
0.7pt\hbar}(\hskip 0.5pt\mathfrak{g})$ generated by all the $E_{i}$’s, resp.
all the $F_{i}$’s, and all the $S_{i}$’s is isomorphic to the “standard”
positive, resp. negative, Borel subalgebra in the derived version of
Drinfeld’s QUEA $U_{\hbar}(\mathfrak{g})\,$. On the other hand, both
subalgebras $U_{\hbar}(\mathfrak{b}_{\pm})$ are not Hopf subalgebras inside
$U^{\,{\mathcal{R}}}_{\\!P,\hskip 0.7pt\hbar}(\hskip 0.5pt\mathfrak{g})\,$,
contrary to what happens in Drinfeld’s setup.

###  4.4. Further results on FoMpQUEAs

We present now a few more techniques, which provide alternative proofs for our
results about the structure of quantum nilpotent, Cartan and Borel
subalgebras, as well as the triangular decomposition results. This mainly
follows in the footpath of a standard strategy, already used for one-parameter
QUEA’s.

######  4.4.1.

Preformal multiparameter QUEAs and special representations. We introduce now
some “preliminary versions of FoMpQUEAs”, essentially defined like the
FoMpQUEAs but for dropping from their definition the quantum Serre relations.
These “pre-FoMpQUEAs” will be a key tool in our analysis, as well as some
special representations of them that we also introduce presently. Let $U$ be
the unital, associative ${\Bbbk[[\hbar]]}$-algebra generated by the
${\Bbbk[[\hbar]]}$–submodule $\mathfrak{h}$ together with elements
$\,E_{i}\,$, $F_{i}\,$ (for all $\,i\in I\,$), subject to the same relations
as in (4.5), except the last two — the quantum Serre relations. Let
$\,\widetilde{U}^{\,{\mathcal{R}}}_{\\!P,\hbar}(\mathfrak{g})\,$ be the
$\hbar$–adic completion of $U$. From the proof of Theorem 4.3.2, it follows at
once that $\,\widetilde{U}^{\,{\mathcal{R}}}_{\\!P,\hbar}(\mathfrak{g})\,$ is
a topological Hopf algebra over ${\Bbbk[[\hbar]]}\,$.

Let $\,V:=\bigoplus_{i\in I}{\Bbbk[[\hbar]]}.v_{i}\,$ be the free
${\Bbbk[[\hbar]]}$–module with basis $\,{\\{v_{i}\\}}_{i\in I}\,$, let
$\,T_{\hbar}(V)\,$ be the tensor algebra of $\,V$ over $\,{\Bbbk[[\hbar]]}\,$,
and let $\,\widehat{T}_{\hbar}(V)\,$ be the $\hbar$–adic completion of
$\,T_{\hbar}(V)\,$. Then, $\,T_{\hbar}(V)\,$ is a free
${\Bbbk[[\hbar]]}$–module with basis $\,{\\{v_{J}\\}}_{J\in\mathcal{J}}\,$,
where $\mathcal{J}$ is the set of all finite sequences of elements in $I$ and
$\,v_{J}:=v_{j_{1}}\otimes\cdots\otimes v_{j_{r}}\,$ — or simply
$\,v_{J}:=v_{j_{1}}\cdots v_{j_{r}}\,$ — is standard monomial notation for all
$\,J:=(j_{1},\dots,j_{r})\in\mathcal{J}\,$. For
$\,J=(j_{1},\ldots,j_{r})\in\mathcal{J}\,$ and $\,1\leq k\leq r\,$, write
$\,\widehat{J}_{k}:=(j_{1},\dots,j_{k-1},j_{k+1},\ldots,j_{r})\,$,
$\,J_{k}:=(j_{k+1},\ldots,j_{r})\,$ and
$\,\alpha_{J}:=\sum_{\ell=1}^{r}\alpha_{j_{\ell}}\,$.

######  Lemma 4.4.2.

For every $\,\lambda\in\mathfrak{h}^{*}\,$, there exists a unique
representation of $\,U\,$ onto $\,T_{\hbar}(V)$ — which is then denoted
$\,T_{\hbar}^{\lambda}(V)$ — such that, for all
$\,J=(j_{1},\dots,j_{r})\in\mathcal{J}\,$

$\halign{\hbox to\displaywidth{$\hfil\displaystyle#\hfil$}\cr
0.0pt{\hfil$\displaystyle F_{i}\,.\,v_{J}\;=\;v_{(i,\,J)}\quad,\qquad
T\,.\,v_{J}\;=\;\big{(}\lambda(T)-\alpha_{J}(T)\big{)}v_{J}\cr
0.0pt{\hfil$\displaystyle E_{i}\,.\,v_{J}\;=\sum_{1\leq\ell\leq
r\,,\;j_{\ell}=i}\hskip-5.0pt\frac{\;q^{+\lambda(T_{i}^{+})-\alpha_{J_{\ell}}(T_{i}^{+})}\\!-q^{-\lambda(T_{i}^{-})+\alpha_{J_{\ell}}(T_{i}^{-})}\;}{q_{i}^{+1}-q_{i}^{-1}}\;v_{\widehat{J}_{\ell}}\crcr}}}$

In addition, this representation — of $\,U$ onto $\,T_{\hbar}(V)$ — induces by
continuity a unique representation
$\,\widetilde{U}^{\,{\mathcal{R}}}_{\\!P,\hbar}(\mathfrak{g})\,$ onto
$\,\widehat{T}_{\hbar}(V)\,$, which is hereafter denoted by
$\,\widehat{T}_{\hbar}^{\lambda}(V)\,$.

###### Proof.

We must prove that the equalities above do endow $T_{\hbar}(V)$ with a
structure of $U$–module: to this end, let us performe a quick check to show
that such an action is well-defined. For every
$\,T,T^{\prime}\in\mathfrak{h}\,$ we have

$T^{\prime}.\big{(}T.\,v_{J}\big{)}=\big{(}\lambda(T)-\alpha_{J}(T)\big{)}T^{\prime}.v_{J}\,=\big{(}\lambda(T)-\alpha_{J}(T)\big{)}\big{(}\lambda(T^{\prime})-\alpha_{J}(T^{\prime})\big{)}\,v_{J}\,=\,T.\big{(}T^{\prime}.v_{J}\big{)}$

Take now $\,T\in\mathfrak{h}\,$ and $\,F_{i}\,$ with $\,i\in I\,$. Then

$\halign{\hbox to\displaywidth{$\hfil\displaystyle#\hfil$}\cr
0.0pt{\hfil$\displaystyle
T.\big{(}F_{i}\,.v_{J}\big{)}-F_{i}\,.\big{(}T.v_{J}\big{)}\;=\;T.v_{(i,\,J)}-F_{i}\,.\big{(}\big{(}\lambda(T)-\alpha_{J}(T)\big{)}\,v_{J}\big{)}\;=\hfill\cr
0.0pt{\hfil$\displaystyle=\;\big{(}\lambda(T)-\alpha_{(i,\,J)}(T)-\lambda(T)+\alpha_{J}(T)\big{)}v_{(i,J)}\;=\;\big{(}\\!-\alpha_{(i,J)}(T)+\alpha_{J}(T)\big{)}\,v_{(i,J)}\;=\cr
0.0pt{\hfil$\displaystyle\hfill=\;-\alpha_{i}(T)\,v_{(i,J)}\;=\;-\alpha_{i}(T)\,F_{i}\,.v_{J}\crcr}}}}$

Similarly, for $\,T\in\mathfrak{h}\,$ and $E_{i}$ with $\,i\in I\,$ we find

$\halign{\hbox to\displaywidth{$\hfil\displaystyle#\hfil$}\cr
0.0pt{\hfil$\displaystyle
T.\big{(}E_{i}\,.v_{J}\big{)}\;=\;\sum_{1\leq\ell\leq
r\,,\;i=j_{\ell}}\hskip-5.0pt\big{(}\,\lambda(T)-\alpha_{\widehat{J}_{\ell}}(T)\big{)}\,\frac{\;q^{+\lambda(T_{i}^{+})-\alpha_{J_{\ell}}(T_{i}^{+})}-q^{-\lambda(T_{i}^{-})+\alpha_{J_{\ell}}(T_{i}^{-})}\;}{q_{i}^{+1}\\!-q_{i}^{-1}}\;v_{\widehat{J}_{\ell}}\cr
0.0pt{\hfil$\displaystyle
E_{i}\,.\big{(}T.v_{J}\big{)}\;=\;\sum_{1\leq\ell\leq
r\,,\;i=j_{\ell}}\hskip-5.0pt\big{(}\,\lambda(T)-\alpha_{J}(T)\big{)}\,\frac{\;q^{+\lambda(T_{i}^{+})-\alpha_{J_{\ell}}(T_{i}^{+})}-q^{-\lambda(T_{i}^{-})+\alpha_{J_{\ell}}(T_{i}^{-})}\;}{q^{+1}_{i}\\!-q_{i}^{-1}}\;v_{\widehat{J}_{\ell}}\crcr}}}$

Since
$\,\alpha_{J}(T)-\alpha_{\widehat{J}_{\ell}}(T)=\alpha_{j_{\ell}}(T)=\alpha_{i}(T)\,$
for all $\,j_{\ell}=i\,$, we obtain

$[T,E_{i}]\,.v_{J}\;=\sum_{1\leq\ell\leq
r\,,\;i=j_{\ell}}\hskip-5.0pt\alpha_{i}(T)\,\frac{\;q^{+\lambda(T_{i}^{+})-\alpha_{J_{\ell}}(T_{i}^{+})}-q^{-\lambda(T_{i}^{-})+\alpha_{J_{\ell}}(T_{i}^{-})}\;}{q_{i}^{+1}\\!-q_{i}^{-1}}\;v_{\widehat{J}_{\ell}}\;=\;\alpha_{i}(T)\,E_{i}\,.v_{J}$

Finally, to check the commuting relation between $E_{i}$ and $F_{j}$ we note
first that
$\;\sum_{n=0}^{m}\frac{\,\hbar^{n}\,}{\,n!\,}\,T^{n}.v_{J}\,=\,\sum_{n=0}^{m}\frac{\,\hbar^{n}\,}{\,n!\,}\,{\big{(}\,\lambda(T)-\alpha_{J}(T)\big{)}}^{n}.v_{J}\;$
for all $\,T\in\mathfrak{h}\,$ and $\,m\geq 1\,$. Then, by the continuity of
the linear action, we get that

$e^{+\hbar\,T^{+}_{i}}.\,v_{J}\;=\;q^{+(\lambda(T^{+}_{i})-\alpha_{J}(T^{+}_{i}))}\,v_{J}\qquad\text{
and }\qquad
e^{-\hbar\,T^{-}_{i}}.\,v_{J}\;=\;q^{-(\lambda(T^{-}_{i})-\alpha_{J}(T^{-}_{i}))}\,v_{J}$

Then, for $\,i,j\in I\,$ with $\,i\neq j\,$, we have

$\halign{\hbox to\displaywidth{$\hfil\displaystyle#\hfil$}\cr
0.0pt{\hfil$\displaystyle
E_{i}\,.\big{(}F_{j}\,.v_{J}\big{)}\,-\,F_{j}\,.\big{(}E_{i}\,.v_{J}\big{)}\;=\hfill\cr
0.0pt{\hfil$\displaystyle=\;E_{i}\,.v_{(j,J)}\,-\,F_{j}\,.\left(\sum_{1\leq\ell\leq
r\,,\;i=j_{\ell}}\hskip-5.0pt\frac{\;q^{+\lambda(T_{i}^{+})-\alpha_{J_{\ell}}(T_{i}^{+})}-q^{-\lambda(T_{i}^{-})+\alpha_{J_{\ell}}(T_{i}^{-})}\;}{q_{i}^{+1}\\!-q_{i}^{-1}}\;v_{\widehat{J}_{\ell}}\right)\;=\cr
0.0pt{\hfil$\displaystyle\hfill=\;E_{i}\,.v_{(j,J)}\,-\,\sum_{1\leq\ell\leq
r\,,\;i=j_{\ell}}\hskip-5.0pt\frac{\;q^{+\lambda(T_{i}^{+})-\alpha_{J_{\ell}}(T_{i}^{+})}-q^{-\lambda(T_{i}^{-})+\alpha_{J_{\ell}}(T_{i}^{-})}\;}{q_{i}^{+1}\\!-q_{i}^{-1}}\;v_{(j,\widehat{J}_{\ell})}\crcr}}}}$

First, if $\,i\neq j\,$, then $\,j_{\ell}=i\neq j\,$ and
$\,\widehat{(\,j\,,J\,)}_{\ell}=\big{(}\,j\,,\widehat{J}_{\ell}\,\big{)}\,$,
$\,\alpha_{{(\,j,\,J\,)}_{\ell}}=\alpha_{J_{\ell}}\,$ for all $\,1\leq\ell\leq
r\,$. Hence

$E_{i}\,.v_{(j,J)}\;\;=\sum_{1\leq\ell\leq
r\,,\;i=j_{\ell}}\hskip-9.0pt{\big{(}q_{i}^{+1}\\!-q_{i}^{-1}\big{)}}^{-1}\big{(}q^{+\lambda(T_{i}^{+})-\alpha_{J_{\ell}}(T_{i}^{+})}-q^{-\lambda(T_{i}^{-})+\alpha_{J_{\ell}}(T_{i}^{-})}\big{)}\,v_{(j,\widehat{J}_{\ell})}$

which implies that
$\;E_{i}\,.\big{(}F_{j}\,.v_{J}\big{)}\,-\,F_{j}\,.\big{(}E_{i}\,.v_{J}\big{)}\,=\,0\;$.

Assume now that $\,i=j\,$. Then for $\,i=j_{\ell}\,$ we have
$\,\widehat{(i\,,J)}_{\ell}=J\,$ and $\,\alpha_{(i,J)_{\ell}}=\alpha_{J}\,$
for $\,\ell=1\,$, while
$\,\widehat{(i\,,J)}_{\ell}=\big{(}i\,,\widehat{J}_{\ell}\big{)}\,$ and
$\,\alpha_{(i,J)_{\ell}}=\alpha_{J_{\ell}}\,$ for $\,\ell>1\,$. This implies
that

$\halign{\hbox to\displaywidth{$\hfil\displaystyle#\hfil$}\cr
0.0pt{\hfil$\displaystyle
E_{i}\,.v_{(i,J)}\,\;=\;\,{\big{(}q_{i}^{+1}\\!-q_{i}^{-1}\big{)}}^{-1}\big{(}q^{+\lambda(T_{i}^{+})-\alpha_{J}(T_{i}^{+})}-q^{-\lambda(T_{i}^{-})+\alpha_{J}(T_{i}^{-})}\big{)}\,v_{J}\,\;+\hfill\cr
0.0pt{\hfil$\displaystyle\hfill+\sum_{1\leq\ell\leq
r\,,\;i=j_{\ell}}\hskip-9.0pt{\big{(}q_{i}^{+1}\\!-q_{i}^{-1}\big{)}}^{-1}\big{(}q^{+\lambda(T_{i}^{+})-\alpha_{J_{\ell}}(T_{i}^{+})}-q^{-\lambda(T_{i}^{-})+\alpha_{J_{\ell}}(T_{i}^{-})}\big{)}\,v_{(i,\widehat{J}_{\ell})}\crcr}}}$

and consequently

$E_{i}.\big{(}F_{i}.v_{J}\big{)}-F_{i}.\big{(}E_{i}.v_{J}\big{)}\,=\,\frac{q^{\lambda(T_{i}^{+})-\alpha_{J}(T_{i}^{+})}-q^{-\lambda(T_{i}^{-})+\alpha_{J}(T_{i}^{-})}}{q_{i}^{+1}\\!-q_{i}^{-1}}\,.\,v_{J}\,=\,\frac{\,e^{\hbar\,T_{i}^{+}}\\!-e^{\hbar\,T_{i}^{-}}\,}{q_{i}^{+1}\\!-q_{i}^{-1}}\,.\,v_{J}$

Therefore, the formulas above define indeed an action of $U$ onto
$T_{\hbar}(V)\,$.

Finally, since this action is ${\Bbbk[[\hbar]]}$–linear, it induces a unique
(topological) action of
$\widetilde{U}^{\,{\mathcal{R}}}_{\\!P,\hbar}(\mathfrak{g})$ on
$\widehat{T}_{\hbar}(V)$ by completion. This completes the proof. ∎

With an entirely analogous proof, we obtain the following lemma:

######  Lemma 4.4.3.

For every $\,\lambda\in\mathfrak{h}^{*}\,$, there exists a unique
representation of $U$ onto ${T}_{\hbar}(V)$ — which is then denoted
$\,{}^{\lambda}T_{\hbar}(V)$ — such that, for all
$\,J=(j_{1},\dots,j_{r})\in\mathcal{J}\,$

$\halign{\hbox to\displaywidth{$\hfil\displaystyle#\hfil$}\cr
0.0pt{\hfil$\displaystyle E_{i}\,.\,v_{J}\;=\;v_{(i,\,J)}\quad,\qquad
T\,.\,v_{J}\;=\;\big{(}\lambda(T)+\alpha_{J}(T)\big{)}v_{J}\cr
0.0pt{\hfil$\displaystyle F_{i}\,.\,v_{J}\;=\sum_{1\leq\ell\leq
r\,,\;j_{\ell}=i}\hskip-5.0pt\frac{\;q^{-\lambda(T_{i}^{-})-\alpha_{J_{\ell}}(T_{i}^{-})}\\!-q^{+\lambda(T_{i}^{+})+\alpha_{J_{\ell}}(T_{i}^{+})}\;}{q_{i}^{+1}-q_{i}^{-1}}\;v_{\widehat{J}_{\ell}}\crcr}}}$

In addition, this representation, of $\,U$ onto $\,T_{\hbar}(V)\,$, induces by
continuity a unique representation
$\,\widetilde{U}^{\,{\mathcal{R}}}_{\\!P,\hbar}(\mathfrak{g})\,$ onto
$\,\widehat{T}_{\hbar}(V)\,$, which is hereafter denoted by
$\,{}^{\lambda}\widehat{T}_{\hbar}(V)\,$. ∎

Denote by $U^{0}$, resp. $U^{+}$, resp. $U^{-}$, the unital associative
${\Bbbk[[\hbar]]}$–subalgebras of $U$ generated by $\mathfrak{h}\,$, resp. by
all the $E_{i}$’s, resp. by all the $F_{i}$’s ($\,i\in I\,$). Write
$\,\widetilde{U}^{0}\,\big{(}=\widetilde{U}_{\hbar}(\mathfrak{h}):=\widetilde{U}_{P,\hbar}^{\,{\mathcal{R}}}(\mathfrak{h})\big{)}\,$,
resp. $\widetilde{U}^{+}$, resp. $\widetilde{U}^{-}$, for the $\hbar$–adic
completion of them: clearly, all these are topological
${\Bbbk[[\hbar]]}$–subalgebras of
$\widetilde{U}^{\,{\mathcal{R}}}_{\\!P,\hbar}(\mathfrak{g})\,$. As a first
consequence of Lemma 4.4.2, we get the following:

######  Proposition 4.4.4.

The Cartan subalgebra
$\,\widetilde{U}^{0}=\widetilde{U}_{\hbar}(\mathfrak{h})=\widetilde{U}_{P,\hbar}^{\,{\mathcal{R}}}(\mathfrak{h})\,$
is the $\hbar$–adic completion of $\,S_{\hbar}(\mathfrak{h})\,$, the symmetric
algebra of $\,\mathfrak{h}$ over $\,{\Bbbk[[\hbar]]}\,$. In particular,
$\,\widetilde{U}_{\hbar}(\mathfrak{h})\,$ is independent of $\,P$ and
$\,{\mathcal{R}}$ (though not of $\,\mathfrak{h}$) — whence the simplified
notation.

###### Proof.

We provide two different, independent proofs. $\underline{\text{First
Proof}}\,$: Let $T_{\hbar}(\mathfrak{h})\,$, resp.
$\,S_{\hbar}(\mathfrak{h})\,$, be the tensor algebra, resp. the symmetric
algebra, of $\mathfrak{h}$ over ${\Bbbk[[\hbar]]}\,$, and let
$\widehat{T}_{\hbar}(\mathfrak{h})\,$, resp.
$\,\widehat{S}_{\hbar}(\mathfrak{h})$ be the $\hbar$–adic completion of them.
By the commutation relations among elements of $\mathfrak{h}$ in
$\widetilde{U}^{\,{\mathcal{R}}}_{\\!P,\hbar}(\mathfrak{g})\,$, we have a
diagram of morphisms

$\textstyle{\;\widehat{T}_{\hbar}(\mathfrak{h})\;\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{p_{\hbar,S}}$$\textstyle{\;\widehat{S}_{\hbar}(\mathfrak{h})\;\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{p_{\hbar,U}}$$\textstyle{\;\widetilde{U}_{\hbar}(\mathfrak{h})\;\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\iota_{\hbar}}$$\textstyle{\;\widetilde{U}^{\,{\mathcal{R}}}_{\\!P,\hbar}(\mathfrak{g})}$$\textstyle{\;T_{\hbar}(\mathfrak{h})\;\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{p_{S}}$$\textstyle{\;S_{\hbar}(\mathfrak{h})\;\ignorespaces\ignorespaces\ignorespaces\ignorespaces\;\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{p_{U^{0}}}$$\textstyle{\;U^{0}\;\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\;\mathfrak{h}\;\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$

where the maps $p_{\hbar,S}$, $p_{\hbar,U}$, $p_{S}$ and $p_{U^{0}}$ are the
canonical epimorphisms, $\iota_{\hbar}$ is the canonical inclusion, all
vertical arrows are canonical embeddings, and the diagonal arrows
$\,\mathfrak{h}\longrightarrow S_{\hbar}(\mathfrak{h})\,$ too. We want to show
that $p_{\hbar,U}$ is in fact an isomomorphism. Let $\,V:=\bigoplus_{i\in
I}{\Bbbk[[\hbar]]}.v_{i}\,$ be the free ${\Bbbk[[\hbar]]}$–module with basis
$\,{\\{v_{i}\\}}_{i\in I}\,$. Choosing $\,\lambda\in\mathfrak{h}^{*}\,$ and
restricting the action defined in Lemma 4.4.2 to the image of $i_{\hbar}\,$,
we have that $\widetilde{U}_{\hbar}(\mathfrak{h})$ acts on
$\widehat{T}_{\hbar}^{\lambda}(V)$ by the character $\lambda$ via
$\;T.v_{J}\,=\,\big{(}\lambda(T)-\alpha_{J}(T)\big{)}\,v_{J}\,$, for all
$\,T\in\mathfrak{h}\,$ and for all
$\,J=\big{(}j_{1}\,,\cdots,j_{r}\big{)}\in\mathcal{J}\,$. In particular,
$\,T.v_{\emptyset}=T.1=\lambda(T)\,$ for all $\,T\in\mathfrak{h}\,$. This
action induces an action of $\widehat{S}_{\hbar}(\mathfrak{h})\,$, and of its
${\Bbbk[[\hbar]]}$–subalgebra $S_{\hbar}(\mathfrak{h})\,$, on
$\widehat{T}_{\hbar}^{\lambda}(V)$ via the epimorphism $p_{\hbar,U}\,$. By the
very definition, this action coincides with the unique action of
$S_{\hbar}(\mathfrak{h})$ on $\widehat{T}_{\hbar}^{\lambda}(V)$ defined by the
character $\,\lambda\in\mathfrak{h}^{*}\,$ by the universal properties of the
symmetric and the tensor algebras.

Let $\,t\in S_{\hbar}(\mathfrak{h})\,$ be such that $p_{\hbar,U}(t)=0\,$: then
$\,0=t.v_{\emptyset}=\lambda_{S}(t)\,$ — extending
$\,\lambda\in\mathfrak{h}^{*}\,$ to a ${\Bbbk[[\hbar]]}$–algebra character
$\lambda_{S}$ of $S_{\hbar}(\mathfrak{h})\,$. Since $\lambda$ is arbitrary, we
get $\,\lambda_{S}(t)=0\,$ for all $\,\lambda\in\mathfrak{h}^{*}\,$, so that
$\,t=0\,$. Similarly, $\lambda_{S}$ further extends, canonically and uniquely,
to a $\hbar$–adically continuous character of
$\widehat{S}_{\hbar}(\mathfrak{h})\,$, denoted $\lambda_{\widehat{S}}\,$. Then
for any $\,\widehat{t}\in\widehat{S}_{\hbar}(\mathfrak{h})\,$ such that
$p_{\hbar,U}\big{(}\,\widehat{t}\;\big{)}=0\,$ we have
$\,0=t.v_{\emptyset}=\lambda_{\widehat{S}}\big{(}\,\widehat{t}\;\big{)}\,$,
thus $\,\lambda_{\widehat{S}}\big{(}\,\widehat{t}\;\big{)}=0\,$ for all
$\,\lambda\in\mathfrak{h}^{*}\,$, which implies $\,\widehat{t}=0\,$. Hence
$p_{\hbar,U}$ is injective and
$\,\widehat{S}_{\hbar}(\mathfrak{h})\cong\widetilde{U}_{\hbar}(\mathfrak{h})\,$.
$\underline{\text{Second Proof}}\,$: By definition there is a
${\Bbbk[[\hbar]]}$–linear morphism from $\mathfrak{h}$ to
$U^{\,{\mathcal{R}}}_{\\!P,\hskip 0.7pt\hbar}(\hskip 0.5pt\mathfrak{g})\,$,
whose image we denote by $\mathfrak{h}^{\prime}\,$; in other words,
$\mathfrak{h}^{\prime}$ is the ${\Bbbk[[\hbar]]}$–submodule of
$U^{\,{\mathcal{R}}}_{\\!P,\hskip 0.7pt\hbar}(\hskip 0.5pt\mathfrak{g})$
spanned by the generators $\,T\in\mathfrak{h}\,$. By definition,
$U_{P,\hbar}^{\,{\mathcal{R}}}(\mathfrak{h})$ is (topologically) generated by
$\mathfrak{h}^{\prime}\,$, which in turn is a Lie subalgebra inside the Lie
algebra of primitive elements of the Hopf algebra
$U_{P,\hbar}^{\,{\mathcal{R}}}(\mathfrak{h})\,$. Since we are in
characteristic zero, by Milnor-Moore’s Theorem we deduce that
$U_{P,\hbar}^{\,{\mathcal{R}}}(\mathfrak{h})$ is indeed nothing but the
$\hbar$–adic completion of the universal enveloping algebra
$\,U\big{(}\mathfrak{h}^{\prime}\big{)}$ of $\,\mathfrak{h}^{\prime}\,$; in
turn, the latter coincides with the symmetric ${\Bbbk[[\hbar]]}$–algebra
$S_{\hbar}\big{(}\mathfrak{h}^{\prime}\big{)}$ — hence its completion
coincides with $\widehat{S}_{\hbar}\big{(}\mathfrak{h}^{\prime}\big{)}$ —
because the multiplication therein is commutative, by construction!

Finally, we observe that the built-in epimorphism
$\,\mathfrak{h}\relbar\joinrel\twoheadrightarrow\mathfrak{h}^{\prime}\,$ is
indeed an isomorphism, so that $\,\mathfrak{h}^{\prime}\cong\mathfrak{h}\,$.
This is proved again making use of Lemma 4.4.2 above, in particular looking at
how $\mathfrak{h}$ acts on each representation
$\widehat{T}_{\hbar}^{\lambda}(V)\,$, for all
$\,\lambda\in\mathfrak{h}^{*}\,$, along the same lines as in the last part of
the First Proof above. ∎

Let $U^{\prime}$ be the ${\Bbbk[[\hbar]]}$–subalgebra of
$\widetilde{U}^{\,{\mathcal{R}}}_{\\!P,\hbar}(\mathfrak{g})$ generated by the
${\Bbbk[[\hbar]]}$–subalgebras $U^{-}$, $U^{+}$ and
$\,\widetilde{U}^{0}=\widetilde{U}_{\hbar}(\mathfrak{h})\,$ — the $\hbar$–adic
completion of $U^{0}$. By Theorem 4.3.2, it follows that $U^{\prime}$ is a
(topological) Hopf ${\Bbbk[[\hbar]]}$–subalgebra of
$\,\widetilde{U}^{\,{\mathcal{R}}}_{\\!P,\hbar}(\mathfrak{g})\,$. We fix some
more notation. For any finite sequence
$\,J=\big{(}j_{1}\,,j_{2}\,,\ldots,j_{r}\big{)}\,$ — with $\,r\geq 1\,$ — of
elements in $I$, we set $\,T^{\pm}_{\emptyset}:=0\,$ and