0.0pt{\hfil$\displaystyle\quad\quad\quad=\;\delta_{i,r}\,\delta_{\ell,h}\,{\textstyle\sum\limits_{m=0}^{+\infty}}{{\,\hbar^{m}\,2^{-m}\,}\over{\,m!\,}}\,{\left(\,{\textstyle\sum_{t,\,k=1}^{n}}\,\phi_{t,k}\,\delta_{i,t}\,\delta_{\ell,k}\right)}^{m}\;=\hfill\cr
0.0pt{\hfil$\displaystyle\quad\quad\quad\quad=\;\delta_{i,r}\,\delta_{\ell,h}\,{\textstyle\sum\limits_{m=0}^{+\infty}}{{\,\hbar^{m}\,2^{-m}\,}\over{\,m!\,}}\,{(\phi_{i,\ell})}^{m}\;=\;\delta_{i,r}\,\delta_{\ell,h}\,\exp\big{(}\hbar\,2^{-1}\phi_{i,\ell}\big{)}\;=\;\delta_{i,r}\,\delta_{\ell,h}\,e^{\hbar\,\phi_{i,\ell}/2}\hfill\crcr}}}}}}$

in short

$\sigma_{{}_{\mathcal{F}}}\big{(}x_{i,\,r}\,,x_{\ell,\,h}\big{)}\;=\;\delta_{i,r}\,\delta_{\ell,h}\,e^{\hbar\,\phi_{i,\ell}/2}\qquad\qquad\forall\;\;i\,,r,\ell,h\in\\{1,\dots,n\\}$
(3.19)

Using this formula, the deformed product in ${{F_{\hbar}[[G\hskip
1.5pt]]}}_{\sigma_{{}_{\mathcal{F}}}}$ can be described as follows:

$\halign{\hbox to\displaywidth{$\hfil\displaystyle#\hfil$}\cr
0.0pt{\hfil$\displaystyle
x_{i,\,j}\raise-1.0pt\hbox{$\;\scriptstyle\dot{\sigma}_{{}_{\mathcal{F}}}$}\,x_{\ell,\,t}\;:=\;\sigma_{{}_{\mathcal{F}}}\big{(}{(x_{i,\,j})}_{(1)}\,,\,{(x_{\ell,\,t})}_{(1)}\big{)}\;{(x_{i,\,j})}_{(2)}\,{(x_{\ell,\,t})}_{(2)}\;\sigma_{{}_{\mathcal{F}}}^{-1}\big{(}{(x_{i,\,j})}_{(3)}\,,\,{(x_{\ell,\,t})}_{(3)}\big{)}\;=\hfill\cr
0.0pt{\hfil$\displaystyle=\;{\textstyle\sum\limits_{r,\,s,\,h,\,k=1}^{n}}\sigma_{{}_{\mathcal{F}}}\big{(}x_{i,\,r}\,,\,x_{\ell,\,h}\big{)}\;x_{r,\,s}\,x_{h,\,k}\;\sigma_{{}_{\mathcal{F}}}^{-1}\big{(}x_{s,\,j}\,,\,x_{k,\,t}\big{)}\;=\quad\qquad\qquad\cr
0.0pt{\hfil$\displaystyle\hfill=\;\sigma_{{}_{\mathcal{F}}}\big{(}x_{i,\,i}\,,\,x_{\ell,\,\ell}\big{)}\;x_{i,\,j}\,x_{\ell,\,t}\;\sigma_{{}_{\mathcal{F}}}^{-1}\big{(}x_{j,\,j}\,,\,x_{t,\,t}\big{)}\;=\;e^{\hbar\,(\phi_{i,\ell}\,-\,\phi_{j,t})/2}\;x_{i,\,j}\,x_{\ell,\,t}\crcr}}}}$

— where we used (3.19); in short, we get

$x_{i,\,j}\raise-1.0pt\hbox{$\;\scriptstyle\dot{\sigma}_{{}_{\mathcal{F}}}$}\,x_{\ell,\,t}\;=\;e^{\hbar\,(\phi_{i,\ell}\,-\,\phi_{j,t})/2}\;x_{i,\,j}\,x_{\ell,\,t}\qquad\qquad\forall\;\;i\,,j,\ell,t\in\\{1,\dots,n\\}$
(3.20)

Note that this formula shows how the new, deformed product is equivalent
modulo $\,\hbar\,$ to the old one: this is a general fact, due to the very
construction, namely because we are working with 2–cocycles of the form
$\,\exp\big{(}\hbar\,\varsigma\big{)}\,$ where $\,\varsigma\,$ is some
bilinear form on the QFSHA to be deformed. By this same reason, any set of
elements which generate, as an algebra, the QFSHA under exam, will also
generate it (as an algebra again) w.r.t. the new, deformed product. For this
reason, the formula (3.20) is enough to describe the deformed algebra
${{F_{\hbar}[[G\hskip 1.5pt]]}}_{\sigma_{{}_{\mathcal{F}}}}$ as the latter is
(again) generated — w.r.t. the new product — by the $x_{i,\,j}$’s, just like
${F_{\hbar}[[G\hskip 1.5pt]]}$ was (with the old product). Let us now see how
(3.20) yields a modified Poisson bracket in the semiclassical limit of
$\,{{F_{\hbar}[[G\hskip 1.5pt]]}}_{\sigma_{{}_{\mathcal{F}}}}\,$. Using such
notation as $\;\overline{x}_{r,\,s}:=x_{r,\,s}\ \big{(}\,\text{mod}\,\
\hbar\,{{F_{\hbar}[[G\hskip 1.5pt]]}}_{\sigma_{{}_{\mathcal{F}}}}\big{)}\,$,
the Poisson bracket inherited by $\,{{F_{\hbar}[[G\hskip
1.5pt]]}}_{\sigma_{{}_{\mathcal{F}}}}\,$ is given, by definition, by

${\big{\\{}\overline{x}_{i,\,j}\,,\overline{x}_{\ell,\,t}\big{\\}}}_{\sigma_{{}_{\mathcal{F}}}}\,\;:=\;\;{{\;{\big{[}{x}_{i,\,j}\,,{x}_{\ell,\,t}\big{]}}_{\sigma_{{}_{\mathcal{F}}}}\;}\over{\;\hbar\;}}\
\ \Big{(}\,\text{mod}\,\ \hbar\,{{F_{\hbar}[[G\hskip
1.5pt]]}}_{\sigma_{{}_{\mathcal{F}}}}\Big{)}$

Now

$\halign{\hbox to\displaywidth{$\hfil\displaystyle#\hfil$}\cr
0.0pt{\hfil$\displaystyle{\big{[}{x}_{i,\,j}\,,{x}_{\ell,\,t}\big{]}}_{\sigma_{{}_{\mathcal{F}}}}\,=\;x_{i,\,j}\raise-1.0pt\hbox{$\;\scriptstyle\dot{\sigma}_{{}_{\mathcal{F}}}$}\,x_{\ell,\,t}\,-\,x_{\ell,\,t}\raise-1.0pt\hbox{$\;\scriptstyle\dot{\sigma}_{{}_{\mathcal{F}}}$}\,x_{i,\,j}\;=\hfill\cr
0.0pt{\hfil$\displaystyle=\;e^{\hbar\,(\phi_{i,\ell}\,-\,\phi_{j,t})/2}\;x_{i,\,j}\,x_{\ell,\,t}\,-\,e^{\hbar\,(\phi_{j,t}\,-\,\phi_{i,\ell})/2}\;x_{\ell,\,t}\,x_{i,\,j}\;=\qquad\qquad\qquad\qquad\cr
0.0pt{\hfil$\displaystyle\hfill=\;e^{\hbar\,(\phi_{i,\ell}\,-\,\phi_{j,t})/2}\,\big{[}x_{i,\,j}\,,x_{\ell,\,t}\big{]}\,+\,\big{(}e^{\hbar\,(\phi_{i,\ell}\,-\,\phi_{j,t})/2}-e^{\hbar\,(\phi_{j,t}\,-\,\phi_{i,\ell})/2}\big{)}\,x_{\ell,\,t}\,x_{i,\,j}\crcr}}}}$

hence expanding the exponentials we get

${\big{[}{x}_{i,\,j}\,,{x}_{\ell,\,t}\big{]}}_{\sigma_{{}_{\mathcal{F}}}}\,=\;\big{(}1+\hbar\,(\phi_{i,\ell}\,-\,\phi_{j,t})/2\big{)}\,\big{[}x_{i,\,j}\,,x_{\ell,\,t}\big{]}\,+\,\hbar\,\big{(}\phi_{i,\ell}\,-\,\phi_{j,t}\big{)}\,x_{\ell,\,t}\,x_{i,\,j}\,+\,\mathcal{O}\big{(}\hbar^{2}\big{)}$

from which we eventually get

${\big{\\{}\overline{x}_{i,\,j}\,,\overline{x}_{\ell,\,t}\big{\\}}}_{\sigma_{{}_{\mathcal{F}}}}\,:=\;\big{\\{}\overline{x}_{i,\,j}\,,\overline{x}_{\ell,\,t}\big{\\}}\,+\,\Big{(}\,\overline{\phi}_{i,\ell}\,-\,\overline{\phi}_{j,t}\Big{)}\,\overline{x}_{\ell,\,t}\,\overline{x}_{i,\,j}$
(3.21)

where $\,\big{\\{}\overline{x}_{i,\,j}\,,\overline{x}_{\ell,\,t}\big{\\}}\,$
denotes the old (undeformed) Poisson bracket and we took into account that
$\;\big{[}\overline{x}_{i,\,j}\,,\overline{x}_{\ell,\,t}\big{]}\,=\,0\;$ and
that the deformed and undeformed product do coincide modulo $\hbar\,$.

In addition, the formula (3.21) for the Poisson bracket also induces,
following the general recipe, a concrete description of the modified Lie
bracket in the cotangent Lie bialgebra
$\,{\mathfrak{g}}^{*}:={\mathfrak{m}}\big{/}{\mathfrak{m}}^{2}\,$, where
${\mathfrak{m}}$ is the augmentation ideal of ${{F_{\hbar}[[G\hskip
1.5pt]]}}_{\sigma_{{}_{\mathcal{F}}}}\,$. Indeed, the latter has as
$\Bbbk$–basis the set of cosets $\,\big{(}\text{modulo\
}{\mathfrak{m}}^{2}\,\big{)}$

$\Big{\\{}\,\text{x}_{i,\,j}:=\big{(}\overline{x}_{i,\,j}-\delta_{i,\,j}\big{)}\,\
\text{mod}\,\ {\mathfrak{m}}^{2}\,\Big{|}\,i\,,j=1,\dots,n\,\Big{\\}}$

and for these elements from (3.21) we deduce the deformed Lie bracket as given
by

$\halign{\hbox to\displaywidth{$\hfil\displaystyle#\hfil$}\cr
0.0pt{\hfil$\displaystyle{\big{[}\text{x}_{i,\,j}\,,\text{x}_{\ell,\,t}\big{]}}_{\sigma_{{}_{\mathcal{F}}}}=\;\big{[}\text{x}_{i,\,j}\,,\text{x}_{\ell,\,t}\big{]}\;\;\;,\quad{\big{[}\text{x}_{i,\,i}\,,\text{x}_{\ell,\,\ell}\big{]}}_{\sigma_{{}_{\mathcal{F}}}}=\;\big{[}\text{x}_{i,\,i}\,,\text{x}_{\ell,\,\ell}\big{]}\quad\qquad\forall\;\;i\not=j\,,\;\ell\not=t\cr
0.0pt{\hfil$\displaystyle{\big{[}\text{x}_{i,\,i}\,,\text{x}_{\ell,\,t}\big{]}}_{\sigma_{{}_{\mathcal{F}}}}\,=\;\big{[}\text{x}_{i,\,i}\,,\text{x}_{\ell,\,t}\big{]}\,+\,\Big{(}\,\overline{\phi}_{i,\ell}\,-\,\overline{\phi}_{i,t}\Big{)}\,\text{x}_{\ell,\,t}\qquad\qquad\forall\;\;\ell\not=t\cr
0.0pt{\hfil$\displaystyle{\big{[}\text{x}_{i,\,j}\,,\text{x}_{\ell,\,\ell}\big{]}}_{\sigma_{{}_{\mathcal{F}}}}\,:=\;\big{[}\text{x}_{i,\,i}\,,\text{x}_{\ell,\,t}\big{]}\,+\,\Big{(}\,\overline{\phi}_{i,\ell}\,-\,\overline{\phi}_{j,\ell}\Big{)}\,\text{x}_{i,\,j}\qquad\qquad\forall\;\;i\not=j\crcr}}}}$

###  3.3. Deformations by quasi-2-cocycle of QUEA’s

This subsection is dedicated to deformations by 2-cocycle of QUEA’s. In this
case the result that we achieve is somewhat surprising, in that we are indeed
“stretching the standard recipe”, as the 2-cocycles that we use to deform our
Hopf ${\Bbbk[[\hbar]]}$–algebras are valued in the field
${\Bbbk(\hskip-1.7pt(\hbar)\hskip-1.7pt)}$ rather than in our ground ring
${\Bbbk[[\hbar]]}\,$. Therefore, a priori nothing even guarantees that the
recipe would just work and produce a new Hopf algebra over
${\Bbbk[[\hbar]]}\,$; nonetheless, we eventually find quite a meaningful
result, which also says that the standard procedure of deformation by twist
for QUEA’s can be extended somewhat beyond its natural borders. We begin with
two ancillary results.

######  Lemma 3.3.1.

Let $\,U_{\hbar}:={U_{\hbar}(\hskip 0.5pt{\mathfrak{g}})}\,$ be any QUEA, and
$\,J_{\hbar}:=\textsl{Ker}\big{(}\epsilon_{U_{\hbar}}\big{)}\,$. For every
$\,z\in U_{\hbar}\,$, there exists $\,N\in\mathbb{N}\,$ such that
$\;\delta_{n}(z)\,\in\,\hbar^{\,\max(n,N)-N}J_{\hbar}^{\,\otimes n}\;$ for
every $\,n\in\mathbb{N}\,$.

###### Proof.

As Drinfeld’s functors are inverse to each other — cf. Theorem 2.4.2(a) —
applying ${(\ )}^{\vee}$ after ${(\ )}^{\prime}$ to the QUEA $U_{\hbar}$ we
get $\,U_{\hbar}={\big{(}U_{\hbar}^{\,\prime}\big{)}}^{\\!\vee}\,$: letting
$\,I_{\hbar}^{\,\prime}:=\hbar\,U_{\hbar}^{\,\prime}+\textsl{Ker}\\!\big{(}\epsilon_{U_{\hbar}^{\,\prime}}\big{)}\,$,
this last identity reads

$U_{\hbar}\,=\,\hbar\text{-adic completion of\ }\sum\limits_{n\geq
0}\hbar^{-n}{\big{(}I_{\hbar}^{\,\prime}\big{)}}^{n}\,=\,\hbar\text{-adic
completion of\ }\bigcup\limits_{n\geq
0}\hbar^{-n}{\big{(}I_{\hbar}^{\,\prime}\big{)}}^{n}$

In particular, this implies that for our $\,z\in{U_{\hbar}(\hskip
0.5pt{\mathfrak{g}})}\,$ there exist some $\,N\in\mathbb{N}\,$ and
$\,z^{\prime}\in{\big{(}I_{\hbar}^{\,\prime}\big{)}}^{N}\,$ such that
$\;z\,\equiv\hbar^{-N}z^{\prime}\,\big{(}\\!\\!\\!\mod\hbar\,{U_{\hbar}(\hskip
0.5pt{\mathfrak{g}})}\big{)}\;$. Now, given $\,n\in\mathbb{N}\,$ we have
$\,\delta_{n}\big{(}z^{\prime}\big{)}\in\hbar^{n}U_{\hbar}^{\,\otimes n}\,$
because $\,z^{\prime}\in{\big{(}I_{\hbar}^{\,\prime}\big{)}}^{N}\subseteq
U_{\hbar}^{\,\prime}\,$, and also
$\,\delta_{n}\big{(}z^{\prime}\big{)}\in\\!\\!{\textstyle\sum\limits_{s_{1}+\cdots+s_{n}=N}}\\!\\!\\!\otimes_{i=1}^{n}\\!{\big{(}I_{\hbar}^{\,\prime}\big{)}}^{s_{i}}\,$
because $I_{\hbar}^{\,\prime}$ is a Hopf ideal; moreover,
$\,\textsl{Ker}\\!\big{(}\epsilon_{U_{\hbar}^{\,\prime}}\big{)}\subseteq\hbar\,U_{\hbar}^{\,\prime}\,$
again by construction, hence
$\,I_{\hbar}^{\,\prime}\subseteq\hbar\,U_{\hbar}^{\,\prime}\,$. In the end,
all this yields
$\;\delta_{n}\big{(}z^{\prime}\big{)}\in\hbar^{\,\max(n,N)}\,J_{\hbar}^{\,\otimes
n}\,$, therefore
$\;\delta_{n}(z)\,\in\,\hbar^{\,\max(n,N)-N}J_{\hbar}^{\,\otimes n}\;$ as
claimed. ∎

For the second, auxiliary result, we fix some more notation: namely, hereafter
by “$\log_{*}$” and “$\exp_{*}$” we denote the logarithm and the exponential
with respect to the convolution product, whenever defined.

######  Lemma 3.3.2.

Let ${U_{\hbar}(\hskip 0.5pt{\mathfrak{g}})}$ be any QUEA, and let $\chi$ be a
${\Bbbk[[\hbar]]}$–bilinear form on ${U_{\hbar}(\hskip 0.5pt{\mathfrak{g}})}$
such that $\,\chi(z\,,1)=0=\chi(1\,,z)\,$ for any $\,z\in{U_{\hbar}(\hskip
0.5pt{\mathfrak{g}})}\,$; denote also by the same symbol $\chi$ the scalar
extension of $\chi$ to a ${\Bbbk(\hskip-1.7pt(\hbar)\hskip-1.7pt)}$–bilinear
form for the ${\Bbbk(\hskip-1.7pt(\hbar)\hskip-1.7pt)}$–vector space
$\,{\mathbb{U}_{\hbar}(\hskip
0.5pt{\mathfrak{g}})}:={\Bbbk(\hskip-1.7pt(\hbar)\hskip-1.7pt)}\otimes_{\Bbbk[[\hbar]]}{U_{\hbar}(\hskip
0.5pt{\mathfrak{g}})}\,$. Then: (a) the formal expression
$\,\sigma:=\exp_{*}\\!\big{(}\hbar^{-1}\chi\big{)}\,$ uniquely provides a
well-defined, ${\Bbbk(\hskip-1.7pt(\hbar)\hskip-1.7pt)}$–valued bilinear form
for $\,{\mathbb{U}_{\hbar}(\hskip 0.5pt{\mathfrak{g}})}\,$; (b)
$\,\sigma(z\,,1)=\epsilon(z)=\sigma(1\,,z)\,$ for any $\,z\in{U_{\hbar}(\hskip
0.5pt{\mathfrak{g}})}\,$; (c) $\sigma$ is orthogonal, i.e.
$\,\sigma_{2,1}\\!=\sigma^{\,-1}$, iff $\chi$ is antisymmetric, i.e.
$\,\chi_{2,1}=-\chi\,$.

###### Proof.

(a) Fix notation $\,U_{\hbar}:={U_{\hbar}(\hskip 0.5pt{\mathfrak{g}})}\,$ and
$\,J_{\hbar}:=\textsl{Ker}\big{(}\epsilon_{U_{\hbar}}\big{)}\,$. For any
$\,z\in U_{\hbar}\,$, set

$\hat{z}:=\epsilon(z)\,,\;\;\;z^{+}:=z-\epsilon(z)=z-\hat{z}\,\in\,J_{\hbar}\;,\quad\text{hence}\quad
z=z^{+}+\hat{z}$ (3.22)

The assumption $\,\chi(z\,,1)=0=\chi(1\,,z)\,$ for $\,z\in{U_{\hbar}(\hskip
0.5pt{\mathfrak{g}})}\,$ implies (for all $\,u,v\in U_{\hbar}\,$)

$\chi(u,v)\,=\,\chi\big{(}u^{+}\\!+\hat{u}\,,v^{+}\\!+\hat{v}\big{)}\,=\,\chi\big{(}u^{+},v^{+}\big{)}$
(3.23)

Now, for any $\,a\,,b\in U_{\hbar}\,$, the formula
$\;\sigma=\exp_{*}\\!\big{(}\hbar^{-1}\chi\,\big{)}\,=\,\sum_{n\,\geq
0}\hbar^{-n}\chi^{*\,n}\big{/}n!\;$ gives

$\displaystyle\hskip-5.0pt\sigma(a\,,b\,)\;=\,{\textstyle\sum_{n\,\geq
0}}\,\hbar^{-n}\chi^{*\,n}(a\,,b\,)\Big{/}n!\,\;=$ (3.24) $\displaystyle\hskip
23.0pt=\;{\textstyle\sum_{n\,\geq
0}}\,\hbar^{-n}{\textstyle\prod_{i=1}^{n}}\chi\big{(}a_{(i)},b_{(i)}\big{)}\Big{/}n!\;=\;{\textstyle\sum_{n\,\geq
0}}\,\hbar^{-n}{\textstyle\prod_{i=1}^{n}}\chi\big{(}a^{+}_{(i)},b^{+}_{(i)}\big{)}\Big{/}n!$

where we took into account that
$\,\chi^{*\,k}(u\,,v\,)\,=\,\prod_{s=1}^{k}\\!\chi\big{(}u_{(s)},v_{(s)}\big{)}\,=\,\prod_{s=1}^{k}\\!\chi\big{(}u^{+}_{(s)},v^{+}_{(s)}\big{)}\,$
for each $\,u,v\in U_{\hbar}\,$, $\,k\in\mathbb{N}\,$, by definitions along
with (3.23). Now we notice that
$\,\otimes_{i=1}^{n}a^{+}_{(i)}=\delta_{n}(a)\,$ and
$\,\otimes_{i=1}^{n}b^{+}_{(i)}=\delta_{n}(b)\,$, hence Lemma 3.3.1 guarantees
that

$h^{-n}{\textstyle\prod\limits_{i=1}^{n}}\chi\big{(}a^{+}_{(i)},b^{+}_{(i)}\big{)}\;\in\;\hbar^{-n+\,\max(n,A)-A+\max(n,B)-B}$
$None$

whence in particular

$\displaystyle\;\;h^{-n}{\textstyle\prod\limits_{i=1}^{n}}\chi\big{(}a^{+}_{(i)},b^{+}_{(i)}\big{)}\;\in\;\hbar^{-\min(A,B)}\,{\Bbbk[[\hbar]]}\qquad\forall\;\;n\in\mathbb{N}_{+}$
(3.25) $\displaystyle
h^{-n}{\textstyle\prod\limits_{i=1}^{n}}\chi\big{(}a^{+}_{(i)},b^{+}_{(i)}\big{)}\;\in\;\hbar^{n-(A+B)}\,{\Bbbk[[\hbar]]}\qquad\forall\;\;n\geq
A+B$

where $\,A\in\mathbb{N}\,$, resp. $\,B\in\mathbb{N}\,$, plays for $a\,$, resp.
for $b\,$, the role of $N$ for $z$ in Lemma 3.3.1 above; by this, the formal
expansion for $\sigma(a\,,b\,)$ in (3.24) yields a well defined element in
${\Bbbk[[\hbar]]}\,$, hence $\sigma$ is a well-defined
${\Bbbk(\hskip-1.7pt(\hbar)\hskip-1.7pt)}$–bilinear form of
${\mathbb{U}_{\hbar}(\hskip 0.5pt{\mathfrak{g}})}$ as claimed. (b–c) Both
claims are obvious, by construction, as they follow from standard identities
for formal exponentials. ∎

The previous result leads us to introduce the following notion:

######  Definition 3.3.3.

Let ${U_{\hbar}(\hskip 0.5pt{\mathfrak{g}})}$ be a QUEA, and
$\,{\mathbb{U}_{\hbar}(\hskip
0.5pt{\mathfrak{g}})}:={\Bbbk(\hskip-1.7pt(\hbar)\hskip-1.7pt)}\otimes_{\Bbbk[[\hbar]]}{U_{\hbar}(\hskip
0.5pt{\mathfrak{g}})}\,$. Note that ${\mathbb{U}_{\hbar}(\hskip
0.5pt{\mathfrak{g}})}$ has a natural “Hopf algebra structure” of
${\mathbb{U}_{\hbar}(\hskip 0.5pt{\mathfrak{g}})}$ induced by scalar extension
from ${U_{\hbar}(\hskip 0.5pt{\mathfrak{g}})}$ — so that, in particular, the
“coproduct” takes values in
$\,{\Bbbk(\hskip-1.7pt(\hbar)\hskip-1.7pt)}\otimes_{\Bbbk[[\hbar]]}\\!\big{(}{U_{\hbar}(\hskip
0.5pt{\mathfrak{g}})}\,\widehat{\otimes}_{\Bbbk[[\hbar]]}{U_{\hbar}(\hskip
0.5pt{\mathfrak{g}})}\big{)}\,$ rather than in $\,{\mathbb{U}_{\hbar}(\hskip
0.5pt{\mathfrak{g}})}\otimes_{\Bbbk(\hskip-1.7pt(\hbar)\hskip-1.7pt)}{\mathbb{U}_{\hbar}(\hskip
0.5pt{\mathfrak{g}})}\,$.

We call quasi-2-cocycle of ${U_{\hbar}(\hskip 0.5pt{\mathfrak{g}})}$ any
${\Bbbk(\hskip-1.7pt(\hbar)\hskip-1.7pt)}$–bilinear form $\,\sigma$ of
${\mathbb{U}_{\hbar}(\hskip 0.5pt{\mathfrak{g}})}$ which has the form
$\,\sigma:=\exp_{*}\\!\big{(}\hbar^{-1}\chi\big{)}\,$ for some
${\Bbbk[[\hbar]]}$–bilinear form $\,\chi\in{\Big{(}{U_{\hbar}(\hskip
0.5pt{\mathfrak{g}})}^{\widehat{\otimes}2}\Big{)}}^{*}\,$ of
${U_{\hbar}(\hskip 0.5pt{\mathfrak{g}})}$ such that
$\,\chi(z\,,1)=0=\chi(1\,,z)\,$ for all $\,z\in{U_{\hbar}(\hskip
0.5pt{\mathfrak{g}})}\,$, and in addition enjoys the 2–cocycle properties with
respect to the above “Hopf algebra structure” of ${\mathbb{U}_{\hbar}(\hskip
0.5pt{\mathfrak{g}})}\,$.

######  Remark 3.3.4.

The notion of “quasi-2-cocycle” for a QUEA ${U_{\hbar}(\hskip
0.5pt{\mathfrak{g}})}$ can also be cast in the following, equivalent shape.
Recall that $\,{F_{\hbar}[[G\hskip 1.5pt]]}:={{U_{\hbar}(\hskip
0.5pt{\mathfrak{g}})}}^{*}\,$ is a QFSHA (cf. §2.3.4), and then
$\,{\Big{(}{U_{\hbar}(\hskip
0.5pt{\mathfrak{g}})}^{\widehat{\otimes}2}\Big{)}}^{*}={{U_{\hbar}(\hskip
0.5pt{\mathfrak{g}})}}^{*}\,\widetilde{\otimes}\,{{U_{\hbar}(\hskip
0.5pt{\mathfrak{g}})}}^{*}={F_{\hbar}[[G\hskip
1.5pt]]}\,\widetilde{\otimes}\,{F_{\hbar}[[G\hskip 1.5pt]]}\,$. Given
$\,\chi\in{\Big{(}{U_{\hbar}(\hskip
0.5pt{\mathfrak{g}})}^{\widehat{\otimes}2}\Big{)}}^{*}\,$ as in Definition
3.3.3 above, the condition $\,\chi(z\,,1)=0=\chi(1\,,z)\,$ for all
$\,z\in{U_{\hbar}(\hskip 0.5pt{\mathfrak{g}})}\,$ means that $\,\chi\in
J_{{F_{\hbar}[[G\hskip 1.5pt]]}}\,\widetilde{\otimes}\,J_{{F_{\hbar}[[G\hskip
1.5pt]]}}\,$, with $\,J_{{F_{\hbar}[[G\hskip
1.5pt]]}}:=\textsl{Ker}\big{(}\epsilon_{{F_{\hbar}[[G\hskip
1.5pt]]}}\big{)}\,$, hence we have
$\;\chi\,\in\,\hbar^{2}\,{\big{(}J_{{F_{\hbar}[[G\hskip
1.5pt]]}}^{\,\vee}\big{)}}^{\widehat{\otimes}\,2}\subseteq\,\hbar^{2}\,{\big{(}{{F_{\hbar}[[G\hskip
1.5pt]]}}^{\vee}\big{)}}^{\widehat{\otimes}\,2}\;$ where
$\,J_{{F_{\hbar}[[G\hskip 1.5pt]]}}^{\,\vee}:=\hbar^{-1}J_{{F_{\hbar}[[G\hskip
1.5pt]]}}\,$ and $\,{{F_{\hbar}[[G\hskip 1.5pt]]}}^{\vee}$ is the QUEA defined
in §2.4 out of ${F_{\hbar}[[G\hskip 1.5pt]]}\,$. Thus, it follows that
$\;\hbar^{-1}\chi\,\in\,\hbar\,{\big{(}{{F_{\hbar}[[G\hskip
1.5pt]]}}^{\vee}\big{)}}^{\widehat{\otimes}\,2}$, so
$\,\sigma:=\exp_{*}\\!\big{(}\hbar^{-1}\chi\big{)}\,$ is a well-defined
element in ${\big{(}{{F_{\hbar}[[G\hskip
1.5pt]]}}^{\vee}\big{)}}^{\widehat{\otimes}\,2}\,$.

Now, the requirement that
$\,\sigma:=\exp_{*}\\!\big{(}\hbar^{-1}\chi\big{)}\,$ be a quasi-2-cocycle for
${U_{\hbar}(\hskip 0.5pt{\mathfrak{g}})}$ in the sense of Definition 3.3.3
above is equivalent to the property of $\sigma$ being a twist element for
${{F_{\hbar}[[G\hskip 1.5pt]]}}^{\vee}\,$ — which makes perfectly sense in
sight of Proposition 2.2.7.

Clearly, every 2-cocycle for ${U_{\hbar}(\hskip 0.5pt{\mathfrak{g}})}$ is a
quasi-2-cocycle as well; the converse, instead, is not true, in general
(counterexamples do exist). However, the key point is that every
quasi-2-cocycle still provides a well-defined deformation by 2-cocycle of
$\,{U_{\hbar}(\hskip 0.5pt{\mathfrak{g}})}$ — in short, the construction of
deformations by 2-cocycle does properly extend to “deformations by
quasi-2-cocycle” as well: this is indeed our next result.

######  Theorem 3.3.5.

Let ${U_{\hbar}(\hskip 0.5pt{\mathfrak{g}})}$ be a QUEA, and
$\,\sigma=\exp_{*}\\!\big{(}\hbar^{-1}\chi\big{)}\,$ a quasi-2-cocycle for it,
as in Definition 3.3.3. Then the procedure of 2-cocycle deformation by
$\,\sigma$ applied to $\,{\mathbb{U}_{\hbar}(\hskip 0.5pt{\mathfrak{g}})}$
actually restricts to $\,{U_{\hbar}(\hskip 0.5pt{\mathfrak{g}})}$, making the
latter into a new QUEA.

###### Proof.

First of all, we have to explain the statement itself. To begin with, note
that, by definitions and by Lemma 3.3.2, we can perform the deformation by the
2–cocycle $\sigma$ onto the “Hopf
$\,{\Bbbk(\hskip-1.7pt(\hbar)\hskip-1.7pt)}$–algebra”
$\,{\mathbb{U}_{\hbar}(\hskip
0.5pt{\mathfrak{g}})}:={\Bbbk(\hskip-1.7pt(\hbar)\hskip-1.7pt)}\,\widehat{\otimes}\,{U_{\hbar}(\hskip
0.5pt{\mathfrak{g}})}\,$. Our statement then claims the resulting deformed
Hopf structure onto ${\mathbb{U}_{\hbar}(\hskip 0.5pt{\mathfrak{g}})}$
actually “restricts” to a deformation of $\,{U_{\hbar}(\hskip
0.5pt{\mathfrak{g}})}$ itself: in turn, this amounts to claiming that
${U_{\hbar}(\hskip 0.5pt{\mathfrak{g}})}$ is closed for the $\sigma$–deformed
product in ${\big{(}{\mathbb{U}_{\hbar}(\hskip
0.5pt{\mathfrak{g}})}\big{)}}_{\sigma}\,$ — so we go and tackle this last
problem. Fix notation $\,U_{\hbar}:={U_{\hbar}(\hskip
0.5pt{\mathfrak{g}})}\,$,
$\,J_{\hbar}:=\textsl{Ker}\big{(}U_{\hbar}\big{)}\,$,
$\,J_{\hbar}^{\,\prime}:=\textsl{Ker}\big{(}U_{\hbar}^{\,\prime}\big{)}\,$ and
$\,\widetilde{J_{\hbar}}:=\hbar^{-1}J_{\hbar}^{\,\prime}\,$, where
$\,U_{\hbar}^{\,\prime}:={U_{\hbar}(\hskip 0.5pt{\mathfrak{g}})}^{\prime}\,$
is given in Definition 2.4.1(a). As it was mentioned in the proof of Lemma
3.3.1, Theorem 2.4.2(a) implies that
$\,U_{\hbar}={\big{(}U_{\hbar}^{\,\prime}\big{)}}^{\\!\vee}\,$, that is

$U_{\hbar}\,=\,\hbar\text{-adic completion of\ }\sum\limits_{n\geq
0}\hbar^{-n}{\big{(}I_{\hbar}^{\,\prime}\big{)}}^{n}$

where
$\,I_{\hbar}^{\,\prime}:=\hbar\,U_{\hbar}^{\,\prime}+\textsl{Ker}\\!\big{(}\epsilon_{U_{\hbar}^{\,\prime}}\big{)}=\hbar\,U_{\hbar}^{\,\prime}+J_{\hbar}^{\,\prime}\,$;
then a moment’s thought shows that the previous expression of $U_{\hbar}$
reads also

$U_{\hbar}\,=\,\hbar\text{-adic completion of\ }{\textstyle\sum\limits_{n\geq
0}}\hbar^{-n}{\big{(}J_{\hbar}^{\,\prime}\big{)}}^{n}\,=\,\hbar\text{-adic
completion of\ }{\textstyle\sum\limits_{n\geq
0}}\widetilde{J}_{\hbar}^{\raisebox{3.0pt}{$\;\scriptstyle n$}}$ (3.26)

Note also that clearly $J_{\hbar}^{\,\prime}$ is a Hopf ideal in
$U_{\hbar}^{\,\prime}$, and moreover
$\;J_{\hbar}^{\,\prime}\,\subseteq\,\hbar\,J_{\hbar}\;$ (by construction);
therefore for $\,z^{\prime}\in
J_{\hbar}^{\,\prime\\!\raisebox{1.0pt}{$\,\scriptstyle N$}}\,$ — with
$\,N\in\mathbb{N}\,$ — acting like in the proof of Lemma 3.3.1, one sees that

$\qquad\delta_{n}\big{(}z^{\prime}\big{)}\;\in\;\hbar^{n}J_{\hbar}^{\,\otimes
n}\;{\textstyle\bigcap}\;\Big{(}\,{\textstyle\sum_{\sum_{i}N_{i}=N}\bigotimes_{i=1}^{n}}J_{\hbar}^{\,\prime\\!\raisebox{1.0pt}{$\,\scriptstyle
N_{i}$}}\Big{)}\;\subseteq\;\hbar^{\,\max(n,N)}J_{\hbar}^{\,\otimes n}$ (3.27)

Again, for any $\,z\in U_{\hbar}\,$ we retain notation as in (3.22) above,
that is

$\hat{z}:=\epsilon(z)\,,\;\;\;z^{+}:=z-\epsilon(z)=z-\hat{z}\,\in\,J_{\hbar}\;,\quad\text{hence}\quad
z=z^{+}+\hat{z}$ (3.28)

and we recall also that for all $\,u,v\in U_{\hbar}\,$ we have

$\chi(u,v)\,=\,\chi\big{(}u^{+}\\!+\hat{u}\,,v^{+}\\!+\hat{v}\big{)}\,=\,\chi\big{(}u^{+},v^{+}\big{)}$
(3.29)

Thanks to (3.26), in order to prove that $\,{U_{\hbar}(\hskip
0.5pt{\mathfrak{g}})}=:U_{\hbar}\,$ is closed for the $\sigma$–deformed
product $\dot{\scriptstyle\sigma}$ it is enough to show that
$\;\widetilde{J}_{\hbar}^{\raisebox{1.0pt}{$\;\scriptstyle
A$}}\,\raisebox{-5.0pt}{$\dot{\scriptstyle\sigma}$}\;\widetilde{J}_{\hbar}^{\raisebox{1.0pt}{$\;\scriptstyle
B$}}\;\subseteq{\textstyle\sum\limits_{n\geq
0}}\widetilde{J}_{\hbar}^{\raisebox{3.0pt}{$\;\scriptstyle n$}}\;$ for any
$\,A\,,B\in\mathbb{N}_{+}\,$.

To begin with, we pick
$\,a\in\widetilde{J}_{\hbar}^{\raisebox{1.0pt}{$\;\scriptstyle
A$}}=\hbar^{-A}J_{\hbar}^{\,\prime\\!\raisebox{1.0pt}{$\;\scriptstyle A$}}\,$
and $\,b\in\widetilde{J}_{\hbar}^{\raisebox{1.0pt}{$\;\scriptstyle
B$}}=\hbar^{-B}J_{\hbar}^{\,\prime\\!\raisebox{1.0pt}{$\;\scriptstyle B$}}\,$;
by definition,

$a\,\raisebox{-5.0pt}{$\dot{\scriptstyle\sigma}$}\,b\;:=\;\sigma\big{(}a_{(1)},b_{(1)}\big{)}\,a_{(2)}\,b_{(2)}\,\sigma^{-1}\big{(}a_{(3)},b_{(3)}\big{)}$

whence expanding the formal formula
$\;\sigma=\exp_{*}\\!\big{(}\hbar^{-1}\chi\,\big{)}\,=\,\sum_{n\,\geq
0}\hbar^{-n}\chi^{*\,n}\big{/}n!\;$ — much like in the proof of Lemma 3.3.2 —
we get

$\displaystyle\hskip-3.0pta\,\raisebox{-5.0pt}{$\dot{\scriptstyle\sigma}$}\,b\,\;=\;\,{\textstyle\sum_{t,\ell\,\geq
0}}\,\hbar^{-(t+\ell)}\,{{(-1)}^{\ell}}\,{(t!\,\ell!)}^{-1}\,\chi^{*\,t}\big{(}a_{(1)},b_{(1)}\big{)}\,a_{(2)}\,b_{(2)}\,\chi^{*\,\ell}\big{(}a_{(3)},b_{(3)}\big{)}\,\;=$
(3.30) $\displaystyle\hskip
2.0pt=\;\,\epsilon\big{(}a^{\prime}_{(1)}\big{)}\,\epsilon\big{(}b^{\prime}_{(1)}\big{)}\,a^{\prime}_{(2)}\,b^{\prime}_{(2)}\,\epsilon\big{(}a^{\prime}_{(3)}\big{)}\,\epsilon\big{(}b^{\prime}_{(3)}\big{)}\;+$
$\displaystyle\hskip
7.0pt+\,{\textstyle\sum_{t+\ell\,>0}}\,\hbar^{-(t+\ell)}\,{(-1)}^{\ell}\,{(t!)}^{-1}{(\ell!)}^{-1}\,\chi^{*\,t}\big{(}a_{(1)},b_{(1)}\big{)}\,a_{(2)}\,b_{(2)}\,\chi^{*\,\ell}\big{(}a_{(3)},b_{(3)}\big{)}\,\;=$
$\displaystyle\hskip-3.0pt=\;\,a\cdot b\,\;+$ $\displaystyle\hskip
15.0pt+\,{\textstyle\sum_{t+\ell\,>0}}\,\hbar^{-(t+\ell)}\,{(-1)}^{\ell}\,{(t!)}^{-1}{(\ell!)}^{-1}\,\chi^{*\,t}\big{(}a_{(1)},b_{(1)}\big{)}\,a_{(2)}\,b_{(2)}\,\chi^{*\,\ell}\big{(}a_{(3)},b_{(3)}\big{)}$

where we took into account coassociativity and counitality properties. Let us
analyze each summand in the very last line in (3.30). From the identities
$\;\chi^{*\,k}(u\,,v\,)\,=\,\prod_{s=1}^{k}\chi\big{(}u_{(s)},v_{(s)}\big{)}\,=\,\
\prod_{s=1}^{k}\chi\big{(}u^{+}_{(s)},v^{+}_{(s)}\big{)}\;$ — cf. (3.29) — we
get

$\halign{\hbox to\displaywidth{$\hfil\displaystyle#\hfil$}\cr
0.0pt{\hfil$\displaystyle\quad\chi^{*\,t}\big{(}a_{(1)},b_{(1)}\big{)}\,a_{(2)}\,b_{(2)}\,\chi^{*\,\ell}\big{(}a_{(3)},b_{(3)}\big{)}\;=\hfill\cr
0.0pt{\hfil$\displaystyle=\;{\textstyle\prod\limits_{i=1}^{t}}\,\chi\big{(}a_{(i)},b_{(i)}\big{)}\,a_{(t+1)}\,b_{(t+1)}\,{\textstyle\prod\limits_{j=1}^{\ell}}\,\chi\big{(}a_{(t+1+j)},b_{(t+1+j)}\big{)}\;=\cr
0.0pt{\hfil$\displaystyle\hfill=\;{\textstyle\prod\limits_{i=1}^{t}}\,\chi\big{(}a^{+}_{(i)},b^{+}_{(i)}\big{)}\,a_{(t+1)}\,b_{(t+1)}\,{\textstyle\prod\limits_{j=1}^{\ell}}\,\chi\big{(}a^{+}_{(t+1+j)},b^{+}_{(t+1+j)}\big{)}\quad\crcr}}}}$

Consider the expansion of $a$ as in (3.28). Then, letting
$\,j_{t+1}:U_{\hbar}^{\,\otimes\,(t+\ell)}\\!\relbar\joinrel\longrightarrow
U_{\hbar}^{\,\otimes\,(t+\ell+1)}\,$ be the map given by
$\;\mathop{\otimes}\limits_{s=1}^{t+\ell}x_{s}\,\mapsto\,\Big{(}\mathop{\otimes}\limits_{s=1}^{t}x_{s}\Big{)}\otimes
1\otimes\Big{(}\mathop{\otimes}\limits_{s=t+1}^{t+\ell}x_{s}\Big{)}\;$, we
have

$\halign{\hbox to\displaywidth{$\hfil\displaystyle#\hfil$}\cr
0.0pt{\hfil$\displaystyle\Big{(}\mathop{\otimes}\limits_{i=1}^{t}\\!a^{+}_{(i)}\Big{)}\otimes
a^{+}_{(t+1)}\otimes\Big{(}\mathop{\otimes}\limits_{j=1}^{\ell}\\!a^{+}_{(t+1+j)}\Big{)}\;=\;\delta_{t+\ell+1}(a)\,\;\in\;\,\hbar^{\,\max(t+\ell+1,A)\,-A}\,U_{\hbar}^{\,\otimes(t+\ell+1)}\cr
0.0pt{\hfil$\displaystyle\Big{(}\mathop{\otimes}\limits_{i=1}^{t}\\!a^{+}_{(i)}\Big{)}\otimes\widehat{a}_{(t+1)}\otimes\Big{(}\mathop{\otimes}\limits_{j=1}^{\ell}\\!a^{+}_{(t+1+j)}\Big{)}\;=\;j_{t+1}\big{(}\delta_{t+\ell}(a)\big{)}\,\;\in\;\,\hbar^{\,\max(t+\ell,A)\,-A}\,U_{\hbar}^{\,\otimes(t+\ell+1)}\crcr}}}$

so that, summing up,

$\Big{(}\mathop{\otimes}\limits_{i=1}^{t}\\!a^{+}_{(i)}\\!\Big{)}\otimes
a_{\,t+1}\otimes\Big{(}\mathop{\otimes}\limits_{j=1}^{\ell}\\!a^{+}_{(t+1+j)}\\!\Big{)}=\delta_{t+\ell+1}(a)+j_{t+1}\big{(}\delta_{t+\ell}(a)\\!\big{)}\in\hbar^{\,\max(t+\ell,A)\,-A}\,U_{\hbar}^{\,\otimes(t+\ell+1)}$

— like in the proof of Lemma 3.3.1 — and similarly with $b\,$, resp. $B\,$,
replacing $a\,$, resp. $A\,$. Eventually, for all $\,t+\ell>0\,$ this gives

$\displaystyle\hskip-5.0pt\chi^{*\,t}\big{(}a_{(1)},b_{(1)}\big{)}\,a_{(2)}\,b_{(2)}\,\chi^{*\,\ell}\big{(}a_{(3)},b_{(3)}\big{)}\;=\hfill$
(3.31)
$\displaystyle\hskip-1.0pt=\,\chi^{*\,t}\big{(}a_{(1)},b_{(1)}\big{)}\,\widehat{a}_{(2)}\,b_{(2)}\,\chi^{*\,\ell}\big{(}a_{(3)},b_{(3)}\big{)}\,+\,\chi^{*\,t}\big{(}a_{(1)},b_{(1)}\big{)}\,a^{+}_{(2)}\,b_{(2)}\,\chi^{*\,\ell}\big{(}a_{(3)},b_{(3)}\big{)}$

where for the two summands in second line, writing $\,n:=t+\ell\,$, we have

$\halign{\hbox to\displaywidth{$\hfil\displaystyle#\hfil$}\cr
0.0pt{\hfil$\displaystyle\chi^{*\,t}\big{(}a_{(1)},b_{(1)}\big{)}\,\widehat{a}_{(2)}\,b_{(2)}\,\chi^{*\,\ell}\big{(}a_{(3)},b_{(3)}\big{)}\;=\hfill\cr
0.0pt{\hfil$\displaystyle\hfill=\;{\textstyle\prod\limits_{i=1}^{t}}\,\chi\big{(}a^{+}_{(i)},b^{+}_{(i)}\big{)}\,\widehat{a}_{(t+1)}\,b_{(t+1)}\,{\textstyle\prod\limits_{k=t+2}^{n+1}}\chi\big{(}a^{+}_{(k)},b^{+}_{(k)}\big{)}\;\in\;\hbar^{\,\max(n,A)\,-A+\,\max(n,B)\,-B}\,U_{\hbar}^{\,\otimes(n+1)}\cr
0.0pt{\hfil$\displaystyle\chi^{*\,t}\big{(}a_{(1)},b_{(1)}\big{)}\,a^{+}_{(2)}\,b_{(2)}\,\chi^{*\,\ell}\big{(}a_{(3)},b_{(3)}\big{)}\;=\hfill\cr
0.0pt{\hfil$\displaystyle\hfill=\;{\textstyle\prod\limits_{i=1}^{t}}\,\chi\big{(}a^{+}_{(i)},b^{+}_{(i)}\big{)}\,a^{+}_{(t+1)}\,b_{(t+1)}\,{\textstyle\prod\limits_{k=t+2}^{n+1}}\chi\big{(}a^{+}_{(k)},b^{+}_{(k)}\big{)}\;\in\;\hbar^{\,\max(n+1,A)\,-A+\,\max(n,B)\,-B}\,U_{\hbar}^{\,\otimes(n+1)}\crcr}}}}}$

Let us now assume that $\,A:=1\,$, so that $\,n:=t+\ell>0\,$ implies
$\,n:=t+\ell\geq 1=A\,$. Then the last estimates read

$\displaystyle\chi^{*\,t}\big{(}a_{(1)},b_{(1)}\big{)}\,\widehat{a}_{(2)}\,b_{(2)}\,\chi^{*\,\ell}\big{(}a_{(3)},b_{(3)}\big{)}\;\in\;\hbar^{\,n-1+\,\max(n,B)\,-B}\,U_{\hbar}^{\,\otimes\,(n+1)}$
(3.32) $\displaystyle\hskip
5.0pt\chi^{*\,t}\big{(}a_{(1)},b_{(1)}\big{)}\,a^{+}_{(2)}\,b_{(2)}\,\chi^{*\,\ell}\big{(}a_{(3)},b_{(3)}\big{)}\;\in\;\hbar^{\,n+\,\max(n,B)\,-B}\,U_{\hbar}^{\,\otimes\,(n+1)}$

The term in the second line, when plugged in (3.31) and then in (3.30), yields
a contribution of the form

${{\,{(-1)}^{\ell}\,}\over{\,t!\,\ell!\,}}\;\hbar^{-n}\,\chi^{*\,t}\big{(}a_{(1)},b_{(1)}\big{)}\,a^{+}_{(2)}\,b_{(2)}\,\chi^{*\,\ell}\big{(}a_{(3)},b_{(3)}\big{)}\;\in\;\hbar^{\,\max(n,B)\,-B}\,U_{\hbar}^{\,\otimes\,(n+1)}$

that belongs to $\,\hbar^{\,\max(n,B)\,-B}\,U_{\hbar}^{\,\otimes\,(n+1)}\,$,
thus for growing $n$ these elements sum up to a convergent series in
$U_{\hbar}\,$, and we are done.

As to the term in the first line, we split it into

$\displaystyle\hskip-5.0pt\chi^{*\,t}\big{(}a_{(1)},b_{(1)}\big{)}\,\widehat{a}_{(2)}\,b_{(2)}\,\chi^{*\,\ell}\big{(}a_{(3)},b_{(3)}\big{)}\;=$
(3.33)
$\displaystyle\hskip-3.0pt=\;\chi^{*\,t}\big{(}a_{(1)},b_{(1)}\big{)}\,\widehat{a}_{(2)}\,\widehat{b}_{(2)}\,\chi^{*\,\ell}\big{(}a_{(3)},b_{(3)}\big{)}\,+\,\chi^{*\,t}\big{(}a_{(1)},b_{(1)}\big{)}\,\widehat{a}_{(2)}\,b^{+}_{(2)}\,\chi^{*\,\ell}\big{(}a_{(3)},b_{(3)}\big{)}$

Then for the first summand we have (almost by definition, or acting as before)

$\chi^{*\,t}\big{(}a_{(1)},b_{(1)}\big{)}\,\widehat{a}_{(2)}\,\widehat{b}_{(2)}\,\chi^{*\,\ell}\big{(}a_{(3)},b_{(3)}\big{)}\;=\;\chi^{*\,(t+\ell\,)}(a\,,b\,)$

so when we plug every such term in (3.31) and then in (3.30), overall they sum
up to give the contribution

$\halign{\hbox to\displaywidth{$\hfil\displaystyle#\hfil$}\cr
0.0pt{\hfil$\displaystyle\quad{\textstyle\sum_{t+\ell\,>0}}\,\hbar^{-(t+\ell\,)}\,{(-1)}^{\ell}\,{(t!)}^{-1}{(\ell!)}^{-1}\,\chi^{*\,t}\big{(}a_{(1)},b_{(1)}\big{)}\,\widehat{a}_{(2)}\,\widehat{b}_{(2)}\,\chi^{*\,\ell}\big{(}a_{(3)},b_{(3)}\big{)}\;=\hfill\cr
0.0pt{\hfil$\displaystyle=\;{\textstyle\sum\limits_{n\,>\,0}}\;{\textstyle\sum_{t+\ell\,=\,n}}\,\hbar^{-(t+\ell\,)}\,{(-1)}^{\ell}\,{(t!)}^{-1}{(\ell!)}^{-1}\,\chi^{*\,(t+\ell)}(a\,,b\,)\;=\cr
0.0pt{\hfil$\displaystyle\hfill=\;{\textstyle\sum\limits_{n\,>\,0}}\;{{\,1\,}\over{\,n!\,}}\,\hbar^{-n}\,\bigg{(}\,{\textstyle\sum\limits_{t+\ell\,=\,n}}{(-1)}^{\ell}\,{n\choose\ell}\\!\bigg{)}\;\chi^{*\,n}(a\,,b\,)\;=\;\,0\quad\crcr}}}}$

just because of the combinatorial identity
$\,\;{\textstyle\sum\limits_{t+\ell=n}}{(-1)}^{\ell}\,\displaystyle{n\choose\ell}\;=\;0\;\,$.
Finally, we have to dispose of the summands of type

$\chi^{*\,t}\big{(}a_{(1)},b_{(1)}\big{)}\,\widehat{a}_{(2)}\,b^{+}_{(2)}\,\chi^{*\,\ell}\big{(}a_{(3)},b_{(3)}\big{)}$
(3.34)

for which the analogue of the first identity in (3.32) holds true, namely

$\chi^{*\,t}\big{(}a_{(1)},b_{(1)}\big{)}\,\widehat{a}_{(2)}\,b^{+}_{(2)}\,\chi^{*\,\ell}\big{(}a_{(3)},b_{(3)}\big{)}\;\in\;\hbar^{\,n-1+\,\max(n+1,B)\,-B}\,U_{\hbar}^{\,\otimes\,(n+1)}$
(3.35)

where $\,n:=t+\ell\,$, taking into account that
$\,\delta_{n+1}(b)\,\in\,\hbar^{\,\max(n+1,B)\,-B}\,U_{\hbar}^{\,\otimes\,(n+1)}\;$.

Then we have to distinguish two cases, depending on $\,n:=t+\ell\,$. First we
assume $\;n:=t+\ell\geq B\;$. Then $\;n-1+\,\max(n+1,B)\,-B\,\geq\,n\;$, hence
the first identity in (3.32) yields
$\,\;\chi^{*\,t}\big{(}a_{(1)},b_{(1)}\big{)}\,\widehat{a}_{(2)}\,b^{+}_{(2)}\,\chi^{*\,\ell}\big{(}a_{(3)},b_{(3)}\big{)}\,\in\,\hbar^{\,n}\,U_{\hbar}^{\,\otimes\,(n+1)}\;$,
and then, when plugged in (3.33), and subsequently in (3.31) and in (3.30),
this provides to the expansion of
$\;a\,\raisebox{-5.0pt}{$\dot{\scriptstyle\sigma}$}\,b\;$ a contribution of
the form

$\hbar^{-n}\,{{\,{(-1)}^{\ell}\,}\over{\,t!\,\ell!\,}}\,\chi^{*\,t}\big{(}a_{(1)},b_{(1)}\big{)}\,\widehat{a}_{(2)}\,b^{+}_{(2)}\,\chi^{*\,\ell}\big{(}a_{(3)},b_{(3)}\big{)}\;\in\;\hbar^{-n}\,\hbar^{\,n}\,U_{\hbar}^{\,\otimes\,(n+1)}\;=\;U_{\hbar}^{\,\otimes\,(n+1)}$

— which is fair! — hence we are done with it. Then we are left with the case
$\;n:=t+\ell\leq B-1\;$. Tracking backwards our construction, all these case
provide to (3.30) a contribution of the form

$\displaystyle{\textstyle\sum\limits_{t+\ell\,=\,1}^{B-1}}\,\hbar^{-(t+\ell)}\,{{\,{(-1)}^{\ell}\,}\over{\,t!\,\ell!\,}}\,\chi^{*\,t}\big{(}a_{(1)},b_{(1)}\big{)}\,\widehat{a}_{(2)}\,b^{+}_{(2)}\,\chi^{*\,\ell}\big{(}a_{(3)},b_{(3)}\big{)}\;=$
(3.36) $\displaystyle\hskip
21.0pt=\;{\textstyle\sum\limits_{n\,=\,1}^{B-1}}\,{{\,1\,}\over{\,n!\,}}\,\hbar^{-n}{\textstyle\sum\limits_{t+\ell\,=\,n}}\,{(-1)}^{\ell}\,{n\choose\ell}\,\chi^{*\,t}\big{(}a_{(1)},b_{(1)}\big{)}\,\widehat{a}_{(2)}\,b^{+}_{(2)}\,\chi^{*\,\ell}\big{(}a_{(3)},b_{(3)}\big{)}$

With no loss of generality, we can assume that $\;a\not\equiv 0\,$,
$\,b\not\equiv 0\;\big{(}\,\text{mod}\ \hbar\,U_{\hbar}\,\big{)}\;$. Then for
their corresponding cosets $\;\overline{a}\,,\overline{b}\in
U_{\hbar}\Big{/}\hbar\,U_{\hbar}\cong{U(\hskip 0.5pt{\mathfrak{g}})}\;$ we
have $\,\overline{a}\in{{U(\hskip 0.5pt{\mathfrak{g}})}}_{1}\,$ and
$\,\overline{b}\in{{U(\hskip 0.5pt{\mathfrak{g}})}}_{\\!B}\,$, where
$\,{\big{\\{}{{U(\hskip
0.5pt{\mathfrak{g}})}}_{n}\big{\\}}}_{n\in\mathbb{N}}\,$ is the standard,
coradical filtration of ${U(\hskip 0.5pt{\mathfrak{g}})}\,$, and also
$\,\delta_{1}(\,\overline{a}\,)\not=0\,$ as well as
$\,\delta_{n}\big{(}\,\overline{b}\,\big{)}\not=0\,$ for $\,1\leq n\leq B\,$ —
cf. [Ga1], Lemma 3.3. Moreover, we recall that $\,U_{\hbar}:={U_{\hbar}(\hskip
0.5pt{\mathfrak{g}})}\,$ is cocommutative modulo $\,\hbar\,U_{\hbar}\,$, as it
is a QUEA: in particular, this implies that
$\,\delta_{n}\big{(}\,\overline{b}\,\big{)}\,$ is a symmetric tensor — for
$\,1\leq n\leq B\,$ — hence we can write $\delta_{n}(b\,)$ in the form

$\delta_{n}(b\,)\;=\;b^{+}_{(1)}\otimes\cdots\otimes
b^{+}_{(n)}\;=\;\beta_{\langle 1\rangle}\otimes\cdots\otimes\\!\beta_{\langle
n\rangle}\,+\,\mathcal{O}_{n}\big{(}\hbar^{1}\big{)}$ (3.37)

(for $\,1\leq n\leq B\,$) where $\,\beta_{\langle
1\rangle}\otimes\cdots\otimes\beta_{\langle n\rangle}\,$ — using some
$\sigma$–notation of sort, as usual — is some symmetric tensor in
$U_{\hbar}^{\,\otimes\,n}$ and hereafter
$\mathcal{O}_{n}\big{(}\hbar^{s}\big{)}$ stands for some element in
$\,\hbar^{s}\,U_{\hbar}^{\,\otimes\,n}\,$, for every $\,s,n\in\mathbb{N}\,$.
Then plugging (3.37) in (3.34) we find

$\halign{\hbox to\displaywidth{$\hfil\displaystyle#\hfil$}\cr
0.0pt{\hfil$\displaystyle\chi^{*\,t}\big{(}a_{(1)},b_{(1)}\big{)}\,\widehat{a}_{(2)}\,b^{+}_{(2)}\,\chi^{*\,\ell}\big{(}a_{(3)},b_{(3)}\big{)}\;=\;{\textstyle\prod\limits_{i=1}^{t}}\,\chi\big{(}a^{+}_{(i)},b^{+}_{(i)}\big{)}\,\widehat{a}_{(t+1)}\,b^{+}_{(t+1)}\,{\textstyle\prod\limits_{k=t+2}^{t+\ell+1}}\chi\big{(}a^{+}_{(k)},b^{+}_{(k)}\big{)}\;=\hfill\cr
0.0pt{\hfil$\displaystyle\hfill=\;{\textstyle\prod\limits_{i=1}^{t}}\,\chi\big{(}a^{+}_{(i)},\beta_{\langle
i\rangle}\big{)}\,\widehat{a}_{(t+1)}\,\beta_{\langle
t+1\rangle}\,{\textstyle\prod\limits_{k=t+2}^{t+\ell+1}}\chi\big{(}a^{+}_{(k)},\beta_{\langle
k\rangle}\big{)}\;+\;\mathcal{O}_{1}\big{(}\hbar^{\,t+\ell\,}\big{)}\crcr}}}$

for all $\,t+\ell\leq B-1\,$, with

${\textstyle\prod\limits_{i=1}^{t}}\,\chi\big{(}a^{+}_{(i)},\beta_{\langle
i\rangle}\big{)}\,\widehat{a}_{(t+1)}\,\beta_{\langle
t+1\rangle}\,{\textstyle\prod\limits_{k=t+2}^{t+\ell+1}}\chi\big{(}a^{+}_{(k)},\beta_{\langle
k\rangle}\big{)}\,\;\in\;\,\hbar^{\,t+\ell-1}\,U_{\hbar}$

Therefore, the contribution to (3.30) given in (3.36) now reads

$\halign{\hbox to\displaywidth{$\hfil\displaystyle#\hfil$}\cr
0.0pt{\hfil$\displaystyle{\textstyle\sum\limits_{n\,=\,1}^{B-1}}\,{{\,1\,}\over{\,n!\,}}\,\hbar^{-n}{\textstyle\sum\limits_{t+\ell\,=\,n}}\,{(-1)}^{\ell}\,{n\choose\ell}\,\chi^{*\,t}\big{(}a_{(1)},b_{(1)}\big{)}\,\widehat{a}_{(2)}\,b^{+}_{(2)}\,\chi^{*\,\ell}\big{(}a_{(3)},b_{(3)}\big{)}\;=\hfill\cr
0.0pt{\hfil$\displaystyle=\;{\textstyle\sum\limits_{n\,=\,1}^{B-1}}\,{{\,1\,}\over{\,n!\,}}\,\hbar^{-n}{\textstyle\sum\limits_{t+\ell\,=\,n}}\,{(-1)}^{\ell}\,{n\choose\ell}\,{\textstyle\prod\limits_{i=1}^{t}}\,\chi\big{(}a^{+}_{(i)},\beta_{\langle
i\rangle}\big{)}\,\widehat{a}_{(t+1)}\,\beta_{\langle
t+1\rangle}\,{\textstyle\prod\limits_{k=t+2}^{n+1}}\chi\big{(}a^{+}_{(k)},\beta_{\langle
k\rangle}\big{)}\,\;+\cr
0.0pt{\hfil$\displaystyle\hfill+\;\,\mathcal{O}_{1}\big{(}\hbar^{0}\big{)}\crcr}}}}$

where in the last formula we have

${\textstyle\sum\limits_{n\,=\,1}^{B-1}}\,{{\,1\,}\over{\,n!\,}}\,\hbar^{-n}\\!{\textstyle\sum\limits_{t+\ell\,=\,n}}\\!{(-1)}^{\ell}\,{n\choose\ell}\,{\textstyle\prod\limits_{i=1}^{t}}\,\chi\big{(}a^{+}_{(i)},\beta_{\langle
i\rangle}\big{)}\,\widehat{a}_{(t+1)}\,\beta_{\langle
t+1\rangle}\,{\textstyle\prod\limits_{k=t+2}^{n+1}}\chi\big{(}a^{+}_{(k)},\beta_{\langle
k\rangle}\big{)}\,\in\,\hbar^{-1}\,U_{\hbar}$

Now, observe that, setting $\,n:=t+\ell\,$, we can re-write

${\textstyle\prod\limits_{i=1}^{t}}\,\chi\big{(}a^{+}_{(i)},\beta_{\langle
i\rangle}\big{)}\,\widehat{a}_{(t+1)}\,\beta_{\langle
t+1\rangle}{\textstyle\prod\limits_{k=t+2}^{n+1}}\\!\chi\big{(}a^{+}_{(k)},\beta_{\langle
k\rangle}\big{)}\,=\,\varPhi\big{(}\delta_{n}(a)\otimes\beta_{\langle
1\rangle}\otimes\cdots\otimes\beta_{\langle n+1\rangle}\big{)}$ (3.38)

with
$\;\varPhi:U_{\hbar}^{\,\otimes\,2(n+1)}\\!\relbar\joinrel\relbar\joinrel\longrightarrow
U_{\hbar}\;$ being the map given by the composition

$\varPhi\;:=\;\mu\circ\big{(}\,\chi^{\otimes\,t}\otimes\operatorname{id}_{U_{\hbar}}^{\,\otimes
2}\otimes\chi^{\otimes(n-t)}\big{)}\circ\varsigma_{\,n+1}$

where (1)
$\;\varsigma_{\,n+1}:U_{\hbar}^{\,\otimes\,(2n+1)}\\!\relbar\joinrel\relbar\joinrel\longrightarrow
U_{\hbar}^{\,\otimes\,(2n+1)}\;$ is the “shuffle” map

$x_{1}\otimes\cdots\otimes x_{n}\otimes y_{1}\otimes\cdots\otimes y_{n}\otimes
y_{n+1}\;\mapsto\;x_{1}\otimes y_{1}\otimes x_{2}\otimes
y_{2}\otimes\cdots\otimes x_{n}\otimes y_{n}\otimes y_{n+1}$

and, considering ${\Bbbk[[\hbar]]}$ as embedded into $U_{\hbar}$ via the unit
map, (2)
$\;\mu:U_{\hbar}^{\otimes\,n}\relbar\joinrel\relbar\joinrel\longrightarrow
U_{\hbar}\;$ is the obvious ($n$–fold iterated) multiplication by scalars. Now
recall that $\,\beta_{\langle 1\rangle}\otimes\cdots\otimes\beta_{\langle
n+1\rangle}\,$ represents a tensor in $\sigma$–notation, so more explicitly we
might write $\,\beta_{\langle 1\rangle}\otimes\cdots\otimes\beta_{\langle
n+1\rangle}\,=\,\sum_{s=1}^{N}\beta_{s,1}\otimes\beta_{s,\,n+1}\,$; so in the
formula we are dealing with what is written as a product
${\textstyle\prod\limits_{i=1}^{t}}\chi\big{(}a^{+}_{(i)},\beta_{\langle
i\rangle}\big{)}\,\widehat{a}_{(t+1)}\,\beta_{\langle
t+1\rangle}{\textstyle\prod\limits_{k=t+2}^{n+1}}\chi\big{(}a^{+}_{(k)},\beta_{\langle
k\rangle}\big{)}$ is actually a sum of several products as
$\;{\textstyle\prod\limits_{i=1}^{t}}\,\chi\big{(}a^{+}_{(i)},\beta_{s,\,i}\big{)}\,\widehat{a}_{(t+1)}\,\beta_{s,\,t+1}\,{\textstyle\prod\limits_{k=t+2}^{n+1}}\chi\big{(}a^{+}_{(k)},\beta_{s,\,k}\big{)}\;$.
But then recall that this tensor $\,\beta_{\langle
1\rangle}\otimes\cdots\otimes\beta_{\langle
n+1\rangle}\,=\,\sum_{s=1}^{N}\beta_{s,1}\otimes\beta_{s,\,n+1}\,$ is
symmetric, therefore, the various products
$\;{\textstyle\prod\limits_{i=1}^{t}}\,\chi\big{(}a^{+}_{(i)},\beta_{s,\,i}\big{)}\,\widehat{a}_{(t+1)}\,\beta_{s,\,t+1}\,{\textstyle\prod\limits_{k=t+2}^{n+1}}\chi\big{(}a^{+}_{(k)},\beta_{s,\,k}\big{)}\;$
actually all coincide: letting $C_{n}$ be their “common value”, we deduce that

$\halign{\hbox to\displaywidth{$\hfil\displaystyle#\hfil$}\cr
0.0pt{\hfil$\displaystyle{\textstyle\sum\limits_{n\,=\,1}^{B-1}}\,{{\,1\,}\over{\,n!\,}}\,\hbar^{-n}\\!{\textstyle\sum\limits_{t+\ell\,=\,n}}\\!{(-1)}^{\ell}\,{n\choose\ell}\,{\textstyle\prod\limits_{i=1}^{t}}\,\chi\big{(}a^{+}_{(i)},\beta_{\langle
i\rangle}\big{)}\,\widehat{a}_{(t+1)}\,\beta_{\langle
t+1\rangle}\,{\textstyle\prod\limits_{k=t+2}^{n+1}}\chi\big{(}a^{+}_{(k)},\beta_{\langle
k\rangle}\big{)}\;=\hfill\cr
0.0pt{\hfil$\displaystyle\hfill=\;\,{\textstyle\sum\limits_{n\,=\,1}^{B-1}}\,{{\,1\,}\over{\,n!\,}}\,\hbar^{-n}\bigg{(}\,{\textstyle\sum\limits_{t+\ell\,=\,n}}\\!{(-1)}^{\ell}\,{n\choose\ell}\\!\bigg{)}\,C_{n}\,\;=\;\,0\crcr}}}$

again because of the identity
$\,\;{\textstyle\sum\limits_{t+\ell=n}}{(-1)}^{\ell}\,\displaystyle{n\choose\ell}\;=\;0\;\,$.

Thus, also the last contributions to (3.30) given in (3.36) actually belong to
$U_{\hbar}\,$. To sum up, we have proved that

$\displaystyle a\,\raisebox{-5.0pt}{$\dot{\scriptstyle\sigma}$}\,b\;=\;a\cdot
b\,+\,z\qquad\qquad\textsl{with \ }\;z\in J_{\hbar}$ (3.39)
$\displaystyle\forall\;\;\;a\in\widetilde{J}_{\hbar}^{\raisebox{1.0pt}{$\;\scriptstyle
1$}}=\hbar^{-1}J_{\hbar}^{\,\prime}\,,\;\;b\in\widetilde{J}_{\hbar}^{\raisebox{1.0pt}{$\;\scriptstyle
B$}}=\hbar^{-B}J_{\hbar}^{\,\prime\\!\raisebox{1.0pt}{$\;\scriptstyle B$}}$

and similarly — just switching roles of $a$ and $b$ — also

$\displaystyle b\,\raisebox{-5.0pt}{$\dot{\scriptstyle\sigma}$}\,a\;=\;b\cdot
a\,+\,x\qquad\qquad\textsl{with \ }\;x\in J_{\hbar}$ (3.40)
$\displaystyle\forall\;\;\;a\in\widetilde{J}_{\hbar}^{\raisebox{1.0pt}{$\;\scriptstyle
1$}}=\hbar^{-1}J_{\hbar}^{\,\prime}\,,\;\;b\in\widetilde{J}_{\hbar}^{\raisebox{1.0pt}{$\;\scriptstyle
B$}}=\hbar^{-B}J_{\hbar}^{\,\prime\\!\raisebox{1.0pt}{$\;\scriptstyle B$}}$

Let
$\,{\big{\langle}\widetilde{J}_{\hbar}\big{\rangle}}_{\Bbbk[[\hbar]]}^{\sigma}\,$
the unital ${\Bbbk[[\hbar]]}$–subalgebra of
$\,{\big{(}{\mathbb{U}_{\hbar}(\hskip
0.5pt{\mathfrak{g}})}\big{)}}_{\\!\sigma}:={\big{(}{\Bbbk(\hskip-1.7pt(\hbar)\hskip-1.7pt)}\otimes_{\Bbbk[[\hbar]]}U_{\hbar}\big{)}}_{\\!\sigma}\,$
generated by $\widetilde{J}_{\hbar}\,$. Now recall that
$\,\widetilde{J}_{\hbar}:=\hbar^{-1}J_{\hbar}^{\,\prime}\,$ with
$\,J_{\hbar}^{\,\prime}:=\textsl{Ker}\big{(}U_{\hbar}^{\,\prime}\big{)}\,$;
then $\,U_{\hbar}^{\,\prime}=J_{\hbar}^{\,\prime}\oplus{\Bbbk[[\hbar]]}\cdot
1\,$, which implies
$\;\Delta\big{(}J_{\hbar}^{\,\prime}\big{)}\,\subseteq\,J_{\hbar}^{\,\prime}\otimes
1+J_{\hbar}^{\,\prime}\otimes J_{\hbar}^{\,\prime}+1\otimes
J_{\hbar}^{\,\prime}\;$. Then we get also
$\;\Delta\big{(}\widetilde{J}_{\hbar}\big{)}\,\subseteq\,\widetilde{J}_{\hbar}\otimes
1+\widetilde{J}_{\hbar}\otimes\widetilde{J}_{\hbar}+1\otimes\widetilde{J}_{\hbar}\;$.
Since the coalgebra structure is the same in ${\mathbb{U}_{\hbar}(\hskip
0.5pt{\mathfrak{g}})}$ and ${\big{(}{\mathbb{U}_{\hbar}(\hskip
0.5pt{\mathfrak{g}})}\big{)}}_{\\!\sigma}\,$, it follows eventually from this
that
$\,{\big{\langle}\widetilde{J}_{\hbar}\big{\rangle}}_{\Bbbk[[\hbar]]}^{\sigma}\,$
is a Hopf ${\Bbbk[[\hbar]]}$–subalgebra inside
${\big{(}{\mathbb{U}_{\hbar}(\hskip
0.5pt{\mathfrak{g}})}\big{)}}_{\\!\sigma}\,$.

By repeated use of (3.39) or (3.40) alike, we find that
$\;{\big{\langle}\widetilde{J}_{\hbar}\big{\rangle}}_{\Bbbk[[\hbar]]}^{\sigma}\subseteq
U_{\hbar}={\big{\langle}\widetilde{J}_{\hbar}\big{\rangle}}_{\Bbbk[[\hbar]]}\;$.

Now observe that the original product “$\,\cdot\,$” in
$\,U_{\hbar}:={U_{\hbar}(\hskip 0.5pt{\mathfrak{g}})}\,$ and
${\mathbb{U}_{\hbar}(\hskip 0.5pt{\mathfrak{g}})}$ can be obtained from “
$\dot{\scriptstyle\sigma}$ ” through deformation via the inverse $2$–cocycle
$\sigma^{-1}\,$. Thanks to this, we can reverse the roles of
$\,U_{\hbar}={\big{\langle}\widetilde{J}_{\hbar}\big{\rangle}}_{\Bbbk[[\hbar]]}\,$
and
$\,{\big{\langle}\widetilde{J}_{\hbar}\big{\rangle}}_{\Bbbk[[\hbar]]}^{\sigma}\,$
in the previous construction (with some care), thus eventually achieving the
converse inclusion
$\;{\big{\langle}\widetilde{J}_{\hbar}\big{\rangle}}_{\Bbbk[[\hbar]]}\subseteq{\big{\langle}\widetilde{J}_{\hbar}\big{\rangle}}_{\Bbbk[[\hbar]]}^{\sigma}\;$.
Therefore
$\;{\big{\langle}\widetilde{J}_{\hbar}\big{\rangle}}_{\Bbbk[[\hbar]]}\subseteq{\big{\langle}\widetilde{J}_{\hbar}\big{\rangle}}_{\Bbbk[[\hbar]]}^{\sigma}\;$,
which in particular implies that
$\;U_{\hbar}={\big{\langle}\widetilde{J}_{\hbar}\big{\rangle}}_{\Bbbk[[\hbar]]}\;$
is closed for the $\sigma$–product, q.e.d. ∎

######  Definition 3.3.6.

With assumptions as in Theorem 3.3.5, the new QUEA obtained from
${U_{\hbar}(\hskip 0.5pt{\mathfrak{g}})}$ through the process of 2–cocycle
deformation by $\sigma$ of ${\mathbb{U}_{\hbar}(\hskip 0.5pt{\mathfrak{g}})}$
followed by restriction will be called the quasi-2-cocycle deformation of
${U_{\hbar}(\hskip 0.5pt{\mathfrak{g}})}$ by $\sigma$, and it will be denoted
by ${{U_{\hbar}(\hskip 0.5pt{\mathfrak{g}})}}_{\sigma}\,$.

To complete our analysis, next result sheds light onto the new,
quasi-2-cocycle deformed QUEA ${{U_{\hbar}(\hskip
0.5pt{\mathfrak{g}})}}_{\sigma}\,$, describing in detail its semiclassical
limit:

######  Theorem 3.3.7.

Let ${U_{\hbar}(\hskip 0.5pt{\mathfrak{g}})}$ be a QUEA over the Lie bialgebra
$\,{\mathfrak{g}}=\big{(}\,{\mathfrak{g}}\,;\,[\,\ ,\
]\,,\,\delta\,\big{)}\,$. Let $\sigma$ be a quasi-2-cocycle for
${U_{\hbar}(\hskip 0.5pt{\mathfrak{g}})}\,$, so
$\,\sigma=\exp_{*}\\!\big{(}\hbar^{-1}\chi\big{)}\,$ for some
$\,\chi\in{\Big{(}{{U_{\hbar}(\hskip
0.5pt{\mathfrak{g}})}}^{\widehat{\otimes}\,2}\Big{)}}^{\\!*}\,$ with
$\,\chi(z\,,1)=0=\chi(1\,,z)\,$ for $\,z\in{U_{\hbar}(\hskip
0.5pt{\mathfrak{g}})}\,$. Set also $\;\chi_{a}:=\chi-\chi_{2,1}\;$. Then: (a)
$\chi$ is antisymmetric, i.e. $\,\chi_{2,1}=-\chi\,$, iff $\sigma$ is
orthogonal, i.e. $\,\sigma_{2,1}=\sigma^{-1}\,$; (b) the $\Bbbk$–linear map
$\,\gamma\,:=\,\chi_{a}\;\Big{(}\,\text{\rm
mod}\;\hbar\,{\big{(}{U_{\hbar}(\hskip
0.5pt{\mathfrak{g}})}^{\widehat{\otimes}\,2}\big{)}}^{\\!*}\,\Big{)}\Big{|}_{{\mathfrak{g}}\otimes{\mathfrak{g}}}\;$
from $\,{\mathfrak{g}}\otimes{\mathfrak{g}}\,$ to $\Bbbk$ is antisymmetric
2–cocycle for the Lie bialgebra ${\mathfrak{g}}\,$; (c) the quasi-2-cocycle
deformation $\,{\big{(}{U_{\hbar}(\hskip
0.5pt{\mathfrak{g}})}\big{)}}_{\sigma}$ of $\,{U_{\hbar}(\hskip
0.5pt{\mathfrak{g}})}$ is a QUEA for the Lie bialgebra
$\,{\mathfrak{g}}_{\gamma}=\big{(}\,{\mathfrak{g}}\,;\,{[\,\ ,\
]}_{\gamma}\,,\,\delta\,\big{)}\,$ which is the deformation of
$\,{\mathfrak{g}}$ by the 2–cocycle $\gamma\,$; in a nutshell, we have
$\;{\big{(}{U_{\hbar}(\hskip
0.5pt{\mathfrak{g}})}\big{)}}_{\sigma}\cong\,U_{\hbar}\big{(}{\mathfrak{g}}_{\gamma}\big{)}\;$.
In particular, if $\sigma$ is ${\Bbbk[[\hbar]]}$–valued — i.e., it is an
ordinary 2–cocycle for the Hopf ${\Bbbk[[\hbar]]}$–algebra ${U_{\hbar}(\hskip
0.5pt{\mathfrak{g}})}\,$ — or equivalently
$\,\chi\in\hbar\,{\big{(}{U_{\hbar}(\hskip
0.5pt{\mathfrak{g}})}^{\widehat{\otimes}\,2}\big{)}}^{\\!*}\,$, then we have
just $\,\gamma=0\,$ and $\;{\big{(}{U_{\hbar}(\hskip
0.5pt{\mathfrak{g}})}\big{)}}_{\sigma}\cong\,U_{\hbar}\big{(}{\mathfrak{g}}_{\gamma}\big{)}={U_{\hbar}(\hskip
0.5pt{\mathfrak{g}})}\;$.

###### Proof.

(a) This follows from claim (c) in Lemma 3.3.2. (b) We are interested in the
restriction to $\,{\mathfrak{g}}\otimes{\mathfrak{g}}\,$ of the specialization
of $\sigma$ modulo $\hbar\,$. So we start with
$\,\text{a}\,,\text{b}\,,\text{c}\in{\mathfrak{g}}\,$, that we realize as
$\;\text{a}=a\;\big{(}\,\text{mod}\;\hbar\,U_{\hbar}\,\big{)}\;$,
$\;\text{b}=b\;\big{(}\,\text{mod}\;\hbar\,U_{\hbar}\,\big{)}\;$ and
$\;\text{c}=c\;\big{(}\,\text{mod}\;\hbar\,U_{\hbar}\,\big{)}\;$ for some
“lifts” $\,a,b,c\in U_{\hbar}\,$. By the identity
$\,U_{\hbar}={\big{(}U_{\hbar}^{\,\prime}\,\big{)}}^{\\!\vee}\,$ and by Lemma
3.3 in [Ga1], we can choose the lifts $a$, $b$ and $c$ belong to
$\,\widetilde{J}_{\hbar}:=\hbar^{-1}J_{\hbar}^{\,\prime}\,$, so that
$\,a^{\,\prime}:=\hbar\,a\,$, $\,b^{\,\prime}:=\hbar\,b\,$ and
$\,c^{\,\prime}:=\hbar\,c\,$ belong to $J_{\hbar}^{\,\prime}\,$.

As $\sigma$ is a normalized Hopf 2–cocycle for $U_{\hbar}\,$, it must obey the
equality

$\sigma\big{(}b^{\,\prime}_{(1)},c^{\,\prime}_{(1)}\big{)}\,\sigma\big{(}a^{\,\prime},b^{\,\prime}_{(2)}c^{\,\prime}_{(2)}\big{)}\;=\;\sigma\big{(}a^{\,\prime}_{(1)},b^{\,\prime}_{(1)}\big{)}\,\sigma\big{(}a^{\,\prime}_{(2)}b^{\,\prime}_{(2)},c^{\,\prime}\,\big{)}$
(3.41)

Let us focus on the left hand side of (3.41). Expanding the exponential we get

$\displaystyle\sigma\big{(}b_{(1)}^{\,\prime},c_{(1)}^{\,\prime}\big{)}\,\sigma\big{(}a^{\,\prime},b_{(2)}^{\,\prime}c_{(2)}^{\,\prime}\big{)}\;=\,\sum_{n,m\geq
0}\frac{\hbar^{-(n+m)}}{n!\,m!}\,\chi^{\ast\,n}\big{(}b_{(1)}^{\,\prime},c_{(1)}^{\,\prime}\big{)}\,\chi^{\ast\,m}\big{(}a^{\,\prime},b_{(2)}^{\,\prime}c_{(2)}^{\,\prime}\big{)}\;=$
$\displaystyle\hskip
7.0pt=\;\epsilon\big{(}a^{\,\prime}\big{)}\,\epsilon\big{(}b^{\,\prime}\big{)}\,\epsilon\big{(}c^{\,\prime}\big{)}\,+\,\hbar^{-1}\,\chi\big{(}b^{\,\prime},c^{\,\prime}\big{)}\,\epsilon\big{(}a^{\,\prime}\big{)}\,+\,\hbar^{-1}\,\chi\big{(}a^{\,\prime},b^{\,\prime}c^{\,\prime}\big{)}\,+$
$\displaystyle\hskip
10.0pt+\,\hbar^{-2}\,\chi\big{(}b_{(1)}^{\,\prime},c_{(1)}^{\,\prime}\big{)}\,\chi\big{(}a^{\,\prime},b_{(2)}^{\,\prime}c_{(2)}^{\,\prime}\big{)}\,+\,\hbar^{-2}\,2^{-1}\,\chi^{\ast\,2}\big{(}b^{\,\prime},c^{\,\prime}\big{)}\,\epsilon\big{(}a^{\,\prime}\big{)}\,+$
$\displaystyle\hskip
15.0pt+\,\hbar^{-2}\,2^{-1}\,\chi^{\ast\,2}\big{(}a^{\,\prime},b^{\,\prime}c^{\,\prime}\big{)}\,+\sum_{n+m\geq
3}\frac{\hbar^{-(n+m)}}{n!\,m!}\,\chi^{\ast\,n}\big{(}b_{(1)}^{\,\prime},c_{(1)}^{\,\prime}\big{)}\,\chi^{\ast\,m}\big{(}a^{\,\prime},b_{(2)}^{\,\prime}c_{(2)}^{\,\prime}\big{)}$

Then, noting that
$\,\epsilon\big{(}a^{\,\prime}\big{)}=\epsilon\big{(}b^{\,\prime}\big{)}=\epsilon\big{(}c^{\,\prime}\big{)}=0\,$
by construction, and analyzing all other summands as in the proof of claim
Theorem 3.3.5, we obtain

$\displaystyle\sigma\big{(}b_{(1)}^{\,\prime},c_{(1)}^{\,\prime}\big{)}\,\sigma\big{(}a^{\,\prime},b_{(2)}^{\,\prime}c_{(2)}^{\,\prime}\big{)}\;=\;\hbar^{-1}\chi\big{(}a^{\,\prime},b^{\,\prime}c^{\,\prime}\big{)}\,+$
(3.42) $\displaystyle\hskip
50.0pt+\,\hbar^{-2}\chi\big{(}b_{(1)}^{\,\prime},c_{(1)}^{\,\prime}\big{)}\,\chi\big{(}a^{\,\prime},b_{(2)}^{\,\prime}c_{(2)}^{\,\prime}\big{)}\,+\,\hbar^{-2}\,2^{-1}\,\chi^{\ast\,2}\big{(}a^{\,\prime},b^{\,\prime}c^{\,\prime}\big{)}\,+$
$\displaystyle\hskip 100.0pt+\,\big{(}\,\textsl{sum of all terms with\
}n+m\geq 3\,\big{)}$

Writing $\,z^{\prime}=z^{\prime+}+\epsilon\big{(}z^{\prime}\big{)}\,$ and
using that $\,\chi(z,1)=0=\chi(1,z)\,$ and

$\chi\big{(}x_{(1)}^{\,\prime},y_{(1)}^{\,\prime}z_{(1)}^{\,\prime}\big{)}\,\chi\big{(}x_{(2)}^{\,\prime},y_{(2)}^{\,\prime}\,\epsilon\big{(}z_{(2)}^{\,\prime}\big{)}\big{)}\,=\,\chi\big{(}x_{(1)}^{\,\prime},y_{(1)}^{\,\prime}z^{\,\prime}\big{)}\,\chi\big{(}x_{(2)}^{\,\prime},y_{(2)}^{\,\prime}\big{)}$

we have that

$\halign{\hbox to\displaywidth{$\hfil\displaystyle#\hfil$}\cr
0.0pt{\hfil$\displaystyle\hskip
5.0pt\chi^{\ast\,2}\big{(}a^{\,\prime},b^{\,\prime}c^{\,\prime}\big{)}\;=\;\chi\big{(}a_{(1)}^{\,\prime},b_{(1)}^{\,\prime}c_{(1)}^{\,\prime}\big{)}\,\chi(a_{(2)}^{\,\prime},b_{(2)}^{\,\prime}c_{(2)}^{\,\prime}\big{)}\;=\hfill\cr
0.0pt{\hfil$\displaystyle\hskip
17.0pt=\;\chi\big{(}a_{(1)}^{\,\prime+},b_{(1)}^{\,\prime+}c_{(1)}^{\,\prime+}\big{)}\,\chi\big{(}a_{(2)}^{\,\prime+},b_{(2)}^{\,\prime+}c_{(2)}^{\,\prime+}\big{)}\,+\,\chi\big{(}a_{(1)}^{\,\prime+},b_{(1)}^{\,\prime+}\big{)}\,\chi\big{(}a_{(2)}^{\,\prime+},b_{(2)}^{\,\prime+}c^{\,\prime}\big{)}\,+\hfill\cr
0.0pt{\hfil$\displaystyle\hskip
29.0pt+\,\chi\big{(}a_{(1)}^{\,\prime+},c_{(1)}^{\,\prime+}\big{)}\,\chi\big{(}a_{(2)}^{\,\prime+},b^{\,\prime}c_{(2)}^{\,\prime+}\big{)}\,+\,\chi\big{(}a_{(1)}^{\,\prime+},b^{\,\prime}c_{(1)}^{\,\prime+}\big{)}\,\chi\big{(}a_{(2)}^{\,\prime+},c_{(2)}^{\,\prime+}\big{)}\,+\hfill\cr
0.0pt{\hfil$\displaystyle\hfill+\,\chi\big{(}a_{(1)}^{\,\prime+},b^{\,\prime}\big{)}\,\chi\big{(}a_{(2)}^{\,\prime+},c^{\,\prime}\big{)}\,+\,\chi\big{(}a_{(1)}^{\,\prime+},b_{(1)}^{\,\prime+}c^{\,\prime}\big{)}\,\chi\big{(}a_{(2)}^{\,\prime+},b_{(2)}^{\,\prime+}\big{)}\,+\,\chi\big{(}a_{(1)}^{\,\prime+},c^{\,\prime}\big{)}\,\chi\big{(}a_{(2)}^{\,\prime+},b^{\,\prime}\big{)}\crcr}}}}}$

Now, taking into account that $\,z^{\prime}=\hbar\,z\,$ and
$\,z_{(i)}^{\prime\ +}\in\hbar\,U_{\hbar}\,$, we may re-write the expression
above as

$\chi^{\ast\,2}\big{(}a^{\,\prime},b^{\,\prime}c^{\,\prime}\big{)}\;=\;\,\hbar^{4}\,\chi\big{(}a_{(1)},b\big{)}\,\chi\big{(}a_{(2)},c\big{)}\,+\,\hbar^{4}\,\chi\big{(}a_{(1)},c\big{)}\,\chi\big{(}a_{(2)},b\big{)}\,+\,\mathcal{O}\big{(}\hbar^{5}\big{)}$

Performing a similar analysis on the term
$\,\chi\big{(}b_{(1)}^{\,\prime},c_{(1)}^{\,\prime}\big{)}\,\chi\big{(}a^{\,\prime},b_{(2)}^{\,\prime}c_{(2)}^{\,\prime}\big{)}\,$
we get

$\chi\big{(}b_{(1)}^{\,\prime},c_{(1)}^{\,\prime}\big{)}\,\chi\big{(}a^{\,\prime},b_{(2)}^{\,\prime}c_{(2)}^{\,\prime}\big{)}\;=\;\,\hbar^{4}\,\chi\big{(}b,c_{(1)}\big{)}\,\chi\big{(}a,c_{(2)}\big{)}\,+\,\hbar^{4}\,\chi\big{(}b_{(1)},c\big{)}\,\chi\big{(}a,b_{(2)}\big{)}\,+\,\mathcal{O}\big{(}\hbar^{5}\big{)}$

Moreover, with a similar (yet easier) analysis one finds also that

$\chi^{*n}\big{(}b^{\prime}_{(1)},c^{\prime}_{(1)}\big{)}\,\chi^{*m}\big{(}a^{\prime},b^{\prime}_{(2)}c^{\prime}_{(2)}\big{)}\,\in\,\hbar^{+2(n+m)}\,{\Bbbk[[\hbar]]}$

for all $\,n+m\geq 3\,$, so that the (last) summand “$\big{(}\,\textsl{sum of
all terms with\ }n+m\geq 3\,\big{)}$” in (3.42) is of type
$\,\mathcal{O}\big{(}\hbar^{n+m}\big{)}=\mathcal{O}\big{(}\hbar^{3}\big{)}\,$.
Putting all together in (3.42) we find

$\displaystyle\hskip-5.0pt\hbar^{3}\,\sigma\big{(}b_{(1)},c_{(1)}\big{)}\,\sigma\big{(}a,b_{(2)}c_{(2)}\big{)}\;=$
$\displaystyle\hskip-3.0pt=\,\hbar^{2}\,\chi(a,bc)+\hbar^{2}\,\chi\big{(}b,c_{(1)}\big{)}\,\chi\big{(}a,c_{(2)}\big{)}+\hbar^{2}\,\chi\big{(}b_{(1)},c\big{)}\,\chi\big{(}a,b_{(2)}\big{)}\,+$
$\displaystyle\hskip
5.0pt+\,\hbar^{2}\,2^{-1}\,\chi\big{(}a_{(1)},b\big{)}\,\chi\big{(}a_{(2)},c\big{)}\,+\,\hbar^{2}\,2^{-1}\,\chi\big{(}a_{(1)},c\big{)}\,\chi\big{(}a_{(2)},b\big{)}\,+\,\mathcal{O}\big{(}\hbar^{3}\big{)}$

An analogous treatment of the right hand side of (3.41) yields

$\displaystyle\hskip-5.0pt\hbar^{3}\,\sigma\big{(}a_{(1)},b_{(1)}\big{)}\,\sigma\big{(}a_{(2)}b_{(2)},c\,\big{)}\;=$
$\displaystyle\hskip-3.0pt=\,\hbar^{2}\,\chi(ab,c\,)+\hbar^{2}\,\chi\big{(}a,b_{(1)}\big{)}\,\chi\big{(}b_{(2)},c\,\big{)}+\hbar^{2}\,\chi\big{(}a_{(1)},b\,\big{)}\,\chi\big{(}a_{(2)},c\,\big{)}\,+$
$\displaystyle\hskip
5.0pt+\,\hbar^{2}\,2^{-1}\,\chi\big{(}a,c_{(1)}\big{)}\,\chi\big{(}b,c_{(2)}\big{)}\,+\,\hbar^{2}\,2^{-1}\,\chi\big{(}b,c_{(1)}\big{)}\,\chi\big{(}a,c_{(2)}\big{)}\,+\,\mathcal{O}\big{(}\hbar^{3}\big{)}$

Altogether, this implies that

$\displaystyle\hskip-7.0pt\chi(a,bc\,)\,+\,\chi\big{(}b,c_{(1)}\big{)}\,\chi\big{(}a,c_{(2)}\big{)}\,+\,\chi\big{(}b_{(1)},c\,\big{)}\,\chi\big{(}a,b_{(2)}\big{)}\,+$
$\displaystyle\hskip-7.0pt\quad+\,2^{-1}\,\chi\big{(}a_{(1)},b\big{)}\,\chi\big{(}a_{(2)},c\,\big{)}\,+\,2^{-1}\,\chi\big{(}a_{(1)},c\,\big{)}\,\chi\big{(}a_{(2)},b\,\big{)}\;\underset{\hbar}{\equiv}$
$\displaystyle\hskip-11.0pt\qquad\qquad\underset{\hbar}{\equiv}\;\chi(ab,c\,)\,+\,\chi\big{(}a,b_{(1)}\big{)}\,\chi\big{(}b_{(2)},c\,\big{)}\,+\,\chi\big{(}a_{(1)},b\,\big{)}\,\chi\big{(}a_{(2)},c\,\big{)}\,+$
$\displaystyle\hskip-11.0pt\qquad\qquad\quad+\,2^{-1}\,\chi\big{(}a,c_{(1)}\big{)}\,\chi\big{(}b,c_{(2)}\big{)}\,+\,2^{-1}\,\chi\big{(}b,c_{(1)}\big{)}\,\chi\big{(}a,c_{(2)}\big{)}$

where (again) $\,\underset{\hbar}{\equiv}\,$ stands for “congruent modulo
$\,\hbar\,{\Bbbk[[\hbar]]}\,$”, that we re-write as

$\displaystyle\hskip-7.0pt\chi(a,bc\,)\,+\,\chi\big{(}b,c_{(1)}\big{)}\,\chi\big{(}a,c_{(2)}\big{)}\,+\,\chi\big{(}b_{(1)},c\,\big{)}\,\chi\big{(}a,b_{(2)}\big{)}\,+$
(3.43)
$\displaystyle\hskip-7.0pt\quad+\,2^{-1}\,\chi\big{(}a_{(1)},b\big{)}\,\chi\big{(}a_{(2)},c\,\big{)}\,+\,2^{-1}\,\chi\big{(}a_{(1)},c\,\big{)}\,\chi\big{(}a_{(2)},b\,\big{)}\;-$
$\displaystyle\hskip-13.0pt\quad\qquad-\;\chi(ab,c\,)\,-\,\chi\big{(}a,b_{(1)}\big{)}\,\chi\big{(}b_{(2)},c\,\big{)}\,-\,\chi\big{(}a_{(1)},b\,\big{)}\,\chi\big{(}a_{(2)},c\,\big{)}\,-$
$\displaystyle\hskip-19.0pt\qquad\qquad\quad-\,2^{-1}\,\chi\big{(}a,c_{(1)}\big{)}\,\chi\big{(}b,c_{(2)}\big{)}\,-\,2^{-1}\,\chi\big{(}b,c_{(1)}\big{)}\,\chi\big{(}a,c_{(2)}\big{)}\;\underset{\hbar}{\equiv}\;0$

Consider now the action of the group algebra
$\,\Bbbk\big{[}\mathbb{S}_{3}\big{]}\,$ of the symmetric group
$\mathbb{S}_{3}\,$ on $\big{(}U_{\hbar}^{\widehat{\otimes}3}\big{)}^{*}\,$
given by
$\,\sigma\,.\,\varphi(a,b,c):=\varphi\big{(}\sigma^{-1}\,.\,(a,b,c)\big{)}\,$,
where the action of $\,\Bbbk\big{[}\mathbb{S}_{3}\big{]}\,$ on
$\,U_{\hbar}^{\widehat{\otimes}3}\,$ is the natural one that permutes the
tensor factors. Then we let the antisymmetrizer $\textsl{Alt}_{3}$ act on both
sides of (3.43): using that $\;\gamma\,:=\,\chi_{a}\;\Big{(}\,\text{\rm
mod}\;\hbar\,{\big{(}{U_{\hbar}(\hskip
0.5pt{\mathfrak{g}})}^{\widehat{\otimes}\,2}\big{)}}^{\\!*}\,\Big{)}\Big{|}_{{\mathfrak{g}}\otimes{\mathfrak{g}}}\;$
and that $\;a\;\big{(}\hskip-7.0pt\mod\hbar\,U_{\hbar}\big{)}=\text{a}\;$,
$\;b\;\big{(}\hskip-7.0pt\mod\hbar\,U_{\hbar}\big{)}=\text{b}\;$ and
$\;c\;\big{(}\hskip-7.0pt\mod\hbar\,U_{\hbar}\big{)}=\text{c}\;$, a
straightforward calculation eventually yields

$\partial_{*}(\gamma)\,+\,\text{c.p.}\,+\,{[[\gamma\,,\gamma\,]]}_{*}\,\;=\;\,0$

This means exactly that $\,\gamma\,$ is a 2-cocycle for the Lie bialgebra
${\mathfrak{g}}$ — according to Definition 2.1.5 — that is obviously
antisymmetric (by construction), q.e.d.

(c) First of all, we start by noting that $\,{{U_{\hbar}(\hskip
0.5pt{\mathfrak{g}})}}_{\sigma}:={\big{(}{U_{\hbar}(\hskip
0.5pt{\mathfrak{g}})}\big{)}}_{\sigma}\,$ is equal to ${U_{\hbar}(\hskip
0.5pt{\mathfrak{g}})}$ as a counital ${\Bbbk[[\hbar]]}$–coalgebra (by
construction), but with the new product defined by

$m_{\sigma}(a,b\,)\,=\,a\,\raisebox{-5.0pt}{$\dot{\scriptstyle\sigma}$}\,b\,=\,\sigma(a_{(1)},b_{(1)})\,a_{(2)}\,b_{(2)}\,\sigma^{-1}(a_{(3)},b_{(3)})$
$None$

In particular, the ${\Bbbk[[\hbar]]}$–module $\,{{U_{\hbar}(\hskip
0.5pt{\mathfrak{g}})}}_{\sigma}={U_{\hbar}(\hskip 0.5pt{\mathfrak{g}})}\,$ is
still topologically free, so that ${{U_{\hbar}(\hskip
0.5pt{\mathfrak{g}})}}_{\sigma}$ is again a Hopf algebra in
$\mathcal{T}_{\,\widehat{\otimes}\,}\,$, cf. §2.3.1. Moreover, its
semiclassical limit $\,\overline{{{U_{\hbar}(\hskip
0.5pt{\mathfrak{g}})}}_{\sigma}}:={{U_{\hbar}(\hskip
0.5pt{\mathfrak{g}})}}_{\sigma}\Big{/}\hbar\,{{U_{\hbar}(\hskip
0.5pt{\mathfrak{g}})}}_{\sigma}\,$ as a coalgebra is the same as that of
${U_{\hbar}(\hskip 0.5pt{\mathfrak{g}})}\,$; hence it is again cocommutative
connected (as these properties are not affected by $2$–cocycle deformations).
Thus by Milnor-Moore Theorem we have $\,\overline{{{U_{\hbar}(\hskip
0.5pt{\mathfrak{g}})}}_{\sigma}}=U\big{(}\widehat{{\mathfrak{g}}}\big{)}\,$,
where
$\,\widehat{{\mathfrak{g}}}=\operatorname{Prim}\big{(}\,\overline{{{U_{\hbar}(\hskip
0.5pt{\mathfrak{g}})}}_{\sigma}}\,\big{)}\,$ is the space of primitive
elements in $\overline{{{U_{\hbar}(\hskip 0.5pt{\mathfrak{g}})}}_{\sigma}}\,$,
and as such it coincides with
$\,\operatorname{Prim}\big{(}\,\overline{{U_{\hbar}(\hskip
0.5pt{\mathfrak{g}})}}\,\big{)}=\operatorname{Prim}\big{(}U({\mathfrak{g}})\big{)}={\mathfrak{g}}\,$
as a Lie coalgebra; its Lie algebra structure, on the other hand, does depend
on $\sigma\,$. Altogether, this shows that ${{U_{\hbar}(\hskip
0.5pt{\mathfrak{g}})}}_{\sigma}$ is indeed a QUEA, whose semiclassical limit
is $U\big{(}\widehat{{\mathfrak{g}}}\big{)}\,$; then we are only left to prove
that the Lie bracket on $\widehat{{\mathfrak{g}}}$ coincides with that of
${\mathfrak{g}}_{\gamma}\,$, while also proving that $\gamma$ is an
antisymmetric 2–cocycle for the Lie bialgebra ${\mathfrak{g}}\,$.

The Lie bracket in $\widehat{{\mathfrak{g}}}$ is given by the commutator
inside $\,U\big{(}\widehat{{\mathfrak{g}}}\big{)}=\overline{{{U_{\hbar}(\hskip
0.5pt{\mathfrak{g}})}}_{\sigma}}\,$, so we denote it by
$\,\;{[\text{a}\,,\text{b}]}_{\sigma}=\text{a}\;\overline{\raisebox{-5.0pt}{$\dot{\scriptstyle\,\sigma}$}}\;\text{b}-\text{b}\;\overline{\raisebox{-5.0pt}{$\dot{\scriptstyle\,\sigma}$}}\;\text{a}\;\,$
(for all $\,\text{a}\,,\text{b}\in{\mathfrak{g}}\,$), where
$\;\overline{\raisebox{-5.0pt}{$\dot{\scriptstyle\,\sigma}$}}\;$ is the
product in
$\,U\big{(}\widehat{{\mathfrak{g}}}\big{)}=\overline{{{U_{\hbar}(\hskip
0.5pt{\mathfrak{g}})}}_{\sigma}}\,$ induced by the ($\sigma$–deformed) product
in ${{U_{\hbar}(\hskip 0.5pt{\mathfrak{g}})}}_{\sigma}\,$. Therefore, we will
compute such a commutator as the coset modulo $\,\hbar\,U_{\hbar}\,$ of a
commutator in $U_{\hbar}\,$, namely
$\,\;{[\text{a}\,,\text{b}]}_{\sigma}=\text{a}\;\overline{\raisebox{-5.0pt}{$\dot{\scriptstyle\,\sigma}$}}\;\text{b}-\text{b}\;\overline{\raisebox{-5.0pt}{$\dot{\scriptstyle\,\sigma}$}}\;\text{a}\;=\;a\,\raisebox{-5.0pt}{$\dot{\scriptstyle\sigma}$}\,b-b\,\raisebox{-5.0pt}{$\dot{\scriptstyle\sigma}$}\,a\;\;\big{(}\,\text{mod}\;\hbar\,U_{\hbar}\,\big{)}\;$,
where $a$ and $b\,$, like in the proof of claim (c), are lifts of a and b —
i.e., $\,a\;\big{(}\,\text{\rm mod}\;\hbar\,U_{\hbar}\,\big{)}=\text{\rm a}\,$
and $\,b\;\big{(}\,\text{\rm mod}\,\hbar\,U_{\hbar}\,\big{)}=\text{\rm b}\,$ —
such that $\;a^{\,\prime}:=\hbar\,a\,\in J_{\hbar}^{\,\prime}\;$ and
$\;b^{\,\prime}:=\hbar\,b\,\in J_{\hbar}^{\,\prime}\;$. We re-start back from
(3.30), which now gives (taking into account all the analysis carried out
there, with $\,A=1=B\,$)

$\displaystyle
a\,\raisebox{-5.0pt}{$\dot{\scriptstyle\sigma}$}\,b-b\,\raisebox{-5.0pt}{$\dot{\scriptstyle\sigma}$}\,a\;\;\underset{\hbar}{\equiv}\;\,a\cdot
b\;-\;b\cdot a\,\;+$ $\displaystyle\hskip
3.0pt+\,\hbar^{-3}\,\Big{(}\,\chi\big{(}a^{\prime+}_{(1)},b^{\prime+}_{(1)}\big{)}\big{(}\,\widehat{a}^{\,\prime}_{(2)}\,b^{\prime+}_{(2)}+a^{\prime+}_{(2)}\,\widehat{b}^{\,\prime}_{(2)}\big{)}\,-\,\chi\big{(}b^{\prime+}_{(1)},a^{\prime+}_{(1)}\big{)}\big{(}\,\widehat{b}^{\,\prime}_{(2)}\,a^{\prime+}_{(2)}+b^{\prime+}_{(2)}\,\widehat{a}^{\,\prime}_{(2)}\big{)}\\!\Big{)}\;-$
$\displaystyle\hskip
6.0pt-\,\hbar^{-3}\,\Big{(}\\!\big{(}\,\widehat{a}^{\,\prime}_{(1)}\,b^{\prime+}_{(1)}+a^{\prime+}_{(1)}\,\widehat{b}^{\,\prime}_{(1)}\big{)}\,\chi\big{(}a^{\prime+}_{(2)},b^{\prime+}_{(2)}\big{)}\,-\,\big{(}\,\widehat{b}^{\,\prime}_{(1)}\,a^{\prime+}_{(1)}+b^{\prime+}_{(1)}\,\widehat{a}^{\,\prime}_{(1)}\big{)}\,\chi\big{(}b^{\prime+}_{(2)},a^{\prime+}_{(2)}\big{)}\\!\Big{)}$

Second, letting $\;\chi_{a}:=\chi-\chi_{\,2,1}\;$, the previous formula
greatly simplifies into

$\displaystyle
a\,\raisebox{-5.0pt}{$\dot{\scriptstyle\sigma}$}\,b-b\,\raisebox{-5.0pt}{$\dot{\scriptstyle\sigma}$}\,a\;\;=\;\,a\cdot
b\,-\,b\cdot
a\;+\;\hbar^{-3}\,\chi_{a}\big{(}a^{\prime+}_{(1)},b^{\prime+}_{(1)}\big{)}\big{(}\,\widehat{a}^{\,\prime}_{(2)}\,b^{\prime+}_{(2)}\,+\,a^{\prime+}_{(2)}\,\widehat{b}^{\,\prime}_{(2)}\big{)}\,\;+$
$\displaystyle\hskip
113.0pt+\;\,\hbar^{-3}\,\big{(}\,\widehat{b}^{\,\prime}_{(1)}\,a^{\prime+}_{(1)}+b^{\prime+}_{(1)}\,\widehat{a}^{\,\prime}_{(1)}\big{)}\,\chi_{a}\big{(}b^{\prime+}_{(2)},a^{\prime+}_{(2)}\big{)}\;+\;\mathcal{O}(\hbar)\,\;=$
$\displaystyle\hskip 30.0pt=\;\,a\cdot b\,-\,b\cdot
a\;+\;\hbar^{-3}\,\Big{(}\,\chi_{a}\big{(}a^{\prime+}_{(1)},b^{\prime}\big{)}\,a^{\prime+}_{(2)}\,-\,\chi_{a}\big{(}a^{\prime+}_{(2)},b^{\prime}\big{)}\,a^{\prime+}_{(1)}\,\Big{)}\,\;-$
$\displaystyle\hskip
126.0pt-\,\hbar^{-3}\,\Big{(}\,\chi_{a}\big{(}b^{\prime+}_{(1)},a^{\prime}\big{)}\,b^{\prime+}_{(2)}\,-\,\chi_{a}\big{(}b^{\prime+}_{(2)},a^{\prime}\big{)}\,b^{\prime+}_{(1)}\Big{)}\;+\;\mathcal{O}_{1\\!}(\hbar)$

Now, let us write $\;z^{\prime}\,=\,\hbar\,z\;$ for all
$\;z\,\in\,\\{a\,,b\,\\}\;$: then the last formula turns into

$\displaystyle
a\,\raisebox{-5.0pt}{$\dot{\scriptstyle\sigma}$}\,b-b\,\raisebox{-5.0pt}{$\dot{\scriptstyle\sigma}$}\,a\;\;=\;\,a\cdot
b$ $\displaystyle-\;b\cdot
a\,\;+\;\hbar^{-1}\Big{(}\,\chi_{a}\big{(}a^{+}_{(1)},b\,\big{)}\,a^{+}_{(2)}\,-\,\chi_{a}\big{(}a^{+}_{(2)},b\,\big{)}\,a^{+}_{(1)}\Big{)}\,\;-$
(3.44)
$\displaystyle-\;\hbar^{-1}\Big{(}\,\chi_{a}\big{(}b^{+}_{(1)},a\,\big{)}\,b^{+}_{(2)}\,-\,\chi_{a}\big{(}b^{+}_{(2)},a\,\big{)}\,b^{+}_{(1)}\Big{)}\;+\;\mathcal{O}_{1\\!}(\hbar)$

Here we recall that, working with a QUEA, for $\,c\in\\{a\,,b\,\\}\,$ we have

$\Delta(c\,)\,\;=\;\,c\otimes 1\,+\,1\otimes c\,+\,c^{+}_{(1)}\otimes
c^{+}_{(2)}\,+\,\mathcal{O}_{2\\!}\big{(}\hbar^{2}\big{)}\;\;,\qquad
c^{+}_{(1)}\otimes c^{+}_{(2)}\,\in\,\hbar\,U_{h}^{\,\widehat{\otimes}2}$

and moreover — for every $\,c\in\\{a\,,b\,\\}\,$ and
$\,\text{c}\in\\{\text{a}\,,\text{b}\\}\,$, so that $c$ is a lift of c —

$\overline{\hbar^{-1}\big{(}c^{+}_{(1)}\\!\otimes
c^{+}_{(2)}-c^{+}_{(2)}\\!\otimes
c^{+}_{(1)}\big{)}}\;=\;\delta(\text{c})\;=:\;\text{c}_{[1]}\otimes\text{c}_{[2]}$
(3.45)

where hereafter any “overlined” object stands for “its coset modulo
$\hbar\,$”; in addition, we recall also that $\chi_{a}$ is antisymmetric. Then
(3.44) and (3.45) altogether yield

$\displaystyle{[\text{a}\,,\text{b}]}_{\sigma}=\text{a}\;\overline{\raisebox{-5.0pt}{$\dot{\scriptstyle\,\sigma}$}}\;\text{b}-\text{b}\;\overline{\raisebox{-5.0pt}{$\dot{\scriptstyle\,\sigma}$}}\;\text{a}\;=\;\overline{a\,\raisebox{-5.0pt}{$\dot{\scriptstyle\sigma}$}\,b-b\,\raisebox{-5.0pt}{$\dot{\scriptstyle\sigma}$}\,a}\,\;=$
$\displaystyle\hskip
35.0pt=\,\overline{a}\cdot\overline{b}-\overline{b}\cdot\overline{a}\,\;+\;\overline{\hbar^{-1}\Big{(}\,\chi_{a}\big{(}a^{+}_{(1)},b\,\big{)}\,a^{+}_{(2)}\,-\,\chi_{a}\big{(}a^{+}_{(2)},b\,\big{)}\,a^{+}_{(1)}\Big{)}}\,\;-$
$\displaystyle\hskip
70.0pt-\;\overline{\hbar^{-1}\Big{(}\,\chi_{a}\big{(}b^{+}_{(1)},a\,\big{)}\,b^{+}_{(2)}\,-\,\chi_{a}\big{(}b^{+}_{(2)},a\,\big{)}\,b^{+}_{(1)}\Big{)}}\,\;=$
$\displaystyle\hskip
105.0pt=\;\,[\text{a}\,,\text{b}]\,+\,\gamma\big{(}\text{a}_{[1]},\text{b}\big{)}\,\text{a}_{[2]}\,-\,\gamma\big{(}\text{b}_{[1]},\text{a}\big{)}\,\text{b}_{[2]}\,\;=$
$\displaystyle\hskip
140.0pt=\;\,[\text{a}\,,\text{b}]\,-\,\gamma\big{(}\text{a}_{[2]},\text{b}\big{)}\,\text{a}_{[1]}\,-\,\gamma\big{(}\text{b}_{[1]},\text{a}\big{)}\,\text{b}_{[2]}\,\;=:\;\,{[\text{a}\,,\text{b}]}_{\gamma}$

hence $\;{[\text{a}\,,\text{b}]}_{\sigma}={[\text{a}\,,\text{b}]}_{\gamma}\;$
for all $\,\text{a}\,,\text{b}\in{\mathfrak{g}}\,$, in the sense of (2.5), and
we are done. Finally, if in particular $\sigma$ is ${\Bbbk[[\hbar]]}$–valued —
i.e., it is an ordinary 2–cocycle for the Hopf ${\Bbbk[[\hbar]]}$–algebra
${U_{\hbar}(\hskip 0.5pt{\mathfrak{g}})}\,$ — then we have
$\;\chi\,=\,\hbar\,\log_{*}(\sigma)\,\in\,\hbar\,{\big{(}{U_{\hbar}(\hskip
0.5pt{\mathfrak{g}})}^{\widehat{\otimes}\,2}\big{)}}^{\\!*}\,$, hence we have
just $\,\gamma=0\,$ and $\;{\big{(}{U_{\hbar}(\hskip
0.5pt{\mathfrak{g}})}\big{)}}_{\sigma}\cong\,U_{\hbar}\big{(}{\mathfrak{g}}_{\gamma}\big{)}={U_{\hbar}(\hskip
0.5pt{\mathfrak{g}})}\;$. ∎

######  Example 3.3.8.

Some concrete examples of quasi-2-cocycles and deformation by them are treated
in full depth in [GaGa2, Section 5.2]: they concern the wide family of formal
multiparameter QUEAs that we already treated in Example 3.1.4. We then resume
notations and formulas from there.

Fix $\,n\in\mathbb{N}_{+}\,$ and $\,I:=\\{1,\ldots,n\\}\,$. We choose a
multiparameter matrix $\,P:={\big{(}\,p_{i,j}\big{)}}_{i,j\in I}\in
M_{n}\big{(}{\Bbbk[[\hbar]]}\big{)}\,$ of Cartan type, with associated Cartan
matrix $A\,$, a realization
$\;\mathcal{R}\,:=\,\big{(}\,{\mathfrak{h}}\,,\Pi\,,\Pi^{\vee}\,\big{)}\;$ of
it and the (topological) Hopf algebra $U^{\,\mathcal{R}}_{\\!P,\hskip
0.7pt\hbar}(\hskip 0.5pt{\mathfrak{g}})\,$. Let
$\,{\big{\\{}H_{g}\big{\\}}}_{g\in\mathcal{G}}\,$ be a
${\Bbbk[[\hbar]]}$–basis in ${\mathfrak{h}}$, where $\mathcal{G}$ is an index
set with $\,|\mathcal{G}|=\operatorname{rk}({\mathfrak{h}})=t\,$.

We consider special quasi-2–cocycles of $\,U^{\,\mathcal{R}}_{\\!P,\hskip
0.7pt\hbar}(\hskip 0.5pt{\mathfrak{g}})\,$, called “toral” as they are induced
from the quantum torus. Fix an antisymmetric, ${\Bbbk[[\hbar]]}$–bilinear map
$\,\chi:{\mathfrak{h}}\times{\mathfrak{h}}\relbar\joinrel\longrightarrow{\Bbbk[[\hbar]]}\,$,
that corresponds to
$\,X={\big{(}\,\chi_{g{}\gamma}=\chi(H_{g}\,,H_{\gamma})\big{)}}_{g,\gamma\in\mathcal{G}}\in\mathfrak{so}_{t}\big{(}{\Bbbk[[\hbar]]}\big{)}\,$.
Any such map $\chi$ also induces uniquely an antisymmetric,
${\Bbbk[[\hbar]]}$–bilinear map

$\tilde{\chi}_{\scriptscriptstyle
U}\,:\,U_{\\!P,\hbar}^{\,\mathcal{R}}({\mathfrak{h}})\times
U_{\\!P,\hbar}^{\,\mathcal{R}}({\mathfrak{h}})\relbar\joinrel\relbar\joinrel\relbar\joinrel\longrightarrow{\Bbbk[[\hbar]]}$

as follows. By definition, $U_{\\!P,\hbar}^{\,\mathcal{R}}({\mathfrak{h}})$ is
an $\hbar$–adically complete topologically free Hopf algebra isomorphic to
$\,\widehat{S}_{\Bbbk[[\hbar]]}({\mathfrak{h}}):=\widehat{\bigoplus\limits_{n\in\mathbb{N}}S_{\Bbbk[[\hbar]]}^{\,n}({\mathfrak{h}})}\,$,
the $\hbar$–adic completion of the symmetric algebra
$\,S_{\Bbbk[[\hbar]]}({\mathfrak{h}})=\bigoplus\limits_{n\in\mathbb{N}}S_{\Bbbk[[\hbar]]}^{\,n}({\mathfrak{h}})\,$.
Then, $\tilde{\chi}_{\scriptscriptstyle U}$ is defined as the unique
${\Bbbk[[\hbar]]}$–linear (hence $\hbar$–adically continuous) map
$\;U_{\\!P,\hbar}^{\,\mathcal{R}}({\mathfrak{h}})\otimes
U_{\\!P,\hbar}^{\,\mathcal{R}}({\mathfrak{h}})\,{\buildrel\tilde{\chi}_{\scriptscriptstyle
U}\over{\relbar\joinrel\relbar\joinrel\longrightarrow}}\,{\Bbbk[[\hbar]]}\;$
such that

$\displaystyle\tilde{\chi}_{\scriptscriptstyle
U}(z,1):=\epsilon(z)=:\tilde{\chi}_{\scriptscriptstyle
U}(1,z)\qquad\qquad\qquad\forall\;\;z\in\widehat{S}_{\Bbbk[[\hbar]]}({\mathfrak{h}})$
(3.46) $\displaystyle\tilde{\chi}_{\scriptscriptstyle
U}(x,y):=\chi(x,y)\qquad\hskip 55.0pt\qquad\forall\;\;x,y\in
S_{\Bbbk[[\hbar]]}^{\,1}({\mathfrak{h}})$
$\displaystyle\hfill\qquad\tilde{\chi}_{\scriptscriptstyle
U}(x,y):=0\quad\qquad\forall\;\;x\in S_{\Bbbk}^{\,r}({\mathfrak{h}})\,,\;y\in
S_{\Bbbk}^{\,s}({\mathfrak{h}})\,:\,r,s\geq 1\,,\;r+s>2\hskip 5.0pt$

By construction, $\tilde{\chi}_{\scriptscriptstyle U}$ is a normalized
Hochschild 2–cocycle on $U_{\\!P,\hbar}^{\,\mathcal{R}}({\mathfrak{h}})\,$,
that is

$\epsilon(x)\,\tilde{\chi}_{\scriptscriptstyle
U}(y,z)\,-\,\tilde{\chi}_{\scriptscriptstyle
U}(xy,z)\,+\,\tilde{\chi}_{\scriptscriptstyle
U}(x,yz)\,-\,\tilde{\chi}_{\scriptscriptstyle
U}(x,y)\,\epsilon(z)\;=\;0\qquad\forall\;\;x,y,z\in
U_{\\!P,\hbar}^{\,\mathcal{R}}({\mathfrak{h}})$

By [GaGa2, Lemma 5.2.3], the convolution powers of
$\tilde{\chi}_{\scriptscriptstyle U}$ satisfy the following property: for all
$\,H_{+},H_{-}\in{\mathfrak{h}}\,$ and $\,k,\ell,m\in\mathbb{N}_{+}\,$, we
have

${\tilde{\chi}_{\scriptscriptstyle
U}}^{\,\ast\,m}\big{(}H_{+}^{\,k},H_{-}^{\,\ell}\,\big{)}\;=\;\begin{cases}\begin{array}[]{lr}\delta_{k,m}\,\delta_{\ell,m}\,{\big{(}m!\big{)}}^{2}\,\chi(H_{+},H_{-})^{m}&\quad\text{for}\;\;\;\;m\geq
1\;,\\\
\delta_{k,0}\,\delta_{\ell,0}&\quad\text{for}\;\;\;\;m=0\;.\end{array}\end{cases}$

This allows one to define a quasi-2-cocycle $\,\chi_{\scriptscriptstyle U}\,$
as the unique ${\Bbbk[[\hbar]]}$–linear map from
$\,U_{\\!P,\hbar}^{\,\mathcal{R}}({\mathfrak{h}})\\!\mathop{\otimes}\limits_{\Bbbk[[\hbar]]}\\!U_{\\!P,\hbar}^{\,\mathcal{R}}({\mathfrak{h}})\,$
to ${\Bbbk(\hskip-1.7pt(\hbar)\hskip-1.7pt)}$ given by the exponentiation of
$\,\hbar^{-1}\,2^{-1}\,\tilde{\chi}_{\scriptscriptstyle U}\,$, i.e.

$\chi_{\scriptscriptstyle
U}\;:=\;e^{\hbar^{-1}2^{-1}\tilde{\chi}_{\scriptscriptstyle
U}}\;=\;{\textstyle\sum_{m\geq
0}}\,\hbar^{-m}\,{\tilde{\chi}_{\scriptscriptstyle
U}}^{\,\ast\,m}\\!\Big{/}2^{m}\,m!$

By [GaGa2, Lemma 5.2.2], this $\,\chi_{\scriptscriptstyle U}$ is in fact a
well-defined quasi-2–cocycle for
$\,U_{\\!P,\hbar}^{\,\mathcal{R}}({\mathfrak{h}})\,$, in the sense of
Definition 3.3.3. Moreover, one has, for all
$H_{+}\,,H_{-}\in{\mathfrak{h}}\,$, and setting
$\,K_{\pm}:=e^{\,\hbar\,H_{\pm}}\,$,

$\chi_{\scriptscriptstyle U}^{\pm
1}(H_{+},H_{-})\,=\,\pm\hbar^{-1}\,2^{-1}\chi(H_{+},H_{-})\quad,\qquad\chi_{\scriptscriptstyle
U}(K_{+},K_{-})\,=\,e^{\hbar\,2^{-1}\chi(H_{+},H_{-})}$

Assume now that $\,\chi$ satisfies the additional requirement
$\;\chi(S_{i}\,,\,-\,)\,=\,0\,=\,\chi(\,-\,,S_{i})\;$ for all $\,i\in I\,$,
where $\,S_{i}:=2^{-1}\big{(}\,T^{+}_{i}+T^{-}_{i}\big{)}\,$ for all $\,i\in
I\,$. In particular, one has that
$\;\chi\big{(}\,T_{i}^{+}\,,\,T\,\big{)}\,=\,\chi\big{(}-T_{i}^{-}\,,\,T\,\big{)}\;$
and
$\;\chi\big{(}\,T\,,T_{i}^{+}\big{)}\,=\,\chi\big{(}\,T\,,-T_{i}^{-}\big{)}\;$
for all $\,i\in I\,$ and $\,T\in{\mathfrak{h}}\,$. Then $\chi$ canonically
induces a ${\Bbbk[[\hbar]]}$–bilinear map
$\;\overline{\chi}:\overline{{\mathfrak{h}}}\times\overline{{\mathfrak{h}}}\relbar\joinrel\longrightarrow{\Bbbk[[\hbar]]}\;$,
where $\,\overline{{\mathfrak{h}}}:={\mathfrak{h}}\big{/}\mathfrak{s}\,$ with
$\,\mathfrak{s}:=\textsl{Span}_{\Bbbk[[\hbar]]}\big{(}{\\{\,S_{i}\,\\}}_{i\in
I}\big{)}\,$, given by

$\overline{\chi}\,\big{(}\,T^{\prime}\\!+\mathfrak{s}\,,T^{\prime\prime}\\!+\mathfrak{s}\,\big{)}\,:=\,\chi\big{(}\,T^{\prime},T^{\prime\prime}\big{)}\qquad\forall\;\;T^{\prime},T^{\prime\prime}\in{\mathfrak{h}}$

Now, replaying the construction above with $\overline{{\mathfrak{h}}}$ and
$\overline{\chi}$ replacing ${\mathfrak{h}}$ and $\chi\,$, we can construct a
normalized Hopf quasi-$2$–cocycle $\,\overline{\chi}_{\scriptscriptstyle
U}\\!:U_{\\!P,\hbar}^{\,\mathcal{R}}\big{(}\,\overline{{\mathfrak{h}}}\,\big{)}\times
U_{\\!P,\hbar}^{\,\mathcal{R}}\big{(}\,\overline{{\mathfrak{h}}}\,\big{)}\\!\relbar\joinrel\longrightarrow{\Bbbk(\hskip-1.7pt(\hbar)\hskip-1.7pt)}\,$.
Moreover, since
$\,U_{\\!P,\hbar}^{\,\mathcal{R}}\big{(}\,\overline{{\mathfrak{h}}}\,\big{)}\cong\widehat{S}_{\Bbbk[[\hbar]]}\big{(}\,\overline{{\mathfrak{h}}}\,\big{)}\,$,
there exists a unique Hopf algebra epimorphism
$\;\pi:U^{\,\mathcal{R}}_{\\!P,\hskip 0.7pt\hbar}(\hskip
0.5pt{\mathfrak{g}})\relbar\joinrel\relbar\joinrel\twoheadrightarrow
U_{\\!P,\hbar}^{\,\mathcal{R}}\big{(}\,\overline{{\mathfrak{h}}}\,\big{)}\;$
given by $\,\pi(E_{i}):=0\,$, $\,\pi(F_{i}):=0\,$ — for $\,i\in I\,$ — and
$\,\pi(T):=(T+\mathfrak{s})\,\in\,\overline{{\mathfrak{h}}}\subseteq
U_{\\!P,\hbar}^{\,\mathcal{R}}\big{(}\,\overline{{\mathfrak{h}}}\,\big{)}\,$ —
for $\,T\in{\mathfrak{h}}\,$. Then we consider

$\sigma_{\chi}\,:=\,\overline{\chi}_{\scriptscriptstyle
U}\circ(\pi\times\pi)\,:\,U^{\,\mathcal{R}}_{\\!P,\hskip 0.7pt\hbar}(\hskip
0.5pt{\mathfrak{g}})\times U^{\,\mathcal{R}}_{\\!P,\hskip 0.7pt\hbar}(\hskip
0.5pt{\mathfrak{g}})\relbar\joinrel\relbar\joinrel\relbar\joinrel\relbar\joinrel\twoheadrightarrow{\Bbbk(\hskip-1.7pt(\hbar)\hskip-1.7pt)}$

which is automatically a normalized,
${\Bbbk(\hskip-1.7pt(\hbar)\hskip-1.7pt)}$–valued Hopf quasi-$2$–cocycle on
$U^{\,\mathcal{R}}_{\\!P,\hskip 0.7pt\hbar}(\hskip 0.5pt{\mathfrak{g}})\,$.

By Theorem 3.3.7, one may define a “deformed product” on
$U^{\,\mathcal{R}}_{\\!P,\hskip 0.7pt\hbar}(\hskip 0.5pt{\mathfrak{g}})$ using
$\sigma_{\chi}$ hereafter denoted by $\,\scriptstyle\dot{\sigma}_{\chi}\,$.
Write
$\,X^{(n)_{\sigma_{\chi}}}=X\raise-1.0pt\hbox{$\,\scriptstyle\dot{\sigma}_{\chi}$}\cdots\raise-1.0pt\hbox{$\,\scriptstyle\dot{\sigma}_{\chi}$}X\,$
for the $n$–th power of any $\,X\in U^{\,\mathcal{R}}_{\\!P,\hskip
0.7pt\hbar}(\hskip 0.5pt{\mathfrak{g}})\,$ with respect to this deformed
product.

Directly from definitions, sheer computation yields the following formulas,
relating the deformed product with the old one (for all
$\;T^{\prime},T^{\prime\prime},T\in{\mathfrak{h}}\;$, $\;i\,,\,j\in I\,$):

$\halign{\hbox to\displaywidth{$\hfil\displaystyle#\hfil$}\cr
0.0pt{\hfil$\displaystyle
T^{\prime}\raise-1.0pt\hbox{$\,\scriptstyle\dot{\sigma}_{\chi}$}T^{\prime\prime}\,=\;T^{\prime}\,T^{\prime\prime}\quad,\qquad
E_{i}\raise-1.0pt\hbox{$\,\scriptstyle\dot{\sigma}_{\chi}$}F_{j}\;=\,E_{i}\,F_{j}\quad,\qquad
F_{j}\raise-1.0pt\hbox{$\,\scriptstyle\dot{\sigma}_{\chi}$}E_{i}\;=\,F_{j}\,E_{i}\cr
0.0pt{\hfil$\displaystyle
T\raise-1.0pt\hbox{$\,\scriptstyle\dot{\sigma}_{\chi}$}E_{j}\;=\;T\,E_{j}\,+\,2^{-1}\chi\big{(}T,T_{j}^{+}\big{)}\,E_{j}\quad,\quad
E_{j}\raise-1.0pt\hbox{$\,\scriptstyle\dot{\sigma}_{\chi}$}T\;=\;E_{j}\,T\,+\,2^{-1}\chi\big{(}T_{j}^{+},T\big{)}\big{)}\,E_{j}\cr
0.0pt{\hfil$\displaystyle
T\raise-1.0pt\hbox{$\,\scriptstyle\dot{\sigma}_{\chi}$}F_{j}\;=\;T\,F_{j}\,+\,2^{-1}\chi\big{(}T,T_{j}^{-}\big{)}\,F_{j}\quad,\quad
F_{j}\raise-1.0pt\hbox{$\,\scriptstyle\dot{\sigma}_{\chi}$}T\;=\;F_{j}\,T\,+\,2^{-1}\chi\big{(}T_{j}^{-},T\big{)}\,F_{j}\cr
0.0pt{\hfil$\displaystyle
E_{i}^{\,(m)_{\sigma_{\chi}}}\,=\;{\textstyle\prod_{\ell=1}^{m-1}}\sigma_{\chi}\Big{(}e^{\,+\hbar\,\ell\,T_{i}^{+}}\\!,\,e^{\,+\hbar\,T_{i}^{+}}\Big{)}\,E_{i}^{\,m}\;=\;E_{i}^{\,m}\cr
0.0pt{\hfil$\displaystyle
E_{i}^{\,m}\raise-1.0pt\hbox{$\,\scriptstyle\dot{\sigma}_{\chi}$}E_{j}^{\,n}\;=\;\sigma_{\chi}\Big{(}e^{\,+\hbar\,m\,T_{i}^{+}}\\!,\,e^{\,+\hbar\,n\,T_{j}^{+}}\Big{)}\,E_{i}^{\,m}\,E_{j}^{\,n}\;=\;e^{\,+\hbar\,m\,n\,2^{-1}\mathring{\chi}_{ij}}E_{i}^{\,m}\,E_{j}^{\,n}\cr
0.0pt{\hfil$\displaystyle
E_{i}^{\,(m)_{\sigma_{\chi}}}\raise-1.0pt\hbox{$\,\scriptstyle\dot{\sigma}_{\chi}$}\,E_{j}\,\raise-1.0pt\hbox{$\,\scriptstyle\dot{\sigma}_{\chi}$}\,E_{k}^{\,(n)_{\sigma_{\chi}}}=\,\Big{(}{\textstyle\prod_{\ell=1}^{m-1}}\sigma_{\chi}\Big{(}e^{\,+\hbar\,\ell\,T_{i}^{+}}\\!,\,e^{\,+\hbar\,T_{i}^{+}}\Big{)}\\!\Big{)}\Big{(}{\textstyle\prod_{t=1}^{n-1}}\sigma_{\chi}\Big{(}e^{\,+\hbar\,t\,T_{k}^{+}}\\!,\,e^{\,+\hbar\,T_{k}^{+}}\Big{)}\\!\Big{)}\,\cdot\hfill\cr
0.0pt{\hfil$\displaystyle\hfill\cdot\;\sigma_{\chi}\Big{(}e^{\,+\hbar\,m\,T_{i}^{+}}\\!,\,e^{\,+\hbar\,T_{j}^{+}}\Big{)}\,\sigma_{\chi}\Big{(}e^{\,+\hbar\,(m\,T_{i}^{+}+T_{j}^{+})}\\!,\,e^{\,+\hbar\,n\,T_{k}^{+}}\Big{)}\,E_{i}^{\,m}\,E_{j}\,E_{k}^{\,n}\cr
0.0pt{\hfil$\displaystyle
F_{i}^{\,(m)_{\sigma_{\chi}}}\,=\;{\textstyle\prod_{\ell=1}^{m-1}}\sigma_{\chi}^{-1}\Big{(}e^{\,-\hbar\,\ell\,T_{i}^{-}}\\!,\,e^{\,-\hbar\,T_{i}^{-}}\Big{)}\,F_{i}^{\,m}\;=\;F_{i}^{\,m}\cr
0.0pt{\hfil$\displaystyle
F_{i}^{\,m}\raise-1.0pt\hbox{$\,\scriptstyle\dot{\sigma}_{\chi}$}F_{j}^{\,n}\;=\;\sigma_{\chi}^{-1}\Big{(}e^{\,-\hbar\,m\,T_{i}^{-}}\\!,\,e^{\,-\hbar\,n\,T_{j}^{-}}\Big{)}\,F_{i}^{\,m}\,F_{j}^{\,n}\;=\;e^{\,-\hbar\,m\,n\,2^{-1}\mathring{\chi}_{ij}}F_{i}^{\,m}\,F_{j}^{\,n}\cr
0.0pt{\hfil$\displaystyle\qquad
F_{i}^{\,(m)_{\sigma_{\chi}}}\raise-1.0pt\hbox{$\,\scriptstyle\dot{\sigma}_{\chi}$}\,F_{j}\,\raise-1.0pt\hbox{$\,\scriptstyle\dot{\sigma}_{\chi}$}\,F_{k}^{\,(n)_{\sigma_{\chi}}}\,=\hfill\cr
0.0pt{\hfil$\displaystyle=\;\Big{(}{\textstyle\prod_{\ell=1}^{m-1}}\sigma_{\chi}^{-1}\Big{(}e^{\,-\hbar\,\ell\,T_{i}^{-}}\\!,\,e^{\,-\hbar\,T_{i}^{-}}\Big{)}\Big{)}\Big{(}{\textstyle\prod_{t=1}^{n-1}}\sigma_{\chi}^{-1}\Big{(}e^{\,-\hbar\,t\,T_{k}^{-}}\\!,\,e^{\,-\hbar\,T_{k}^{-}}\Big{)}\Big{)}\,\cdot\cr
0.0pt{\hfil$\displaystyle\hfill\cdot\;\sigma_{\chi}^{-1}\Big{(}e^{\,-\hbar\,m\,T_{i}^{-}}\\!,\,e^{\,-\hbar\,T_{j}^{-}}\Big{)}\,\sigma_{\chi}^{-1}\Big{(}e^{\,-\hbar\,(m\,T_{i}^{-}+T_{j}^{-})}\\!,\,e^{\,-\hbar\,n\,T_{k}^{-}}\Big{)}\,F_{i}^{\,m}\,F_{j}\,F_{k}^{\,n}\quad\cr
0.0pt{\hfil$\displaystyle
F_{i}^{\,(m)_{\sigma_{\chi}}}\raise-1.0pt\hbox{$\,\scriptstyle\dot{\sigma}_{\chi}$}\,F_{j}\,\raise-1.0pt\hbox{$\,\scriptstyle\dot{\sigma}_{\chi}$}\,F_{k}^{\,(n)_{\sigma_{\chi}}}=\Big{(}{\textstyle\prod_{\ell=1}^{m-1}}\sigma_{\chi}^{-1}\Big{(}e^{\,-\hbar\,\ell\,T_{i}^{-}}\\!,\,e^{\,-\hbar\,T_{i}^{-}}\Big{)}\\!\Big{)}\Big{(}{\textstyle\prod_{t=1}^{n-1}}\sigma_{\chi}^{-1}\Big{(}e^{\,-\hbar\,t\,T_{k}^{-}}\\!,\,e^{\,-\hbar\,T_{k}^{-}}\Big{)}\\!\Big{)}\cdot\hfill\cr
0.0pt{\hfil$\displaystyle\hfill\cdot\;\sigma_{\chi}^{-1}\Big{(}e^{\,-\hbar\,m\,T_{i}^{-}}\\!,\,e^{\,-\hbar\,T_{j}^{-}}\Big{)}\,\sigma_{\chi}^{-1}\Big{(}e^{\,-\hbar\,(m\,T_{i}^{-}+T_{j}^{-})}\\!,\,e^{\,-\hbar\,n\,T_{k}^{-}}\Big{)}\,F_{i}^{\,m}\,F_{j}\,F_{k}^{\,n}\crcr}}}}}}}}}}}}}}}$

Fix now
$\,\mathring{X}:=\\!{\Big{(}\mathring{\chi}_{i{}j}=\chi\big{(}\,T_{i}^{+},T_{j}^{+}\big{)}\\!\Big{)}}_{i,j\in
I}\,$ and define the multiparameter matrix

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

It turns out that $P_{(\chi)}$ is a matrix of Cartan type — the same of $P$
indeed — and
$\,\mathcal{R}_{(\chi)}\,=\,\big{(}\,{\mathfrak{h}}\,,\Pi_{(\chi)}\,,\Pi^{\vee}\,\big{)}\,$
is a realization of it. Moreover, by [GaGa2, Theorem 5.2.12], there exists an
isomorphism of topological Hopf algebras

${\big{(}U^{\,\mathcal{R}}_{\\!P,\hskip 0.7pt\hbar}(\hskip
0.5pt{\mathfrak{g}})\big{)}}_{\sigma_{\chi}}\;\cong\;\,U_{\\!P_{(\chi)},\,\hbar}^{\,\mathcal{R}_{(\chi)}}({\mathfrak{g}})$

which is the identity on generators. In short, every toral 2–cocycle
deformation of a FoMpQUEA is another FoMpQUEA, whose multiparameter
$\,P_{(\chi)}$ and realization $\mathcal{R}_{(\chi)}$ depend on the original
$P$ and $\mathcal{R}\,$, as well as on $\chi\,$. Moreover, under some mild
restrictions on the realizations, one proves that the FoMpQUEA
$U^{\,\mathcal{R}}_{\\!P,\hskip 0.7pt\hbar}(\hskip 0.5pt{\mathfrak{g}})$ is
isomorphic to a toral 2–cocycle deformation of the Drinfeld’s standard double
QUEA, see [GaGa2, Theorem 5.2.14].

About the semiclassical limit, we have the following. Taking everything modulo
$\hbar\,$, the map
$\,\chi:{\mathfrak{h}}\times{\mathfrak{h}}\relbar\joinrel\longrightarrow{\Bbbk[[\hbar]]}\,$
defines a similar antisymmetric, $\Bbbk$–bilinear map
$\;\gamma:=\big{(}\,\chi\\!\mod\hbar\,\big{)}:{\mathfrak{h}}_{0}\times{\mathfrak{h}}_{0}\relbar\joinrel\longrightarrow\Bbbk\;$
— where
$\,{\mathfrak{h}}_{0}:={\mathfrak{h}}\Big{/}\hbar\,{\mathfrak{h}}\,=\,\overline{{\mathfrak{h}}}\,$.
Out of $\gamma$ one constructs a toral $2$–cocycle $\gamma_{\mathfrak{g}}$ for
the Lie bialgebra
${\mathfrak{g}}^{\bar{{\scriptscriptstyle\mathcal{R}}}}_{\bar{{\scriptscriptstyle
P}}}\,$, and out of it the $2$–cocycle deformed Lie bialgebra
$\,{\big{(}{\mathfrak{g}}^{\bar{{\scriptscriptstyle\mathcal{R}}}}_{\bar{{\scriptscriptstyle
P}}}\big{)}}_{\gamma\raisebox{-1.0pt}{${}_{\mathfrak{g}}$}}\,$. Similarly as
above, out of $\gamma$ we get the multiparameter matrix $P_{(\gamma)}$ and its
realization $\,\mathcal{R}_{(\gamma)}\,$: then by construction
$\,P_{(\gamma)}=\bar{P}_{(\chi)}\,$ and
$\,\mathcal{R}_{(\gamma)}=\bar{\mathcal{R}}_{(\chi)}\,$. Attached to these we
have $\,U_{\\!P_{(\chi)},\,\hbar}^{\,\mathcal{R}_{(\chi)}}({\mathfrak{g}})\,$
and $\,{\mathfrak{g}}_{{\scriptscriptstyle
P}_{(\gamma)}}^{{\scriptscriptstyle\mathcal{R}}_{(\gamma)}}={\mathfrak{g}}_{\bar{{\scriptscriptstyle
P}}_{(\chi)}}^{\bar{{\scriptscriptstyle\mathcal{R}}}_{(\chi)}}\,$, again
connected via quantization/specialization, and
$\,{\mathfrak{g}}_{{\scriptscriptstyle
P}_{(\gamma)}}^{{\scriptscriptstyle\mathcal{R}}_{(\gamma)}}\cong\,{\big{(}{\mathfrak{g}}^{\bar{{\scriptscriptstyle\mathcal{R}}}}_{\bar{{\scriptscriptstyle
P}}}\big{)}}_{\gamma\raisebox{-1.0pt}{${}_{\mathfrak{g}}$}}\,$ as Lie
bialgebras. Actually, one can prove that “deformation by 2–cocycle commutes
with specialization”, see [GaGa2, Theorem 6.2.4]: with assumptions as above,
we have that ${\big{(}\,U^{\,\mathcal{R}}_{\\!P,\hskip 0.7pt\hbar}(\hskip
0.5pt{\mathfrak{g}})\big{)}}_{\\!\sigma_{\chi}}$ is a quantized universal
enveloping algebra, with semiclassical limit
$\,U\Big{(}\\!{{\big{(}{\mathfrak{g}}^{\bar{{\scriptscriptstyle\mathcal{R}}}}_{\bar{{\scriptscriptstyle
P}}}\big{)}}_{\gamma\raisebox{-1.0pt}{${}_{\mathfrak{g}}$}}}\Big{)}\,\cong\,U\Big{(}{\mathfrak{g}}_{{\scriptscriptstyle
P}_{(\gamma)}}^{{\scriptscriptstyle\mathcal{R}}_{(\gamma)}}\Big{)}\,$.

######  Remark 3.3.9.

It is important to stress that our notion of quasi–2–cocycle did not come out
of the blue, but rather was suggested by the previous example. Indeed, the
authors first “met” these objects when studying polynomial-type QUEAs
$U_{q}({\mathfrak{g}})$ — i.e., QUEAs “à la Jimbo-Lusztig”, defined over
$\Bbbk\big{[}q\,,q^{-1}\big{]}\,$: these are standard Hopf algebras (no
topology is involved), to which one can apply deformation by 2–cocycles and
then obtain some “multiparameter QUEAs” — cf. [GaGa1], §4.2. Every such
polynomial $U_{q}({\mathfrak{g}})$ can be realized as a Hopf subalgebra of a
formal ${U_{\hbar}(\hskip 0.5pt{\mathfrak{g}})}\,$, hence it makes sense to
try and extend the 2–cocycle and the associated deformation procedure used for
$U_{q}({\mathfrak{g}})$ to the larger Hopf algebra ${U_{\hbar}(\hskip
0.5pt{\mathfrak{g}})}\,$. When we fulfilled this task — in [GaGa2] — what we
actually found was that the unique extension of the 2–cocycle of
$U_{q}({\mathfrak{g}})$ to ${U_{\hbar}(\hskip 0.5pt{\mathfrak{g}})}$ actually
is a quasi–2–cocycle (and not a 2–cocycle any more), yet despite this the
deformation procedure does extend from $U_{q}({\mathfrak{g}})$ to the whole
${U_{\hbar}(\hskip 0.5pt{\mathfrak{g}})}\,$. Thus the very notion of
“quasi–2–cocycle” and the associated deformation procedure showed up as
something real from this concrete example.

###  3.4. Deformations by quasi-twist of QFSHA’s

In this subsection we consider deformations by twist of QFSHA’s, but again
“stretching the standard recipe”, much like in §3.3: in fact, rather than
twists in the usual sense we consider some special twist elements belonging to
the scalar extension from ${\Bbbk[[\hbar]]}$ to
${\Bbbk(\hskip-1.7pt(\hbar)\hskip-1.7pt)}$ of our QFSHA. For these elements —
that we call “quasi-twists” — nothing ensures a priori that the deformation
recipe would properly work on the given QFSHA — as it is defined on
${\Bbbk[[\hbar]]}\,$; nevertheless, we eventually find that this is indeed the
case. In other words, we prove — with a parallel result to Theorem 3.3.5 —
that the standard procedure of deformation by twist for QFSHA’s can be
extended (beyond its natural borders) to the case of quasi-twist elements. We
begin with a couple of technical lemmas:

######  Lemma 3.4.1.

Let ${F_{\hbar}[[G\hskip 1.5pt]]}$ be a QFSHA, and $\,{{F_{\hbar}[[G\hskip
1.5pt]]}}^{\vee}$ the associated QUEA defined in §2.4. Let $\,\varphi\in
J_{\hbar}^{\,2}\,$, with
$\,J_{\hbar}:=\textsl{Ker}\big{(}\epsilon_{{F_{\hbar}[[G\hskip
1.5pt]]}}\big{)}\,$. Then: (a) $F:=\exp\big{(}\hbar^{-1}\varphi\big{)}\,$ is a
well-defined element in $\,{{F_{\hbar}[[G\hskip 1.5pt]]}}^{\vee}\;$; (b)
$\,\operatorname{Ad}(F)\big{(}f\big{)}:=F\cdot f\cdot
F^{-1}\,\in\,{F_{\hbar}[[G\hskip 1.5pt]]}\,$ for all
$\,f\in{F_{\hbar}[[G\hskip 1.5pt]]}\,$, so that the adjoint action of $\,F$
onto $\,{{F_{\hbar}[[G\hskip 1.5pt]]}}^{\vee}$ actually restricts to
$\,{F_{\hbar}[[G\hskip 1.5pt]]}\;$.

###### Proof.

(a) The assumption $\,\varphi\in J_{\hbar}^{\;2}\,$ implies
$\;\varphi\,\in\,\hbar^{2}\,{\big{(}J_{\hbar}^{\,\vee}\big{)}}^{2}\subseteq\,\hbar^{2}\,{{F_{\hbar}[[G\hskip
1.5pt]]}}^{\vee}\,$, where
$\;J_{\hbar}^{\,\vee}:=\hbar^{-1}J_{\hbar}\,\subseteq\,{{F_{\hbar}[[G\hskip
1.5pt]]}}^{\vee}\,$. Therefore
$\;\hbar^{-1}\varphi\,\in\,\hbar\,{{F_{\hbar}[[G\hskip 1.5pt]]}}^{\vee}\,$,
hence $\,F:=\exp\big{(}\hbar^{-1}\varphi\big{)}\,$ is indeed a well-defined
element in ${{F_{\hbar}[[G\hskip 1.5pt]]}}^{\vee}\,$, q.e.d. (b) We compute
$\,\operatorname{Ad}(F)\big{(}f\big{)}\,$, $\,f\in{F_{\hbar}[[G\hskip
1.5pt]]}\,$: using the identity
$\;\operatorname{Ad}\\!\big{(}\\!\exp(X)\big{)}(Y)\,=\,\exp\\!\big{(}\\!\operatorname{ad}(X)\big{)}(Y)\;$
and expanding the exponential into a power series we get

$\operatorname{Ad}(F)\big{(}f\big{)}\;=\;\operatorname{Ad}\\!\big{(}\\!\exp\\!\big{(}\hbar^{-1}\varphi\big{)}\big{)}(f)\;=\;\exp\\!\big{(}\operatorname{ad}\\!\big{(}\hbar^{-1}\varphi\big{)}\big{)}(f)\;=\;{\textstyle\sum\limits_{n=0}^{+\infty}}\,\frac{\,1\,}{\,n!\,}\;{\operatorname{ad}\\!\big{(}\hbar^{-1}\varphi\big{)}}^{n}(f)$

Now Lemma 2.3.5(c) and the assumption $\,\varphi\in J_{\\!F_{\hbar}}^{\;2}\,$
together guarantee that

${\operatorname{ad}\\!\big{(}\hbar^{-1}\varphi\big{)}}^{n}(f)\,=\,{\operatorname{ad}\\!\big{(}\hbar^{-1}\varphi\big{)}}^{n}\big{(}f_{+}\big{)}\,\in\,(1-\delta_{s,0})\,J_{\\!F_{\hbar}}^{\;n+s}$
$None$

with $\,s\in\mathbb{N}\,$ such that $\,f\in J_{\\!F_{\hbar}}^{\;s}\,$, hence
$\;\operatorname{Ad}(F)\big{(}f\big{)}={\textstyle\sum\limits_{n=0}^{+\infty}}\,\frac{\,1\,}{\,n!\,}\,{\operatorname{ad}\\!\big{(}\hbar^{-1}\varphi\big{)}}^{n}(f)\;$
is indeed a well-defined element — a convergent series! — of
${F_{\hbar}[[G\hskip 1.5pt]]}\,$. ∎

######  Lemma 3.4.2.

Let ${F_{\hbar}[[G\hskip 1.5pt]]}$ be a QFSHA, and $\,{{F_{\hbar}[[G\hskip
1.5pt]]}}^{\vee}$ the associated QUEA defined in §2.4. Let
$\,\phi\in{F_{\hbar}[[G\hskip 1.5pt]]}^{\widetilde{\otimes}\,2}\,$ be such
that
$\,(\operatorname{id}\otimes\epsilon)(\phi)=0=(\epsilon\otimes\operatorname{id})(\phi)\,$.
Then: (a) $\mathcal{F}=\exp\big{(}\hbar^{-1}\phi\big{)}\,$ is a well-defined
element in $\,{\big{(}{{F_{\hbar}[[G\hskip
1.5pt]]}}^{\vee}\big{)}}^{\widehat{\otimes}\,2}\;$; (b)
$\,\mathcal{F}\cdot(x\otimes y)\cdot\mathcal{F}^{-1}\,\in\,{F_{\hbar}[[G\hskip
1.5pt]]}^{\,\widetilde{\otimes}\,2}\;$ for all $\,x,y\in{F_{\hbar}[[G\hskip
1.5pt]]}\,$, so that the adjoint action of $\,\mathcal{F}$ onto
$\,{\big{(}{{F_{\hbar}[[G\hskip
1.5pt]]}}^{\vee}\big{)}}^{\widehat{\otimes}\,2}$ actually restricts to
$\,{{F_{\hbar}[[G\hskip 1.5pt]]}}^{\widetilde{\otimes}\,2}\;$; (c)
$(\operatorname{id}\otimes\epsilon)(\mathcal{F}\,)=1=(\epsilon\otimes\operatorname{id})(\mathcal{F}\,)\;$;
(d) $\mathcal{F}$ is orthogonal, i.e.
$\,\mathcal{F}_{2,1}=\mathcal{F}^{-1}\,$, iff $\phi$ is antisymmetric, i.e.
$\,\phi_{\,2,1}=-\phi\,$;

###### Proof.

(a)–(b) The assumption
$\,(\operatorname{id}\otimes\epsilon)(\phi)=0=(\epsilon\otimes\operatorname{id})(\phi)\,$
means that $\,\phi\in J_{\hbar}^{\,\widetilde{\otimes}\,2}\,$ where
$\,J_{\hbar}:=\textsl{Ker}\big{(}\epsilon_{{F_{\hbar}[[G\hskip
1.5pt]]}}\big{)}\,$. Now note that $\,F_{\hbar}[[G\times G\hskip
1.5pt]]:={{F_{\hbar}[[G\hskip 1.5pt]]}}^{\,\widetilde{\otimes}\,2}\,$ is in
turn a QFSHA, for the group $\,G\times G\,$; furthermore, its augmentation
ideal is

$\textsl{Ker}\big{(}\epsilon_{F_{\hbar}[[G\times G\hskip
1.5pt]]}\big{)}\;=\;\textsl{Ker}\big{(}\epsilon_{{F_{\hbar}[[G\hskip
1.5pt]]}}\big{)}\otimes{F_{\hbar}[[G\hskip 1.5pt]]}\,+\,{F_{\hbar}[[G\hskip
1.5pt]]}\otimes\textsl{Ker}\big{(}\epsilon_{{F_{\hbar}[[G\hskip
1.5pt]]}}\big{)}$

so that
$\,J_{\hbar}^{\,\widetilde{\otimes}\,2}\subseteq{\textsl{Ker}\big{(}\epsilon_{F_{\hbar}[[G\times
G\hskip 1.5pt]]}\big{)}}^{2}\,$, and finally, we have also

${F_{\hbar}[[G\times G\hskip 1.5pt]]}^{\vee}\;=\;\,{\big{(}{F_{\hbar}[[G\hskip
1.5pt]]}\,\widetilde{\otimes}\,{F_{\hbar}[[G\hskip
1.5pt]]}\big{)}}^{\vee}\;=\;\,{\big{(}{{F_{\hbar}[[G\hskip
1.5pt]]}}^{\vee}\big{)}}^{\widehat{\otimes}\,2}$

Therefore, applying Lemma 3.4.1 above to the QFSHA $F_{\hbar}[[G\times G\hskip
1.5pt]]$ and to $\,\varphi:=\phi\,$ we get both claims (a) and (b). (c)–(d)
Both these claims follow at once from definitions, along with the assumption
that
$\,(\operatorname{id}\otimes\epsilon)(\phi)=0=(\epsilon\otimes\operatorname{id})(\phi)\,$.
∎

The previous result leads us to introduce the notion of “quasi-twist”, as
follows:

######  Definition 3.4.3.

Let ${F_{\hbar}[[G\hskip 1.5pt]]}$ be a QFSHA, and $\,{{F_{\hbar}[[G\hskip
1.5pt]]}}^{\vee}\,$ as in 2.4.1(b). We call quasi-twist (element) of
${F_{\hbar}[[G\hskip 1.5pt]]}$ any element in $\,\big{(}{{F_{\hbar}[[G\hskip
1.5pt]]}}^{\widetilde{\otimes}\,2}\,\big{)}^{\\!\vee}={\big{(}{{F_{\hbar}[[G\hskip
1.5pt]]}}^{\vee}\,\big{)}}^{\widehat{\otimes}\,2}\,$ of the form
$\,\mathcal{F}:=\exp\\!\big{(}\hbar^{-1}\phi\big{)}\,$ — for some
$\,\phi\in{{F_{\hbar}[[G\hskip 1.5pt]]}}^{\widetilde{\otimes}\,2}\,$ such that
$\,(\operatorname{id}\otimes\epsilon)(\phi)=0=(\epsilon\otimes\operatorname{id})(\phi)\,$
— which have the property of a twist element for the QUEA
${{F_{\hbar}[[G\hskip 1.5pt]]}}^{\vee}\,$.

Of course, every twist for ${F_{\hbar}[[G\hskip 1.5pt]]}$ is a quasi-twist
too; the converse, instead, in general, is false (counterexamples do exist).
However, every quasi-twist still provides a well-defined deformation by twist
of ${F_{\hbar}[[G\hskip 1.5pt]]}$ — in other words, the construction of
deformations by twist properly extends to “deformations by quasi-twist” as
well:

######  Theorem 3.4.4.

Let ${F_{\hbar}[[G\hskip 1.5pt]]}$ be a QFSHA, and
$\,\mathcal{F}=\exp\big{(}\hbar^{-1}\phi\big{)}\,$ a quasi-twist for it, as in
Definition 3.4.3. Then the procedure of twist deformation by $\mathcal{F}$
applied to the QUEA ${{F_{\hbar}[[G\hskip 1.5pt]]}}^{\vee}$ restricts to
${F_{\hbar}[[G\hskip 1.5pt]]}\,$, making the latter into a new QFSHA.

###### Proof.

When deforming ${{F_{\hbar}[[G\hskip 1.5pt]]}}^{\vee}$ by the twist
$\mathcal{F}$ one introduces on ${{F_{\hbar}[[G\hskip 1.5pt]]}}^{\vee}$ the
new coproduct $\Delta^{\mathcal{F}}$ given by
$\,\Delta^{\mathcal{F}}:=\operatorname{Ad}(\mathcal{F}\,)\circ\Delta\,$. Then
Lemma 3.4.2(b) ensures that $\Delta^{\mathcal{F}}$ restricts to
${F_{\hbar}[[G\hskip 1.5pt]]}\,$, in that it maps the latter into
$\,{{F_{\hbar}[[G\hskip 1.5pt]]}}^{\widetilde{\otimes}\,2}\,$. The antipode
can be dealt with similarly, whence we conclude that the (deformed) Hopf
structure of ${\big{(}{{F_{\hbar}[[G\hskip
1.5pt]]}}^{\vee\,}\big{)}}^{\mathcal{F}}$ does restrict to
${F_{\hbar}[[G\hskip 1.5pt]]}\,$, q.e.d. ∎

######  Definition 3.4.5.

With assumptions as in Theorem 3.4.4, the new QFSHA obtained from
${F_{\hbar}[[G\hskip 1.5pt]]}$ through the process of twist deformation (by
$\mathcal{F}\,$) of ${{F_{\hbar}[[G\hskip 1.5pt]]}}^{\vee}$ followed by
restriction will be called the quasi-twist deformation of ${F_{\hbar}[[G\hskip
1.5pt]]}$ by $\mathcal{F}$, and denoted by ${{F_{\hbar}[[G\hskip
1.5pt]]}}^{\mathcal{F}}\,$.

Finally, the next result describes in detail what exactly is the nature of the
new, quasi-twist deformed QFSHA ${{F_{\hbar}[[G\hskip
1.5pt]]}}^{\mathcal{F}}$, shedding light onto its semiclassical limit:

######  Theorem 3.4.6.

Let ${F_{\hbar}[[G\hskip 1.5pt]]}$ be a QFSHA over the Lie bialgebra
$\,{\mathfrak{g}}=\big{(}\,{\mathfrak{g}}\,;\,[\,\ ,\
]\,,\,\delta\,\big{)}\,$. Set
$\,{\mathfrak{m}}:=\textsl{Ker}\big{(}\epsilon_{{}_{F[[G\hskip
1.5pt]]}}\big{)}\,$, so
$\;{\mathfrak{m}}\Big{/}{\mathfrak{m}}^{2}\,\cong\,{\mathfrak{g}}^{*}\,$ and
$\;\big{(}{\mathfrak{m}}\otimes{\mathfrak{m}}\big{)}\Big{/}\big{(}{\mathfrak{m}}^{2}\otimes{\mathfrak{m}}+{\mathfrak{m}}\otimes{\mathfrak{m}}^{2}\,\big{)}\,\cong\,{\mathfrak{g}}^{*}\otimes{\mathfrak{g}}^{*}\,$
as Lie bialgebras. Let $\mathcal{F}$ be a quasi-twist for ${F_{\hbar}[[G\hskip
1.5pt]]}\,$, of the form $\,\mathcal{F}=\exp\big{(}\hbar^{-1}\phi\big{)}\,$
for some $\,\phi\in{F_{\hbar}[[G\hskip 1.5pt]]}^{\widetilde{\otimes}\,2}\,$,
and set also $\;\phi_{a}:=\phi-\phi_{\,2,1}\;$. Then: (a) $\phi$ is
antisymmetric, i.e. $\,\phi_{\,2,1}=-\phi\,$, iff $\mathcal{F}$ is orthogonal,
i.e. $\,\mathcal{F}_{2,1}=\mathcal{F}^{-1}\,$; (b) the element
$\,c\,:=\,\bigg{(}\,\phi_{a}\,\Big{(}\,\text{\rm
mod}\;\hbar\,{F_{\hbar}[[G\hskip
1.5pt]]}^{\widetilde{\otimes}\,2\,}\Big{)}\\!\\!\mod\big{(}{\mathfrak{m}}^{2}\otimes{\mathfrak{m}}+{\mathfrak{m}}\otimes{\mathfrak{m}}^{2}\,\big{)}\bigg{)}\;$
in
$\;\big{(}{\mathfrak{m}}\otimes{\mathfrak{m}}\big{)}\Big{/}\big{(}{\mathfrak{m}}^{2}\otimes{\mathfrak{m}}+{\mathfrak{m}}\otimes{\mathfrak{m}}^{2}\,\big{)}\,\cong\,{\mathfrak{g}}^{*}\otimes{\mathfrak{g}}^{*}\,$
is an antisymmetric twist for ${\mathfrak{g}}^{*}\,$; (c) the quasi-twist
deformation $\,{\big{(}{F_{\hbar}[[G\hskip 1.5pt]]}\big{)}}^{\mathcal{F}}$ of
$\,{F_{\hbar}[[G\hskip 1.5pt]]}$ is a QFSHA for the Poisson group $G^{\,c}$
whose cotangent Lie bialgebra is
$\,\;{\textsl{Lie}\,\big{(}G^{\,c}\big{)}}^{*}=\,{\big{(}\,{\mathfrak{g}}^{*}\big{)}}^{c}\,=\,\big{(}\,{\mathfrak{g}}^{*}\,;\,{[\,\
,\ ]}_{*}\,,\,\delta_{*}^{\,c}\,\big{)}\;\,$ that is the deformation of
$\,{\mathfrak{g}}^{*}$ by the twist $c\,$; in short,
$\;{\big{(}{F_{\hbar}[[G\hskip
1.5pt]]}\big{)}}^{\mathcal{F}}\cong\,F_{\hbar}\big{[}\big{[}G^{\,c}\big{]}\big{]}\;$.

In particular, if $\mathcal{F}$ is ${\Bbbk[[\hbar]]}$–valued — i.e., it is an
ordinary twist for the Hopf ${\Bbbk[[\hbar]]}$–algebra ${F_{\hbar}[[G\hskip
1.5pt]]}\,$ — or equivalently $\,\phi\in\hbar\,{F_{\hbar}[[G\hskip
1.5pt]]}^{\widetilde{\otimes}\,2}\,$, then we have just $\,c=0\,$ and
$\;{\big{(}{F_{\hbar}[[G\hskip
1.5pt]]}\big{)}}^{c}\cong\,F_{\hbar}\big{[}\big{[}G^{\,c}\big{]}\big{]}={F_{\hbar}[[G\hskip
1.5pt]]}\;$.

###### Proof.

(a) This is a special case of Lemma 3.4.2(d).

(b) We start from the twist identity
$\,\;\mathcal{F}_{1{}2}\,\big{(}\Delta\otimes\text{id}\big{)}(\mathcal{F}\,)\;=\,\mathcal{F}_{2{}3}\,\big{(}\text{id}\otimes\Delta\big{)}(\mathcal{F}\,)\;\,$
that we re-write in the equivalent form

$\big{(}\Delta\otimes\text{id}\big{)}(\mathcal{F}\,)\,{\big{(}\text{id}\otimes\Delta\big{)}(\mathcal{F}\,)}^{-1}\;=\;\,\mathcal{F}_{1{}2}^{\,-1}\,\mathcal{F}_{2{}3}$
(3.47)

Replacing $\,\mathcal{F}=\exp\\!\big{(}\hbar^{-1}\phi\big{)}\,$, we find

$\halign{\hbox to\displaywidth{$\hfil\displaystyle#\hfil$}\cr
0.0pt{\hfil$\displaystyle\quad\big{(}\Delta\otimes\text{id}\big{)}(\mathcal{F}\,)\big{)}\cdot{\big{(}\text{id}\otimes\Delta\big{)}(\mathcal{F}\,)}^{-1}\,\;=\hfill\cr
0.0pt{\hfil$\displaystyle=\;\,\big{(}\Delta\otimes\text{id}\big{)}\big{(}\exp\\!\big{(}\hbar^{-1}\phi\big{)}\big{)}\cdot{\big{(}\text{id}\otimes\Delta\big{)}\big{(}\exp\\!\big{(}\hbar^{-1}\phi\big{)}\big{)}}^{-1}\;=\cr
0.0pt{\hfil$\displaystyle\hfill=\;\,\exp\big{(}\,\hbar^{-1}\big{(}\Delta\otimes\text{id}\big{)}(\phi)\big{)}\cdot\exp\big{(}\\!-\hbar^{-1}\big{(}\text{id}\otimes\Delta\big{)}(\phi)\big{)}\quad\crcr}}}}$

Now we recall the Baker-Campbell-Hausdorff’s formula, that is the formal
identity

$\exp(X)\cdot\exp(Y)\;=\;\exp\big{(}\mathcal{B\,C\,H}(X,Y\,)\big{)}$ (3.48)

which allows to express the product of two exponential as a single
exponential: in it, $\,\mathcal{B\,C\,H}(X,Y\,)\,$ is an explicit formal
series given by

$\displaystyle\mathcal{B\,C\,H}(X,Y\,)\,\;:=\;\,\log\big{(}\exp(X)\,\exp(Y\,)\big{)}\,\;=$
(3.49)
$\displaystyle\qquad\hfill\,\;=\;\,{\textstyle\sum\limits_{n=1}^{+\infty}}\,\frac{\;{(-1)}^{n-1}\;}{\;n\;}\,{\textstyle\sum\limits_{\begin{subarray}{c}r_{i}+s_{i}>0\\\
1\leq i\leq n\end{subarray}}}\frac{\;\big{[}\,X^{\bullet r_{1}}Y^{\bullet
s_{1}}X^{\bullet r_{2}}Y^{\bullet s_{2}}\cdots X^{\bullet r_{n}}Y^{\bullet
s_{n}}\big{]}\;}{\;\big{(}\sum_{i=1}^{n}(r_{i}+s_{i})\big{)}\cdot\prod_{j=1}^{n}r_{i}!\,s_{i}!\;}$

where we use notation

$\halign{\hbox to\displaywidth{$\hfil\displaystyle#\hfil$}\cr
0.0pt{\hfil$\displaystyle\big{[}\,X^{\bullet r_{1}}Y^{\bullet s_{1}}\cdots
X^{\bullet r_{n}}Y^{\bullet s_{n}}\big{]}\,\;:=\hfill\cr
0.0pt{\hfil$\displaystyle\hfill=\;\,\big{[}\underbrace{X,\big{[}X,\cdots\big{[}X}_{r_{1}},\big{[}\underbrace{Y,\big{[}Y,\cdots\big{[}Y}_{s_{1}},\cdots,\big{[}\underbrace{X,\big{[}X,\cdots\big{[}X}_{r_{n}},\big{[}\underbrace{Y,\big{[}Y,\cdots,Y}_{s_{n}}\big{]}\big{]}\big{]}\cdots\big{]}\big{]}\cdots\big{]}\big{]}\crcr}}}$

with the silent assumption that the Lie monomial $\,\big{[}\,X^{\bullet
r_{1}}Y^{\bullet s_{1}}\cdots X^{\bullet r_{n}}Y^{\bullet s_{n}}\big{]}\,$ is
just $X$, respectively $Y$, when $\,n=1\,$ and $\,s_{1}=0\,$, respectively
$\,r_{1}=0\,$, while it is zero whenever $\,s_{n}>1\,$ or $\,s_{n}=0\,$ and
$\,r_{n}>1\,$. In words, when $\,S:=\sum_{i=1}^{n}(r_{i}+s_{i})>1\,$ the Lie
monomial $\,\big{[}\,X^{\bullet r_{1}}Y^{\bullet s_{1}}\cdots X^{\bullet
r_{n}}Y^{\bullet s_{n}}\big{]}\,$ is the composition of several operators
${\operatorname{ad}(X)}^{r_{i}}$ or ${\operatorname{ad}(Y)}^{s_{i}}$ to $Y$ —
when $\,s_{n}=1\,$ — or to $X$ — when $\,s_{n}=0\,$ and $\,r_{n}=1\,$. Looking
up to second order, (3.49) reads

$\mathcal{B\,C\,H}(X,Y\,)\,\;:=\;\,X\,+\,Y\,+\,\frac{\,1\,}{\,2\,}\,[X,Y\,]\,+\,\mathcal{O}_{\mathcal{L}}(3)$
(3.50)

where $\,\mathcal{O}_{\mathcal{L}}(3)\,$ denotes a (formal) infinite linear
combination of Lie monomials in $X$ and $Y$ of degree at least 3. Setting now
$\,X:=\hbar^{-1}\big{(}\Delta\otimes\text{id}\big{)}(\phi)\,$ and
$\,Y:=-\hbar^{-1}\big{(}\text{id}\otimes\Delta\big{)}(\phi)\,$, the above
analysis yields, rewriting (3.48),

$\displaystyle\big{(}\Delta\otimes\text{id}\big{)}(\mathcal{F}\,)$
$\displaystyle\cdot{\big{(}\text{id}\otimes\Delta\big{)}(\mathcal{F}\,)}^{-1}\,\;=$
(3.51)
$\displaystyle=\;\,\exp\Big{(}\,\mathcal{B\,C\,H}\Big{(}\,\hbar^{-1}\big{(}\Delta\otimes\text{id}\big{)}(\phi)\,,\,-\hbar^{-1}\big{(}\text{id}\otimes\Delta\big{)}(\phi)\Big{)}\Big{)}$

where the BCH series has to be expanded as in (3.49). To this end, writing
$\,\phi=\phi_{1}\otimes\phi_{2}\,$ (a sum being tacitly intended) with
$\,\phi_{1},\phi_{2}\in J_{\hbar}\,$ (by Lemma 3.1.1) we have

$\halign{\hbox to\displaywidth{$\hfil\displaystyle#\hfil$}\cr
0.0pt{\hfil$\displaystyle\big{(}\Delta\otimes\text{id}\big{)}(\phi)\,=\,\Delta\big{(}\phi_{1}\big{)}\otimes\phi_{2}\,=\,\phi_{1}\otimes
1\otimes\phi_{2}\,+\,1\otimes\phi_{1}\otimes\phi_{2}\,+\,{\big{(}{(\phi_{1})}_{(1)}\big{)}}^{+}\otimes{\big{(}{(\phi_{1})}_{(2)}\big{)}}^{+}\otimes\phi_{2}\,=\hfill\cr
0.0pt{\hfil$\displaystyle\hfill=\;\phi_{1,3}\,+\,\phi_{2,3}\,+\,{\big{(}{(\phi_{1})}_{(1)}\big{)}}^{+}\otimes{\big{(}{(\phi_{1})}_{(2)}\big{)}}^{+}\otimes\phi_{2}\crcr}}}$

where we expanded $\,\Delta\big{(}\phi_{1}\big{)}\,$ as in Lemma 2.3.5(d) and
we used that $\,\epsilon\big{(}\phi_{1}\big{)}=0\,$. Note that in the
expansion of $\big{(}\Delta\otimes\text{id}\big{)}(\phi)$ we have

$\big{(}\phi_{1,3}\,+\,\phi_{2,3}\big{)}\,\in\,J_{\hbar}^{(\otimes
3|2)}\;\;,\qquad{\big{(}{(\phi_{1})}_{(1)}\big{)}}^{\\!+}\otimes{\big{(}{(\phi_{1})}_{(2)}\big{)}}^{\\!+}\otimes\phi_{2}\,\in\,J_{\hbar}^{\,\otimes
3}$ (3.52)

where we introduced the notation $\;J_{\hbar}^{(\otimes
3|N)}:=\sum_{\\!\begin{subarray}{c}a,b,c\,\geqslant\,0\\\
a+b+c\,\geqslant\,N\end{subarray}}\\!J_{\hbar}^{\,a}\,\widetilde{\otimes}\,J_{\hbar}^{\,b}\,\widetilde{\otimes}\,J_{\hbar}^{\,c}\;$
(for $\,N\in\mathbb{N}\,$).

A similar analysis for $\,\big{(}\text{id}\otimes\Delta\big{)}(\phi)\,$, just
switching the roles of $\phi_{1}$ and $\phi_{2}\,$, yields

$\halign{\hbox to\displaywidth{$\hfil\displaystyle#\hfil$}\cr
0.0pt{\hfil$\displaystyle\big{(}\text{id}\otimes\Delta\big{)}(\phi)\,=\,\phi_{1}\otimes\Delta\big{(}\phi_{2}\big{)}\,=\,\phi_{1}\otimes\phi_{2}\otimes
1\,+\,\phi_{1}\otimes
1\otimes\phi_{2}\,+\,\phi_{1}\otimes{\big{(}{(\phi_{2})}_{(1)}\big{)}}^{+}\otimes{\big{(}{(\phi_{2})}_{(2)}\big{)}}^{+}\,=\hfill\cr
0.0pt{\hfil$\displaystyle\hfill=\;\phi_{1,2}\,+\,\phi_{1,3}\,+\,\phi_{1}\otimes{\big{(}{(\phi_{2})}_{(1)}\big{)}}^{+}\otimes{\big{(}{(\phi_{2})}_{(2)}\big{)}}^{+}\crcr}}}$

with

$\big{(}\phi_{1,2}\,+\,\phi_{1,3}\big{)}\,\in\,J_{\hbar}^{(\otimes
3|2)}\;\;,\qquad\phi_{1}\otimes{\big{(}{(\phi_{2})}_{(1)}\big{)}}^{\\!+}\otimes{\big{(}{(\phi_{2})}_{(2)}\big{)}}^{\\!+}\,\in\,J_{\hbar}^{\,\otimes
3}$ (3.53)

Now, thanks to Lemma 2.3.5(c), from (3.52) and (3.53) we get

$\halign{\hbox to\displaywidth{$\hfil\displaystyle#\hfil$}\cr
0.0pt{\hfil$\displaystyle\quad\big{[}\hbar\,X,\hbar\,Y\big{]}\;=\;\big{[}\big{(}\Delta\otimes\text{id}\big{)}(\phi)\,,\big{(}\text{id}\otimes\Delta\big{)}(\phi)\big{]}\,\;=\hfill\cr
0.0pt{\hfil$\displaystyle=\;\,\big{[}\,\phi_{1,3}+\phi_{2,3}\,,\,\phi_{1,2}+\phi_{1,3}\,\big{]}\;+\;\hbar\cdot\mathcal{O}\Big{(}J_{\hbar}^{(\otimes
3|4)}\Big{)}\,\;=\quad\cr
0.0pt{\hfil$\displaystyle\hfill=\;\,\big{[}\,\phi_{1,3}\,,\,\phi_{1,2}\,\big{]}+\big{[}\,\phi_{2,3}\,,\,\phi_{1,2}\,\big{]}+\big{[}\,\phi_{2,3}\,,\,\phi_{1,3}\,\big{]}\;+\;\hbar\cdot\mathcal{O}\Big{(}J_{\hbar}^{[\otimes
3|4]}\Big{)}\quad\crcr}}}}$

hence

$\\!\\!\\!\big{[}X,Y\big{]}\,=\,\hbar^{-1}\Big{(}\hbar^{-1}\big{[}\phi_{1,3},\phi_{1,2}\big{]}+\hbar^{-1}\big{[}\phi_{2,3},\phi_{1,2}\big{]}+\hbar^{-1}\big{[}\phi_{2,3},\phi_{1,3}\big{]}\,+\,\mathcal{O}\Big{(}J_{\hbar}^{[\otimes
3|4]}\Big{)}\Big{)}$ (3.54)

for some element $\,\mathcal{O}\Big{(}J_{\hbar}^{[\otimes 3|4]}\Big{)}\in
J_{\hbar}^{[\otimes 3|4]}\,$, where hereafter we use notation

$J_{\hbar}^{[\otimes
3|N]}\,:=\hskip-7.0pt\sum\limits_{\begin{subarray}{c}a,b,c\,\geqslant\,1\\\
a+b+c\,\geqslant\,N\end{subarray}}\hskip-11.0ptJ_{\hbar}^{\otimes a}\otimes
J_{\hbar}^{\otimes b}\otimes J_{\hbar}^{\otimes
c}\;\subseteq\;{\textsl{Ker}\Big{(}\epsilon_{{}_{{{F_{\hbar}[[G\hskip
1.5pt]]}}^{\otimes 3}}}\Big{)}}^{N}\qquad\forall\;\;N\in\mathbb{N}_{+}$ (3.55)

Pushing the analysis further on, we find easily that

$\big{[}\,X^{\bullet r_{1}}Y^{\bullet s_{1}}\cdots X^{\bullet r_{n}}Y^{\bullet
s_{n}}\big{]}\;\in\;\hbar^{-1}\,J_{\hbar}^{[\otimes
3|S+1]}\qquad\forall\;\;S:={\textstyle\sum\limits_{i=1}^{n}}\,(r_{i}+s_{i})>1\,$
(3.56)

looking at (3.49), this tells us that the expansion of the BCH series
occurring in (3.51), when expanded as in (3.49), is actually given by
$\hbar^{-1}$ multiplied by a truly convergent series inside
${{F_{\hbar}[[G\hskip 1.5pt]]}}^{\otimes 3}\,$. In other words, tiding
everything up — from (3.51), (3.49), (3.50), (3.54) and (3.56) altogether — we
find that there exists some $\,\mathcal{Z}\in J_{\hbar}^{\otimes
3}\subseteq{{F_{\hbar}[[G\hskip 1.5pt]]}}^{\otimes 3}\,$ such that
$\;\big{(}\Delta\otimes\text{id}\big{)}(\mathcal{F}\,)\cdot{\big{(}\text{id}\otimes\Delta\big{)}(\mathcal{F}\,)}^{-1}=\,\exp\big{(}\,\hbar^{-1}\mathcal{Z}\,\big{)}\;$
Even more, by (3.49) and (3.50) and the previous analysis we do know the
expansion of this $\mathcal{Z}$ up to second order, whence we find

$\displaystyle\hskip-6.0pt\big{(}\Delta\otimes\text{id}\big{)}(\mathcal{F}\,)\cdot{\big{(}\text{id}\otimes\Delta\big{)}(\mathcal{F}\,)}^{-1}\;=$
(3.57)
$\displaystyle\hskip-5.0pt=\;\,\exp\Big{(}\,\hbar^{-1}\,\Big{(}\big{(}\Delta\otimes\text{id}\big{)}(\phi)\,-\,\big{(}\text{id}\otimes\Delta\big{)}(\phi)\;-$
$\displaystyle\hskip-5.0pt-\hbar^{-1}2^{-1}\big{[}\phi_{1,3},\phi_{1,2}\big{]}-\hbar^{-1}2^{-1}\big{[}\phi_{2,3},\phi_{1,2}\big{]}-\hbar^{-1}2^{-1}\big{[}\phi_{2,3},\phi_{1,3}\big{]}\,+\,\mathcal{O}\Big{(}J_{\hbar}^{[\otimes\,3|4]}\Big{)}\\!\Big{)}\\!\Big{)}$

Now we go and work instead on the right-hand side of (3.47). Again, replacing
$\,\mathcal{F}=\exp\\!\big{(}\hbar^{-1}\phi\big{)}\,$, we find

$\displaystyle\mathcal{F}_{1{}2}^{-1}\cdot\mathcal{F}_{2{}3}\;=\;\exp\\!\big{(}\\!-\\!\hbar^{-1}\phi\otimes
1\big{)}\cdot\exp\\!\big{(}\,\hbar^{-1}1\otimes\phi\big{)}\,\;=$ (3.58)
$\displaystyle\;\;\quad=\;\exp\\!\big{(}\\!-\\!\hbar^{-1}\phi_{1,2}\big{)}\cdot\exp\\!\big{(}\,\hbar^{-1}\phi_{2,3}\big{)}\;=\;\exp\\!\Big{(}\mathcal{B\,C\,H}\big{(}\\!-\\!\hbar^{-1}\,\phi_{1,2}\,,\,\hbar^{-1}\,\phi_{2,3}\,\big{)}\Big{)}$

Now for the computation of
$\;\mathcal{B\,C\,H}\big{(}\\!-\\!\hbar^{-1}\,\phi_{1,2}\,,\hbar^{-1}\,\phi_{2,3}\big{)}\;$;
to avoid possible confusion, we denote the second, right-hand instance of
$\phi$ by $\phi^{\prime}\,$. We begin noting that

$\big{[}\,\phi_{1,2}\,,\phi^{\prime}_{2,3}\,\big{]}\,\;=\;\,\big{[}\,\phi_{1}\otimes\phi_{2}\otimes
1\,,1\otimes\phi^{\prime}_{1}\otimes\phi^{\prime}_{2}\,\big{]}\,\;=\;\,\phi_{1}\otimes\big{[}\,\phi_{2}\,,\phi^{\prime}_{1}\,\big{]}\otimes\phi^{\prime}_{2}\,\;\in\;\,\hbar\,J_{\hbar}^{[\otimes
3|3]}$

so that for $\;X:=-\hbar^{-1}\,\phi_{1,2}\;$ and
$\;Y:=\hbar^{-1}\,\phi_{2,3}\;$ we get, using Lemma 2.3.5(c),

$\big{[}X,Y\big{]}\;=\;\big{[}-\\!\hbar^{-1}\,\phi_{1,2}\,,\hbar^{-1}\,\phi^{\prime}_{2,3}\,\big{]}\;=\;-\\!\hbar^{-2}\big{[}\,\phi_{1,2}\,,\phi^{\prime}_{2,3}\,\big{]}\;\in\;\hbar^{-2}\,\hbar\,J_{\hbar}^{\otimes
3}\;=\;\hbar^{-1}J_{\hbar}^{[\otimes 3|3]}$

A second, similar step gives (with obvious notation $\phi\,$, $\phi^{\prime}$
and $\phi^{\prime\prime}\,$)

$\halign{\hbox to\displaywidth{$\hfil\displaystyle#\hfil$}\cr
0.0pt{\hfil$\displaystyle\big{[}\,\phi_{1,2}\,,\big{[}\,\phi^{\prime}_{1,2}\,,\phi^{\prime\prime}_{2,3}\,\big{]}\big{]}\,\;=\;\,\big{[}\,\phi_{1}\otimes\phi_{2}\otimes
1\,,\phi^{\prime}_{1}\otimes\big{[}\,\phi^{\prime}_{2}\,,\phi^{\prime\prime}_{1}\,\big{]}\otimes\phi^{\prime\prime}_{2}\,\big{]}\,\;=\hfill\cr
0.0pt{\hfil$\displaystyle\hfill=\;\,\big{[}\,\phi_{1}\,,\phi^{\prime}_{1}\,\big{]}\otimes\phi_{2}\cdot\big{[}\,\phi^{\prime}_{2}\,,\phi^{\prime\prime}_{1}\,\big{]}\otimes\phi^{\prime\prime}_{2}\,+\,\phi_{1}\cdot\phi^{\prime}_{1}\otimes\big{[}\,\phi_{2}\,,\big{[}\,\phi^{\prime}_{2}\,,\phi^{\prime\prime}_{1}\,\big{]}\big{]}\otimes\phi^{\prime\prime}_{2}\,\;\in\;\,\hbar^{2}\,J_{\hbar}^{[\otimes
3|4]}\crcr}}}$

so that
$\;\big{[}X,\big{[}X,Y\,\big{]}\big{]}\,\in\,\hbar^{-3}\,\hbar^{2}\,J_{\hbar}^{[\otimes
3|4]}\,=\,\hbar^{-1}\,J_{\hbar}^{[\otimes 3|4]}\;$. More in general, iteration
yields

$\big{[}\,X^{\bullet r_{1}}Y^{\bullet s_{1}}\cdots X^{\bullet r_{n}}Y^{\bullet
s_{n}}\big{]}\;\in\;\hbar^{-S}\,\hbar^{S-1}\,J_{\hbar}^{\,[\otimes
3|S+1]}\;=\;\hbar^{-1}\,J_{\hbar}^{\,[\otimes 3|S+1]}$ (3.59)

with notation as before, still using Lemma 2.3.5(c). Tiding everything up we
find that there exists $\,\mathcal{W}\in J_{\hbar}^{\otimes
3}\subseteq{{F_{\hbar}[[G\hskip 1.5pt]]}}^{\widetilde{\otimes}\,3}\,$ such
that
$\;\mathcal{F}_{1{}2}^{-1}\cdot\mathcal{F}_{2{}3}\,=\,\exp\big{(}\,\hbar^{-1}\mathcal{W}\,\big{)}\;$;
moreover, by (3.49) and (3.50) along with the previous analysis we can write

$\mathcal{F}_{1,2}^{-1}\cdot\mathcal{F}_{2,3}\;=\;\exp\Big{(}\,\hbar^{-1}\Big{(}\\!-\phi_{1,2}\,+\,\phi_{2,3}\,-\,\hbar^{-1}\,2^{-1}\big{[}\,\phi_{1,2}\,,\,\phi_{2,3}\,\big{]}\,+\,\mathcal{O}\Big{(}J_{\hbar}^{[\otimes\,3|4]}\Big{)}\Big{)}$
(3.60)

Finally, comparing (3.57), (3.60) and (3.47) we get the identity in
${{F_{\hbar}[[G\hskip 1.5pt]]}}^{\widetilde{\otimes}\,3}$

$\halign{\hbox to\displaywidth{$\hfil\displaystyle#\hfil$}\cr
0.0pt{\hfil$\displaystyle\big{(}\Delta\otimes\text{id}\big{)}(\phi)\,-\,\big{(}\text{id}\otimes\Delta\big{)}(\phi)\;-\hfill\cr
0.0pt{\hfil$\displaystyle\hskip
11.0pt-\,\hbar^{-1}2^{-1}\big{[}\phi_{1,3}\,,\,\phi_{1,2}\big{]}-\hbar^{-1}2^{-1}\big{[}\phi_{2,3}\,,\,\phi_{1,2}\big{]}-\hbar^{-1}2^{-1}\big{[}\phi_{2,3}\,,\,\phi_{1,3}\big{]}\,+\,\mathcal{O}\Big{(}J_{\hbar}^{[\otimes\,3|4]}\Big{)}\,\;=\hfill\cr
0.0pt{\hfil$\displaystyle\hfill=\;\,-\phi_{1,2}\,+\,\phi_{2,3}\,-\,\hbar^{-1}\,2^{-1}\big{[}\,\phi_{1,2}\,,\,\phi_{2,3}\,\big{]}\,+\,\mathcal{O}\Big{(}J_{\hbar}^{[\otimes\,3|4]}\Big{)}\crcr}}}}$

that in turn, through simplification and reduction modulo
$\,\hbar\,{{F_{\hbar}[[G\hskip 1.5pt]]}}^{\widetilde{\otimes}\,3}\,$, yields
the following identity inside ${{F[[G\hskip
1.5pt]]}}^{\widetilde{\otimes}\,3}\,$

$\displaystyle\big{(}\Delta\otimes\text{id}\big{)}\big{(}\,\overline{\phi}\,\big{)}\,-\,\big{(}\text{id}\otimes\Delta\big{)}\big{(}\,\overline{\phi}\,\big{)}\,+\,\overline{\phi}_{1,2}\,-\,\overline{\phi}_{2,3}\;+$
(3.61) $\displaystyle\hskip
41.0pt+\;2^{-1}\big{\\{}\,\overline{\phi}_{1,2}\,,,\overline{\phi}_{1,3}\,\big{\\}}\,+\,\big{\\{}\,\overline{\phi}_{1,2}\,,\,\overline{\phi}_{2,3}\,\big{\\}}\,+\,2^{-1}\big{\\{}\,\overline{\phi}_{1,3}\,,\,\overline{\phi}_{2,3}\,\big{\\}}\;\underset{{\mathfrak{m}}^{[\otimes\,3\,|4]}}{\equiv}\;0$

where hereafter we adopt the notation for which $\,\overline{\varphi}\,$
denotes the coset modulo $\,\hbar\,$ of any element
$\,\varphi\in{{F_{\hbar}[[G\hskip 1.5pt]]}}^{\widetilde{\otimes}\,3}\,$ with
$\,n\in\mathbb{N}_{+}\,$.

Now let $\,\Bbbk\big{[}\mathbb{S}_{3}\big{]}\,$ act onto ${{F_{\hbar}[[G\hskip
1.5pt]]}}^{\widetilde{\otimes}\,3}$ and consider in particular the action of
the antisymmetrizer
$\;\textsl{Alt}_{\,3}:=\big{(}\operatorname{id}-(1\,2)-(2\,3)-(3\,1)+(1\,2\,3)+(3\,2\,1)\big{)}\;$
onto the above identity: this in turn yields a new identity. Within the
latter, we have a first contribution of the form

$\halign{\hbox to\displaywidth{$\hfil\displaystyle#\hfil$}\cr
0.0pt{\hfil$\displaystyle\textsl{Alt}_{\,3\,}.\Big{(}\big{(}\Delta\otimes\text{id}\big{)}\big{(}\,\overline{\phi}\,\big{)}\,-\,\big{(}\text{id}\otimes\Delta\big{)}\big{(}\,\overline{\phi}\,\big{)}\Big{)}\,\;=\hfill\cr
0.0pt{\hfil$\displaystyle\hfill=\;\,\big{(}\,\nabla\otimes\text{id}\big{)}\big{(}\,\overline{\phi}\,\big{)}-\big{(}\,\nabla\otimes\text{id}\big{)}\big{(}\,\overline{\phi}_{2,1}\big{)}\;+\;\text{c.p}\,\;=\;\,\big{(}\,\nabla\otimes\text{id}\big{)}\big{(}\,\overline{\phi_{a}}\,\big{)}\;+\;\text{c.p}\crcr}}}$

and a second contribution of the form

$\textsl{Alt}_{\,3\,}.\big{(}\,\overline{\phi}_{1,2}\,-\,\overline{\phi}_{2,3}\,\big{)}\,\;=\;\,0$

The third and last contribution is

$\textsl{Alt}_{\,3\,}.\Big{(}\,2^{-1}\big{\\{}\,\overline{\phi}_{1,2}\,,\,\overline{\phi}_{1,3}\,\big{\\}}\Big{)}\,+\,\textsl{Alt}_{\,3\,}.\Big{(}\,\big{\\{}\,\overline{\phi}_{1,2}\,,\,\overline{\phi}_{2,3}\,\big{\\}}\Big{)}\,+\,\textsl{Alt}_{\,3\,}.\Big{(}\,2^{-1}\big{\\{}\,\overline{\phi}_{1,3}\,,\,\overline{\phi}_{2,3}\,\big{\\}}\Big{)}$

We go and compute the first summand, as follows:

$\halign{\hbox to\displaywidth{$\hfil\displaystyle#\hfil$}\cr
0.0pt{\hfil$\displaystyle\textsl{Alt}_{\,3\,}.\Big{(}\,2^{-1}\big{\\{}\,\overline{\phi}_{1,2}\,,\,\overline{\phi}_{1,3}\,\big{\\}}\Big{)}\,\;=\;\,\big{(}\operatorname{id}-(2\,3)\big{)}.\Big{(}\,2^{-1}\big{\\{}\,\overline{\phi}_{1,2}\,,\,\overline{\phi}_{1,3}\,\big{\\}}\Big{)}\;+\;\text{c.p.}\,\;=\hfill\cr
0.0pt{\hfil$\displaystyle\hfill=\;\,\Big{(}\,2^{-1}\big{\\{}\,\overline{\phi}_{1,2}\,,\,\overline{\phi}_{1,3}\,\big{\\}}\;-\;2^{-1}\big{\\{}\,\overline{\phi}_{1,3}\,,\,\overline{\phi}_{1,2}\,\big{\\}}\Big{)}\;+\;\text{c.p.}\,\;=\;\,\big{\\{}\,\overline{\phi}_{1,2}\,,\,\overline{\phi}_{1,3}\,\big{\\}}\;+\;\text{c.p.}\crcr}}}$

A similar analysis applies to the third summand, which yields

$\halign{\hbox to\displaywidth{$\hfil\displaystyle#\hfil$}\cr
0.0pt{\hfil$\displaystyle\textsl{Alt}_{\,3\,}.\Big{(}\,2^{-1}\big{\\{}\,\overline{\phi}_{1,3}\,,\,\overline{\phi}_{2,3}\,\big{\\}}\Big{)}\,\;=\;\,\big{(}\operatorname{id}-(1\,2)\big{)}.\Big{(}\,2^{-1}\big{\\{}\,\overline{\phi}_{1,3}\,,\,\overline{\phi}_{2,3}\,\big{\\}}\Big{)}\;+\;\text{c.p.}\,\;=\hfill\cr
0.0pt{\hfil$\displaystyle\hfill=\;\,\Big{(}\,2^{-1}\big{\\{}\,\overline{\phi}_{1,3}\,,\,\overline{\phi}_{2,3}\,\big{\\}}\;-\;2^{-1}\big{\\{}\,\overline{\phi}_{2,3}\,,\,\overline{\phi}_{1,3}\,\big{\\}}\Big{)}\;+\;\text{c.p.}\,\;=\;\,\big{\\{}\,\overline{\phi}_{1,3}\,,\,\overline{\phi}_{2,3}\,\big{\\}}\;+\;\text{c.p.}\crcr}}}$

whereas for the second summand instead we get

$\halign{\hbox to\displaywidth{$\hfil\displaystyle#\hfil$}\cr
0.0pt{\hfil$\displaystyle\textsl{Alt}_{\,3\,}.\Big{(}\big{\\{}\,\overline{\phi}_{1,2}\,,\,\overline{\phi}_{2,3}\,\big{\\}}\Big{)}\,\;=\;\,\big{(}\operatorname{id}-(1\,3)\big{)}.\Big{(}\big{\\{}\,\overline{\phi}_{1,2}\,,\,\overline{\phi}_{2,3}\,\big{\\}}\Big{)}\;+\;\text{c.p.}\,\;=\hfill\cr
0.0pt{\hfil$\displaystyle\hfill=\;\,\big{\\{}\,\overline{\phi}_{1,2}\,,\,\overline{\phi}_{2,3}\,\big{\\}}\;-\;\big{\\{}\,\overline{\phi}_{3,2}\,,\,\overline{\phi}_{2,1}\,\big{\\}}\;+\;\text{c.p.}\crcr}}}$

Putting all these together we find

$\halign{\hbox to\displaywidth{$\hfil\displaystyle#\hfil$}\cr
0.0pt{\hfil$\displaystyle\textsl{Alt}_{\,3\,}.\Big{(}\,2^{-1}\big{\\{}\,\overline{\phi}_{1,2}\,,\,\overline{\phi}_{1,3}\,\big{\\}}\Big{)}\,+\,\textsl{Alt}_{\,3\,}.\Big{(}\,\big{\\{}\,\overline{\phi}_{1,2}\,,\,\overline{\phi}_{2,3}\,\big{\\}}\Big{)}\,+\,\textsl{Alt}_{\,3\,}.\Big{(}\,2^{-1}\big{\\{}\,\overline{\phi}_{1,3}\,,\,\overline{\phi}_{2,3}\,\big{\\}}\Big{)}\,\;=\hfill\cr
0.0pt{\hfil$\displaystyle=\;\big{\\{}\,\overline{\phi}_{1,2}\,,\,\overline{\phi}_{1,3}\big{\\}}\,+\,\big{\\{}\,\overline{\phi}_{1,2}\,,\,\overline{\phi}_{2,3}\big{\\}}\,-\,\big{\\{}\,\overline{\phi}_{3,2}\,,\,\overline{\phi}_{2,1}\big{\\}}\,+\,\big{\\{}\,\overline{\phi}_{1,3}\,,\,\overline{\phi}_{2,3}\big{\\}}\,+\,\text{c.p.}\;=\;\big{\\{}\\!\big{\\{}\,\overline{\phi_{a}}\,,\,\overline{\phi_{a}}\,\big{\\}}\\!\big{\\}}\crcr}}}$

where the very last identity follows from a routine calculation. Joint with
the previously found identities, the latter gives yet the following, last one,
which is the result of letting $\,\textsl{Alt}_{\,3}\,$ act onto the
congruence (3.61):

$\Big{(}\\!\big{(}\,\nabla\otimes\operatorname{id}\,\big{)}\big{(}\,\overline{\phi_{a}}\,\big{)}\,+\,\text{c.p.}\Big{)}\,+\,\big{\\{}\\!\big{\\{}\,\overline{\phi_{a}}\,,\,\overline{\phi_{a}}\,\big{\\}}\\!\big{\\}}\,\;\underset{{\mathfrak{m}}^{[\otimes\,4]}}{\equiv}\;\,0$

At last, recalling that $\,c:=\overline{\phi_{a}}\,\big{(}\,\text{mod\
}{\mathfrak{m}}^{2}\big{)}\,$ in
$\,{\mathfrak{m}}\Big{/}{\mathfrak{m}}^{2}={\mathfrak{g}}^{*}\,$, and that in
the latter Lie bialgebra the Lie cobracket, resp. the Lie bracket, is given by
$\,\nabla\,$, resp. by $\,[\,\ ,\ ]\,$, reduced modulo ${\mathfrak{m}}^{2}\,$,
the last formula above — in ${\mathfrak{m}}^{\widehat{\otimes}\,3}$ — implies

$\Big{(}\\!\big{(}\,\delta\otimes\operatorname{id}\,\big{)}(c\,)\,+\,\text{c.p.}\Big{)}\;+\;[[\,c\,,c\,]]\,\;=\;\,0$

within ${\big{(}{\mathfrak{g}}^{*}\big{)}}^{\otimes 3}$, which implies exactly
that $\,c\,$ — which is antisymmetric by construction — is an antisymmetric
twist for the Lie bialgebra
$\,{\mathfrak{g}}^{*}={\mathfrak{m}}\Big{/}{\mathfrak{m}}^{2}\,$, q.e.d. (c)
We adopt the following notational convention: any element in
${F_{\hbar}[[G\hskip 1.5pt]]}$ will be denoted by an italic letter, say
$\,f\in{F_{\hbar}[[G\hskip 1.5pt]]}\,$; then its coset modulo
$\,\hbar\,{F_{\hbar}[[G\hskip 1.5pt]]}\,$ will be denoted with a line over
that letter, say $\,\overline{f}:=\big{(}\,f\ \text{mod}\
\hbar\,{F_{\hbar}[[G\hskip 1.5pt]]}\,\big{)}\,$, and finally the coset of the
latter modulo $\,{\mathfrak{m}}^{2}\,$ will be denoted by the corresponding
letter in roman font, say $\,\text{f}:=\big{(}\,\overline{f}\ \text{mod}\
{\mathfrak{m}}^{2}\,\big{)}\,$. Note also that every element in
$\,{\mathfrak{g}}^{*}={\mathfrak{m}}\Big{/}{\mathfrak{m}}^{2}\,$ can be
written as such an $\;\text{f}=\big{(}\,\overline{f}\ \text{mod}\
{\mathfrak{m}}^{2}\,\big{)}\,$ for some $\,f\in
J_{\hbar}\in\textsl{Ker}\big{(}\epsilon_{{}_{F_{\hbar}[[G\hskip
1.5pt]]}}\big{)}\,$.

Similar notation will be used for elements in ${{F_{\hbar}[[G\hskip
1.5pt]]}}^{\widetilde{\otimes}\,2}$ and their coset modulo $\hbar$ and
(further on) modulo $\,{\mathfrak{m}}^{[\otimes
2\,|3]}:={\mathfrak{m}}\otimes{\mathfrak{m}}^{2}+{\mathfrak{m}}^{2}\otimes{\mathfrak{m}}\,$.

Recall that the Lie cobracket induced on
$\,{\mathfrak{m}}\Big{/}{\mathfrak{m}}^{2}={\mathfrak{g}}^{*}\,$ by the
deformed quantization is defined by

$\delta^{\,\mathcal{F}}(\,\text{f}\,)\;:=\;\Big{(}\Delta^{\mathcal{F}}-{\big{(}\Delta^{\mathcal{F}}\big{)}}^{2{}1}\Big{)}\big{(}\,\overline{f}\,\big{)}\
\ \text{mod}\ {\mathfrak{m}}^{[\otimes
2\,|3]}\;=\;\overline{\Delta^{\mathcal{F}}(f)}-\overline{{\big{(}\Delta^{\mathcal{F}}\big{)}}^{\text{op}}(f)}\,\
\ \text{mod}\ {\mathfrak{m}}^{[\otimes 2\,|3]}$

so we start computing $\,\Delta^{\mathcal{F}}(f)\,$. Definitions give

$\halign{\hbox to\displaywidth{$\hfil\displaystyle#\hfil$}\cr
0.0pt{\hfil$\displaystyle\Delta^{\mathcal{F}}(f)\,\;=\;\,\operatorname{Ad}(\,\mathcal{F}\,)\big{(}\Delta(f)\big{)}\,=\,\operatorname{Ad}(\mathcal{F}\,)\big{(}\,f_{(1)}\otimes
f_{(2)}\big{)}\,\;=\hfill\cr
0.0pt{\hfil$\displaystyle\hfill=\;\,\operatorname{Ad}(\mathcal{F}\,)\Big{(}\\!\big{(}\,f_{(1)}\otimes
1\,\big{)}\cdot\big{(}\,1\otimes
f_{(2)}\big{)}\\!\Big{)}\,\;=\;\,\operatorname{Ad}(\mathcal{F}\,)\big{(}\,f_{(1)}\otimes
1\,\big{)}\cdot\operatorname{Ad}(\mathcal{F}\,)\big{(}\,1\otimes
f_{(2)}\big{)}\crcr}}}$

In the last product, we focus on the first factor: thus we get

$\halign{\hbox to\displaywidth{$\hfil\displaystyle#\hfil$}\cr
0.0pt{\hfil$\displaystyle\operatorname{Ad}(\mathcal{F}\,)\big{(}\,f_{(1)}\otimes
1\big{)}\,\;=\;\,\operatorname{Ad}\big{(}\exp\\!\big{(}\hbar^{-1}\phi\big{)}\big{)}\big{(}\,f_{(1)}\otimes
1\,\big{)}\,\;=\;\,\exp\big{(}\operatorname{ad}\\!\big{(}\hbar^{-1}\phi\big{)}\big{)}\big{(}\,f_{(1)}\otimes
1\,\big{)}\,\;=\hfill\cr
0.0pt{\hfil$\displaystyle=\;\,{\textstyle\sum\limits_{n=0}^{+\infty}}\,\frac{\,1\,}{\,n!\,}\;{\operatorname{ad}\\!\big{(}\hbar^{-1}\phi\big{)}}^{n}\big{(}\,f_{(1)}\otimes
1\,\big{)}\,\;=\;\,{\textstyle\sum\limits_{n=0}^{+\infty}}\,\frac{\,1\,}{\,n!\,}\;{\operatorname{ad}\\!\big{(}\hbar^{-1}\phi_{1}\otimes\phi_{2}\big{)}}^{n}\big{(}\,f_{(1)}\otimes
1\,\big{)}\,\;=\cr
0.0pt{\hfil$\displaystyle\hfill=\;\,{\textstyle\sum\limits_{n=0}^{+\infty}}\,\frac{\,1\,}{\,n!\,}\;{\operatorname{ad}\\!\big{(}\hbar^{-1}\phi_{1}\big{)}}^{n}\big{(}\,f_{(1)}\big{)}\otimes\phi_{2}^{\;n}\,\;=\;\,f_{(1)}\otimes
1\,+\,\big{[}\,\hbar^{-1}\phi_{1}\,,f_{(1)}\,\big{]}\otimes\phi_{2}\,+\,\mathcal{O}(2)\crcr}}}}$

that is in short

$\operatorname{Ad}(\mathcal{F}\,)\big{(}\,f_{(1)}\otimes
1\big{)}\,\;=\;\,f_{(1)}\otimes
1\,+\,\big{[}\,\hbar^{-1}\phi_{1}\,,f_{(1)}\,\big{]}\otimes\phi_{2}\,+\,\mathcal{O}(2)$

where hereafter $\,\mathcal{O}(2)\,$ denotes any element in
$\,J_{\hbar}^{[\otimes 2\,|3]}\,$. A similar calculation yields

$\operatorname{Ad}(\mathcal{F}\,)\big{(}\,1\otimes
f_{(2)}\big{)}\,\;=\;\,1\otimes
f_{(2)}\,+\,\phi_{1}\otimes\big{[}\,\hbar^{-1}\phi_{2}\,,f_{(2)}\,\big{]}\,+\,\mathcal{O}(2)$

Pasting together the last two identities we find

$\halign{\hbox to\displaywidth{$\hfil\displaystyle#\hfil$}\cr
0.0pt{\hfil$\displaystyle\Delta^{\mathcal{F}}(f)\,\;=\;\,\operatorname{Ad}(\mathcal{F}\,)\big{(}\,f_{(1)}\otimes
f_{(2)}\big{)}\,\;=\;\,\operatorname{Ad}(\mathcal{F}\,)\big{(}\,f_{(1)}\otimes
1\,\big{)}\cdot\operatorname{Ad}(\mathcal{F}\,)\big{(}\,1\otimes
f_{(2)}\big{)}\,\;=\hfill\cr
0.0pt{\hfil$\displaystyle=\,\big{(}\,f_{(1)}\otimes
1+\big{[}\,\hbar^{-1}\phi_{1}\,,f_{(1)}\,\big{]}\otimes\phi_{2}+\mathcal{O}(2)\big{)}\cdot\big{(}\,1\otimes
f_{(2)}+\phi_{1}\otimes\big{[}\,\hbar^{-1}\phi_{2}\,,f_{(2)}\,\big{]}+\mathcal{O}(2)\big{)}\,=\cr
0.0pt{\hfil$\displaystyle\quad=\;f_{(1)}\otimes
f_{(2)}\,+\,f_{(1)}\,\phi_{1}\otimes\big{[}\,\hbar^{-1}\phi_{2}\,,f_{(2)}\,\big{]}\,+\,\big{[}\,\hbar^{-1}\phi_{1}\,,f_{(1)}\,\big{]}\otimes\phi_{2}\,f_{(2)}\,+\,\mathcal{O}(2)\,\;=\hfill\cr
0.0pt{\hfil$\displaystyle\qquad=\;f_{(1)}\otimes
f_{(2)}\,+\,\epsilon\big{(}f_{(1)}\big{)}\,\phi_{1}\otimes\big{[}\,\hbar^{-1}\phi_{2}\,,f_{(2)}\,\big{]}\,+\,\big{[}\,\hbar^{-1}\phi_{1}\,,f_{(1)}\,\big{]}\otimes\phi_{2}\,\epsilon\big{(}f_{(2)}\big{)}\,+\,\mathcal{O}(2)\,\;=\hfill\cr
0.0pt{\hfil$\displaystyle\hfill=\;f_{(1)}\otimes
f_{(2)}\,+\,\phi_{1}\otimes\big{[}\,\hbar^{-1}\phi_{2}\,,f\,\big{]}\,+\,\big{[}\,\hbar^{-1}\phi_{1}\,,f\,\big{]}\otimes\phi_{2}\,+\,\mathcal{O}(2)\,\;=\cr
0.0pt{\hfil$\displaystyle\hfill=\;\Delta(f)\,+\,\phi_{1}\otimes\big{[}\,\hbar^{-1}\phi_{2}\,,f\,\big{]}\,+\,\big{[}\,\hbar^{-1}\phi_{1}\,,f\,\big{]}\otimes\phi_{2}\,+\,\mathcal{O}(2)\crcr}}}}}}}$

so that, eventually, we get in short

$\Delta^{\mathcal{F}}(f)\,\;\underset{J_{\hbar}^{[\otimes
2\,|3]}}{\equiv}\;\,\Delta(f)\,+\,\phi_{1}\otimes\big{[}\,\hbar^{-1}\phi_{2}\,,f\,\big{]}\,+\,\big{[}\,\hbar^{-1}\phi_{1}\,,f\,\big{]}\otimes\phi_{2}$

Therefore, for
$\,\nabla_{\\!\mathcal{F}}:=\Delta^{\mathcal{F}}-{\big{(}\Delta^{\mathcal{F}}\big{)}}^{2{}1}\,$
we get

$\displaystyle\nabla_{\\!\mathcal{F}}(f)$
$\displaystyle\,\;\underset{J_{\hbar}^{[\otimes
2\,|3]}}{\equiv}\;\,\Delta(f)\,+\,\phi_{1}\otimes\big{[}\,\hbar^{-1}\phi_{2}\,,f\,\big{]}\,+\,\big{[}\,\hbar^{-1}\phi_{1}\,,f\,\big{]}\otimes\phi_{2}\,-$
(3.62) $\displaystyle\hskip
21.0pt-\,\Delta^{\text{op}}\\!(f)\,-\,\big{[}\,\hbar^{-1}\phi_{2}\,,f\,\big{]}\otimes\phi_{1}\,-\,\phi_{2}\otimes\Big{[}\,\hbar^{-1}\phi_{1}\,,f\,\Big{]}\,\;=$
$\displaystyle\hskip
33.0pt=\;\,\nabla(f)\,+\,\phi_{1}^{(a)}\otimes\hbar^{-1}\Big{[}\,\phi_{2}^{(a)},\,f\,\Big{]}\,+\,\hbar^{-1}\Big{[}\,\phi_{1}^{(a)},\,f\,\Big{]}\otimes\phi_{2}^{(a)}\;=$
$\displaystyle\hskip
33.0pt=\;\,\nabla(f)\,-\,\phi_{1}^{(a)}\otimes\hbar^{-1}\Big{[}\,f\,,\,\phi_{2}^{(a)}\Big{]}\,-\,\hbar^{-1}\Big{[}\,f\,,\,\phi_{1}^{(a)}\Big{]}\otimes\phi_{2}^{(a)}$

where we used notation
$\;\phi_{a}:=\phi-\phi_{\,2{}1}=\phi_{1}^{(a)}\otimes\phi_{2}^{(a)}\;$. When
we reduce the last identity in (3.62) modulo $\,\hbar\,J_{\hbar}^{\otimes
2}\,$ we end up with

$\nabla_{\\!\mathcal{F}}\big{(}\,\overline{f}\,\big{)}\,\underset{{\mathfrak{m}}^{[\otimes
2\,|3]}}{\equiv}\,\nabla\big{(}\,\overline{f}\,\big{)}\,-\,\overline{\phi_{1}^{(a)}}\otimes\Big{\\{}\,\overline{f}\,,\,\overline{\phi_{2}^{(a)}}\,\Big{\\}}\,-\,\Big{\\{}\,\overline{f}\,,\,\overline{\phi_{1}^{(a)}}\,\Big{\\}}\otimes\overline{\phi_{2}^{(a)}}$

hence reducing the latter modulo ${\mathfrak{m}}^{[\otimes 2\,|3]}$ we find in
$\;{\mathfrak{m}}^{\otimes 2}\\!\Big{/}{\mathfrak{m}}^{[\otimes
3]}={\mathfrak{g}}^{*}\otimes{\mathfrak{g}}^{*}\,$ the identity

$\halign{\hbox to\displaywidth{$\hfil\displaystyle#\hfil$}\cr
0.0pt{\hfil$\displaystyle\quad\Big{(}\,\nabla_{\\!\mathcal{F}}\text{\ mod\
}{\mathfrak{m}}^{[\otimes
2\,|3]}\Big{)}(\,\text{f}\,)\,\;=\;\,\delta(\,\text{f}\,)\,-\,c_{\,1}\otimes\big{[}\,\text{f}\,,\,c_{\,2}\big{]}\,-\,\big{[}\,\text{f}\,,\,c_{\,1}\big{]}\otimes
c_{\,2}\,\;=\hfill\cr
0.0pt{\hfil$\displaystyle\hfill=\;\,\delta(\,\text{f}\,)\,-\,\big{(}\,\operatorname{ad}(\,\text{f}\,)\big{)}(c)\,\;=\;\,\big{(}\,\delta-\partial_{c}\big{)}(\,\text{f}\,)\,\;=\;\,\delta^{\,c}(\,\text{f}\,)\quad\crcr}}}$

which means that the induced Lie cobracket on
$\,{\mathfrak{m}}\Big{/}{\mathfrak{m}}^{2}={\mathfrak{g}}^{*}\,$ is just
$\delta^{\,c}$, q.e.d. ∎

######  Example 3.4.7.

Let $\,G:=\textit{GL}_{\,n}(\Bbbk)\,$ be the general linear group over
$\Bbbk\,$, and $\,{\mathfrak{g}}:=\mathfrak{gl}_{n}(\Bbbk)\,$. We consider
again the QUEA $\,{U_{\hbar}(\hskip
0.5pt{\mathfrak{g}})}=U_{\hbar}\big{(}\,\mathfrak{gl}_{n}(\Bbbk)\hskip-1.0pt\big{)}\,$
and the QFSHA $\,{F_{\hbar}[[G\hskip
1.5pt]]}=F_{\hbar}\big{[}\big{[}\textit{GL}_{\,n}(\Bbbk)\big{]}\big{]}\,$
introduced in Example 3.2.3. Letting ${\mathfrak{b}}^{-}$ and
${\mathfrak{b}}^{+}$ be the Borel Lie subalgebras in ${\mathfrak{g}}$ of lower
triangular and upper triangular matrices, respectively, the subalgebra
${U_{\hbar}(\hskip 0.5pt{\mathfrak{b}}^{-})}$ of ${U_{\hbar}(\hskip
0.5pt{\mathfrak{g}})}$ generated by the $F_{i}$’s and the $\varGamma_{k}$’s is
a QUEA for ${\mathfrak{b}}^{-}\,$, while the subalgebra ${U_{\hbar}(\hskip
0.5pt{\mathfrak{b}}^{+})}$ generated by the $E_{i}$’s and the
$\varGamma_{k}$’s is a QUEA for ${\mathfrak{b}}^{+}\,$ — both being also Hopf
subalgebras of ${U_{\hbar}(\hskip 0.5pt{\mathfrak{g}})}\,$, indeed. Dually,
the QFSHA $\,{F_{\hbar}[[B^{-}\hskip 1.5pt]]}={{U_{\hbar}(\hskip
0.5pt{\mathfrak{b}}^{-})}}^{*}\,$ identifies with the Hopf algebra quotient of
${F_{\hbar}[[G\hskip 1.5pt]]}$ obtained by modding out the ideal generated by
the $x_{i,\,j}$’s with $\,i<j\,$; similarly, the QFSHA
$\,{F_{\hbar}[[B^{+}\hskip 1.5pt]]}={{U_{\hbar}(\hskip
0.5pt{\mathfrak{b}}^{+})}}^{*}\,$ identifies with the Hopf algebra quotient of
${F_{\hbar}[[G\hskip 1.5pt]]}$ obtained by modding out the ideal generated by
the $x_{i,\,j}$’s with $\,i>j\,$. Therefore, from the presentation of
${F_{\hbar}[[G\hskip 1.5pt]]}$ in Example 3.2.3 one deduces the following
presentations for these quotient opf algebras: ${F_{\hbar}[[B^{-}\hskip
1.5pt]]}$ is generated by the entries of the “lower triangular $q$–matrix”
$\,{\big{(}x^{-}_{i,\,j}\big{)}}_{i=1,\dots,n;}^{j=1,\dots,n;}\,$ with
$\,x^{-}_{i,\,j}:=x_{i,\,j}\,$ for all $\,i\geq j\,$ and
$\,x^{-}_{i,\,j}:=0\,$ for all $\,i<j\,$, and similarly
${F_{\hbar}[[B^{+}\hskip 1.5pt]]}$ is generated by the entries of the “upper
triangular $q$–matrix”
$\,{\big{(}x^{+}_{i,\,j}\big{)}}_{i=1,\dots,n;}^{j=1,\dots,n;}\,$ where
$\,x^{+}_{i,\,j}:=x_{i,\,j}\,$ when $\,i\leq j\,$ and $\,x^{+}_{i,\,j}:=0\,$
for $\,i>j\,$. Now we consider a new group $\,G\,$ which is “double version”
of $\textit{GL}_{\,n}(\Bbbk)\,$, in that it is a Manin double of $B^{-}$ and