# Twisted unipotent groups

Ken A. Brown  School of Mathematics and Statistics, University of Gasgow,
Gasgow G12 8QQ, United Kingdom<EMAIL_ADDRESS>and  Shlomo Gelaki
Department of Mathematics, Iowa State University, Ames, IA 50100, USA.
<EMAIL_ADDRESS>
###### Abstract.

We study the algebraic structure and representation theory of the Hopf
algebras ${}_{J}\mathcal{O}(G)_{J}$ when $G$ is an affine algebraic unipotent
group over $\mathbb{C}$ with $\mathrm{dim}(G)=n$ and $J$ is a Hopf $2$-cocycle
for $G$. The cotriangular Hopf algebras ${}_{J}\mathcal{O}(G)_{J}$ have the
same coalgebra structure as $\mathcal{O}(G)$ but a deformed multiplication. We
show that they are involutive $n$-step iterated Hopf Ore extensions of
derivation type. The 2-cocycle $J$ has as support a closed subgroup $T$ of
$G$, and ${}_{J}\mathcal{O}(G)_{J}$ is a crossed product
$S\\#_{\sigma}U(\mathfrak{t})$, where $\mathfrak{t}$ is the Lie algebra of $T$
and $S$ is a deformed coideal subalgebra. The simple
${}_{J}\mathcal{O}(G)_{J}$-modules are stratified by a family of factor
algebras ${}_{J}\mathcal{O}(Z_{g})_{J}$, parametrised by the double cosets
$TgT$ of $T$ in $G$. The finite dimensional simple
${}_{J}\mathcal{O}(G)_{J}$-modules are all 1-dimensional, so form a group
$\Gamma$, which we prove to be an explicitly determined closed subgroup of
$G$. A selection of examples illustrate our results.

###### Contents

  1. 1 Introduction
    1. 1.1 Hopf algebraic structure
    2. 1.2 Representation theory
    3. 1.3 Special cases and examples
    4. 1.4 Notation
  2. 2 Preliminaries
    1. 2.1 Hopf $2$-cocycles
    2. 2.2 Cotriangular Hopf algebras
    3. 2.3 Unipotent groups
    4. 2.4 Hopf $2$-cocycles for unipotent groups
  3. 3 IHOE structure of ${}_{J}\mathcal{O}(G)_{J}$ for $G$ unipotent
    1. 3.1 ${}_{J}\mathcal{O}(G)_{J}$ is an IHOE
    2. 3.2 Immediate consequences of IHOE structure
  4. 4 Structure of twisted unipotent groups - the minimal case
    1. 4.1 The minimal case - inclusion of $\mathrm{im}(\Psi)$
    2. 4.2 The minimal case - surjectivity of $\Psi$
  5. 5 Structure of twisted unipotent groups - the general case
    1. 5.1 Crossed product decomposition
    2. 5.2 Consequences of Theorem 5.2
    3. 5.3 Stratification of representations by double cosets
    4. 5.4 On the structure of the algebras ${}_{J}\mathcal{O}(Z_{g})_{J}$
  6. 6 Finite dimensional simple ${}_{J}\mathcal{O}(G)_{J}$-modules
    1. 6.1 $\Gamma$ is a subgroup of $G$
    2. 6.2 The determination of $\Gamma$
  7. 7 Special cases and examples
    1. 7.1 Abelian support
    2. 7.2 Examples
  8. 8 Acknowledgements

##  1\. Introduction

Let $G$ be an affine algebraic group over $\mathbb{C}$, and let
$\mathcal{O}(G)$ be its algebra of polynomial functions, an affine commutative
Hopf algebra. Let $J\in\left(\mathcal{O}(G)\otimes\mathcal{O}(G)\right)^{*}$
be a Hopf $2$-cocycle for $G$ (see Definition 2.1). The twisting procedure of
Drinfeld [Dr1] yields a new cotriangular Hopf algebra structure on the
underlying coalgebra of $\mathcal{O}(G)$ by using $J$ to deform the
multiplication, and change the $R$-form $\epsilon\otimes\epsilon$ of
$\mathcal{O}(G)$ to $J_{21}^{-1}\ast J$. See $\S$2.1 for the twisting
procedure, and $\S$2.2 for background on $R$-forms and cotriangular
structures. Let $\mathrm{Rep}(G)$ denote the category of finite dimensional
complex rational representations of $G$. Then Hopf $2$-cocycles for $G$ yield
tensor structures on the forgetful functor
$\mathrm{Rep}(G)\rightarrow\mathrm{Vec}$, as explained for example in [EGNO,
Chapter 5].

The purpose of this paper is to study the algebraic structure and
representation theory of the Hopf algebras ${}_{J}\mathcal{O}(G)_{J}$ when the
group $G$ is unipotent. In particular, this paper extends (and corrects some
of) the results of [G3]. It thus forms part of a sequence of papers which
began with Movshev’s work [Mov] on twisting for finite groups, and continued
with [EG1, EG2, EG3, EG4, EG5, EG6] of Etingof and the second author, and [G1,
G2, G3] of the second author. As is recalled in detail in $\S$2.4, by results
due to Drinfeld, Etingof and the second author, the equivalence classes of
Hopf $2$-cocycles for a unipotent group $G$ with Lie algebra $\mathfrak{g}$
are in bijection with equivalence classes of solutions to the classical Yang-
Baxter equation in $\wedge^{2}\mathfrak{g}$, and to equivalence classes of
pairs $(\mathfrak{t},\omega)$, where $\mathfrak{t}\subseteq\mathfrak{g}$ is a
quasi-Frobenius Lie subalgebra with nondegenerate $2$-cocycle
$\omega\in\wedge^{2}\mathfrak{t}^{*}$.

###  1.1. Hopf algebraic structure

The following result summarises the most basic structural features of the
deformations ${}_{J}\mathcal{O}(G)_{J}$ for $G$ unipotent. Early versions of
parts (1), (2) and (4) were obtained in [G3]. For (6) recall that an ideal $I$
of a ring $A$ is _polycentral_ if it contains elements $y_{1},\ldots,y_{t}$
such that $I=\sum_{j}y_{j}A$ with $y_{1}$ central in $A$ and $y_{j}$ central
_modulo_ $\sum_{i<j}y_{i}A$ for $j=2,\ldots,t$.

######  Theorem 1.1.

(Theorems 3.2, 5.5, Corollaries 3.3, 3.5) Let $G$ be a unipotent algebraic
group over $\mathbb{C}$ of dimension $n$, and let $J$ be a Hopf $2$-cocycle
for $G$.

  1. (1)

${}_{J}\mathcal{O}(G)_{J}$ is an $n$-step iterated Hopf Ore extension of
derivation type. That is, there is a sequence

$\mathbb{C}=H_{0}\subset\cdots\subset{}_{J}(H_{i})_{J}\subset\cdots\subset{}_{J}(H_{n})_{J}={}_{J}\mathcal{O}(G)_{J}$

of Hopf subalgebras ${}_{J}(H_{i})_{J}\subseteq{}_{J}\mathcal{O}(G)_{J}$, and
derivations $\partial_{i}$ of ${}_{J}(H_{i})_{J}$, such that
${}_{J}(H_{i+1})_{J}={}_{J}(H_{i})_{J}[X_{i+1};\partial_{i}]$ is an Ore
extension for all $0\leq i\leq n-1$.

  2. (2)

The formulae for the derivations $\partial_{i}$ can be explicitly stated in
terms of the Hopf $2$-cocycle $J$ and the comultiplication $\Delta$ of
$\mathcal{O}(G)$.

  3. (3)

The associated graded algebra of ${}_{J}\mathcal{O}(G)_{J}$ with respect to
its coradical filtration is the coordinate ring $\mathcal{O}(\widehat{G})$,
the associated graded ring of $\mathcal{O}(G)$, $\widehat{G}$ being the
associated graded unipotent group of $G$.

  4. (4)

${}_{J}\mathcal{O}(G)_{J}$ is a noetherian domain of Gelfand-Kirillov
dimension $n$ with excellent homological properties.

  5. (5)

Every prime ideal $P$ of ${}_{J}\mathcal{O}(G)_{J}$ is completely prime, i.e.,
${}_{J}\mathcal{O}(G)_{J}/P$ is a domain.

  6. (6)

The augmentation ideal $({}_{J}\mathcal{O}(G)_{J})^{+}$ of
${}_{J}\mathcal{O}(G)_{J}$ is polycentral. ∎

Given a cotriangular Hopf algebra $(H,R)$, we say that the $R$-form $R$ is
_nondegenerate_ , and that $(H,R)$ is _minimal_ , if the _radical_

$I_{R}:=\\{a\in H\mid R(a,b)=0,\,\forall b\in H\\}$

is $\\{0\\}$. The radical is a Hopf ideal of $H$, and there is an induced
structure of cotriangular Hopf algebra $(H/I,R)$, on which $R$ is
nondegenerate, so that $(H/I,R)$ is minimal [G2, §2.2]; see $\S$2.2 and Lemma
2.8 for details.

Specialise now to the case where $H$ is the coordinate ring $\mathcal{O}(G)$
of an affine algebraic group $G$, so that, as already mentioned above,
$(\mathcal{O}(G),\epsilon\otimes\epsilon)$ is cotriangular. Let $J$ be a Hopf
$2$-cocycle for $G$, so that, as also noted above,
$({}_{J}\mathcal{O}(G)_{J},R^{J})$ is cotriangular with
$R^{J}:=J_{21}^{-1}\ast J$. Extending the terminology introduced above, we say
that $J$ is _minimal_ if $({}_{J}\mathcal{O}(G)_{J},R^{J})$ is minimal. We can
now appeal to the following result (where, for _gauge equivalence_ , see
Remark 2.3(3)):

######  Proposition 1.2.

([G2, Theorem 3.1]) Let $G$ be an affine algebraic group, and let $J$ be a
Hopf $2$-cocycle for $G$. Then there is a closed subgroup $T\subseteq G$ and a
minimal Hopf $2$-cocycle $\widehat{J}$ for $T$, such that $J$ is gauge
equivalent to $\widehat{J}$ (where $\widehat{J}$ is viewed in the obvious way
as a Hopf $2$-cocycle for $G$). Hence, there is an epimorphism of cotriangular
Hopf algebras from $({}_{J}\mathcal{O}(G)_{J},R^{J})$ to
$({}_{\widehat{J}}\mathcal{O}(T)_{\widehat{J}},R^{\widehat{J}})$ with kernel
$I_{R^{J}}$, the radical of $R^{J}$. ∎

Here, the closed subgroup $T\subseteq G$, which is uniquely determined by $J$
up to conjugation by $G$, is called the _support_ of $J$.

Assume once again that $G$ is unipotent, and let $J$ be a Hopf $2$-cocycle for
$G$ with support $T$, with $R^{J}=J_{21}^{-1}\ast J$ the corresponding
$R$-form for ${}_{J}\mathcal{O}(G)_{J}$. There is an algebra homomorphism
$\Psi$ from ${}_{J}\mathcal{O}(G)_{J}$ to the vector space dual
$\mathcal{O}(G)^{\ast}$, given by

$\Psi(h):=R^{J}(-,\,h),$

as shown in Lemma 2.10. The kernel of $\Psi$ is the radical $I_{R^{J}}$ of
$R^{J}$, so one sees that $\mathrm{im}(\Psi)$ is a Hopf subalgebra of
$({}_{J}\mathcal{O}(T)_{J})^{\circ,\mathrm{op}}$. In fact, $\mathrm{im}(\Psi)$
is also contained in the finite dual $\mathcal{O}(T)^{\circ}$ of
$\mathcal{O}(T)$, which by the Cartier-Gabriel-Kostant theorem is isomorphic
to $U(\mathfrak{t})\\#\mathbb{C}[T]$, where $\mathfrak{t}$ denotes the Lie
algebra of $T$, and we prove that $\mathrm{im}(\Psi)=U(\mathfrak{t})$ (as
predicted in [G3, Theorem 3.4]; see Remark 4.10). These results are
Proposition 4.1 and Theorem 4.4, which together yield Theorem 4.9 and [G3,
Lemma 3.7, Theorems 3.5, 3.8], summarised as follows. Here, and throughout,
for a subspace $V$ of a Hopf algebra, $V^{+}$ denotes the kernel of
$\epsilon_{\mid V}$.

######  Theorem 1.3.

Let $J$ be a Hopf $2$-cocycle for a unipotent affine algebraic group $G$, let
$T$ be (a representative of) the support of $J$, and let $\mathfrak{t}$ be the
Lie algebra of $T$. Denote the antipode of ${}_{J}\mathcal{O}(G)_{J}$ by
$S^{J}$.

  1. (1)

(Theorem 5.2) ${}_{J}\mathcal{O}(G)_{J}$ decomposes as a crossed product
algebra,

${}_{J}\mathcal{O}(G)_{J}\cong({}_{J}\mathcal{O}(G/T)_{J})\\#_{\sigma}{}_{J}\mathcal{O}(T)_{J}\cong({}_{J}\mathcal{O}(G/T)_{J})\\#_{\sigma}U(\mathfrak{t})^{J},$

where ${}_{J}\mathcal{O}(G/T)_{J}={}_{J}\mathcal{O}(G/T)$ is the twist of the
left coideal subalgebra $\mathcal{O}(G/T)$ consisiting of the functions in
$\mathcal{O}(G)$ which are constant on each left coset of $T$ in $G$, where
$\sigma$ is an invertible $2$-cocycle from ${}_{J}\mathcal{O}(T)_{J}$ to
${}_{J}\mathcal{O}(G/T)_{J}$, and where $U(\mathfrak{t})^{J}$ is the
enveloping algebra $U(\mathfrak{t})$ with a twisted comultiplication.

  2. (2)

(Lemma 5.1)
$I_{R^{J}}=({}_{J}\mathcal{O}(G/T)_{J})^{+}{}_{J}\mathcal{O}(G)_{J}=({}_{J}\mathcal{O}(T\backslash
G)_{J})^{+}{}_{J}\mathcal{O}(G)_{J}.$

  3. (3)

(Proposition 5.4) The subalgebra $\mathcal{O}(T\backslash G/T)$ of
$\mathcal{O}(G)$ consisting of functions which are constant on double cosets
of $T$ in $G$ is contained in the centre of ${}_{J}\mathcal{O}(G)_{J}$.

  4. (4)

(Corollary 5.6) $(S^{J})^{2}=\mathrm{id}$, and ${}_{J}\mathcal{O}(G)_{J}$ is
Calabi-Yau. ∎

###  1.2. Representation theory

With regard to representation theory, our focus in this paper is on the simple
${}_{J}\mathcal{O}(G)_{J}$-modules. In view of Theorem 1.3(3), Schur’s lemma
shows that each simple ${}_{J}\mathcal{O}(G)_{J}$-module is annihilated by
$\mathfrak{m}{}_{J}\mathcal{O}(G)_{J}$ for a unique maximal ideal
$\mathfrak{m}$ of $\mathcal{O}(T\backslash G/T)$. Building on this observation
we have the following stratification result, which one can view as
generalising the fact that the simple $\mathcal{O}(G)$-modules are
parametrised by the elements of $G$.

######  Theorem 1.4.

([G3, §4, Proposition 4.1, Corollary 4.2, Theorem 5.1]) Let $G$ be a unipotent
affine algebraic group, and let $J$ be a Hopf $2$-cocycle for $G$ with support
$T$. Let $g\in G$, and let $Z_{g}:=TgT$ be the double coset of $T$ in $G$.

  1. (1)

(Lemma 5.8(1)) $Z_{g}$ is a closed irreducible subset of $G$ with

$\mathrm{dim}(Z_{g})=2\mathrm{dim}(T)-\mathrm{dim}(T\cap gTg^{-1}).$

  2. (2)

(Proposition 5.9) The defining ideal $\mathcal{I}(Z_{g})$ of $Z_{g}$ in
$\mathcal{O}(G)$ is also an ideal of ${}_{J}\mathcal{O}(G)_{J}$.

  3. (3)

(Corollary 5.11) Define
${}_{J}\mathcal{O}(Z_{g})_{J}:={}_{J}\mathcal{O}(G)_{J}/\mathcal{I}(Z_{g})$.
Then ${}_{J}\mathcal{O}(Z_{g})_{J}$ is an affine noetherian domain with

$\mathrm{GKdim}({}_{J}\mathcal{O}(Z_{g})_{J})=2\mathrm{dim}(T)-\mathrm{dim}(T\cap
gTg^{-1}).$

  4. (4)

(Theorem 5.12) Each simple ${}_{J}\mathcal{O}(G)_{J}$-module factors through a
unique quotient ${}_{J}\mathcal{O}(Z_{g})_{J}$. ∎

In view of Theorem 1.4(4) it is natural to explore the representation theory
and algebra structure of the algebras ${}_{J}\mathcal{O}(Z_{g})_{J}$ for $g\in
G$. There is a dichotomy according to whether or not $g$ belongs to the
normaliser $N:=N_{G}(T)$ of $T$. We first consider the strata $Z_{g}$ with
$g\notin N$ \- here there is incomplete information at present except when $g$
is “very far from normalising”, (see (2)):

######  Theorem 1.5.

Keep the notation of Theorem 1.4, let $N:=N_{G}(T)$, and let $g\in G$.

  1. (1)

(Theorem 6.8) If $g\notin N$ then ${}_{J}\mathcal{O}(Z_{g})_{J}$ has no finite
dimensional modules.

  2. (2)

([G3, Remark 4.4], Theorem 5.13) If $T\cap gTg^{-1}=\\{1\\}$ then
${}_{J}\mathcal{O}(Z_{g})_{J}$ is isomorphic to the Weyl algebra
$A_{\mathrm{dim}(T)}(\mathbb{C})$. ∎

Thus finite dimensional simple modules can only occur in the strata $Z_{g}$
where $g\in N$. We consider now these cases. As an immediate corollary of
Theorem 1.1(5), when $G$ is unipotent every finite dimensional simple
${}_{J}\mathcal{O}(G)_{J}$-module is $1$-dimensional. The set of such modules
therefore forms an algebraic group $\Gamma$, whose algebra of regular
functions is the abelianisation
${}_{J}\mathcal{O}(G)_{J}/\langle[{}_{J}\mathcal{O}(G)_{J},{}_{J}\mathcal{O}(G)_{J}]\rangle$
of ${}_{J}\mathcal{O}(G)_{J}$. The following theorem summarises what we know
about the group $\Gamma$. (In particular, it corrects [G3, Theorem 5.2].) Note
first, as explained in Remark 2.3(3), that the group
$(\mathcal{O}(G)^{*})^{\times}_{1}$ of convolution invertible elements of
$\mathcal{O}(G)^{*}$ which map $1$ to $1$ acts on the set of Hopf $2$-cocycles
for $G$. Denote this action by $J\mapsto J^{\chi}$ for
$\chi\in(\mathcal{O}(G)^{*})^{\times}_{1}$. Of course, $G$ itself, and also
the group $\Gamma$, are subgroups of $(\mathcal{O}(G)^{*})^{\times}_{1}$.

######  Theorem 1.6.

(Theorem 6.8) Keep the notation $G$, $J$, $T$ and $N$ of Theorem 1.5, and let
$F$ and $\Gamma$ be the groups of finite dimensional simple
${}_{J}\mathcal{O}(T)_{J}$ and ${}_{J}\mathcal{O}(G)_{J}$-modules,
respectively. Let $[J]$ denote the orbit of $J$ under the action of the
subgroup $(\mathcal{O}(T)^{*})^{\times}_{1}$ of
$(\mathcal{O}(G)^{*})^{\times}_{1}$. Define

$C_{0}:=\\{g\in N\mid J^{g}=J\\}\,\,\,{\rm and}\,\,\,C:=\\{g\in
N\mid[J^{g}]=[J]\\},$

so that $C_{0}\subseteq C$, $T\triangleleft C$, and hence $C_{0}T\subseteq C$.

  1. (1)

$\Gamma=C_{0},$ and $C_{0}=\\{g\in G\mid J^{g}=J\\}$.

  2. (2)

$C=C_{0}T.$

  3. (3)

$F=C_{0}\cap T\subseteq T$ is a closed abelian subgroup of $T$, normal in
$C_{0}$, with

$\mathrm{dim}(F)=\mathrm{dim}(T/[T,T]).$

  4. (4)

$C_{0}/F\cong C/T$, so that there is an exact sequence of unipotent groups

$1\rightarrow F\rightarrow\Gamma\rightarrow C/T\rightarrow 1.$

  5. (5)

$\mathrm{dim}(\Gamma)=\mathrm{dim}(C)-\mathrm{dim}([T,T]).$

  6. (6)

The algebra ${}_{J}\mathcal{O}(Z_{g})_{J}$ has nonzero finite dimensional
representations if and only if $g\in C$. The distinct strata admitting nonzero
finite dimensional modules are $\\{{}_{J}\mathcal{O}(Z_{g})_{J}\mid
g\in\mathcal{C}\\},$ where $\mathcal{C}$ is a set of coset representatives for
$F=C_{0}\cap T$ in $C_{0}$, or equivalently for $T$ in $C$.

  7. (7)

For any $g\in\mathcal{C}$, the set of finite dimensional simple
${}_{J}\mathcal{O}(Z_{g})_{J}$-modules is parametrized by $gF=g(C_{0}\cap T)$.
Specifically, the corresponding functionals have kernels the maximal ideals
$\\{\mathfrak{m}_{gx}\mid x\in C_{0}\cap T\\}$ of $\mathcal{O}(G)$. ∎

###  1.3. Special cases and examples

In $\S$7 we consider the special case when the support $T$ of $J$ is abelian.
Finally we expand [G3, Examples 6.1-6.5] to illustrate our results.

###  1.4. Notation

We work throughout over the complex numbers $\mathbb{C}$, although many of our
results are valid over an arbitrary algebraically closed field of
characteristic $0$. All unadorned tensor products are assumed to be over
$\mathbb{C}$. The notation $(H,m_{H},1_{H},\epsilon_{H},\Delta_{H},S_{H})$
will always denote a Hopf $\mathbb{C}$-algebra and its standard operations,
although we omit the suffix $H$ whenever possible. The augmentation ideal of
$H$ is denoted by $H^{+}$, extending this usage by writing $A^{+}$ for
$H^{+}\cap A$ when $A\subset H$ is a subalgebra. We denote the finite dual of
$H$ by $H^{\circ}$ [Mon, $\S$9.1], and use $\ast$ to denote convolution
product.

##  2\. Preliminaries

###  2.1. Hopf $2$-cocycles

Let $H$ be a Hopf $\mathbb{C}$-algebra.

######  Definition 2.1.

(Doi [Do], Drinfeld [Dr2], Majid [Maj, $\S\S$2.1,2.2,2.3]) A _Hopf
$2$-cocycle_ for $H$ is a convolution-invertible bilinear map $J:H\otimes
H\rightarrow\mathbb{C}$ such that

(2.1) $\sum J(a_{1}b_{1},c)J(a_{2},b_{2})=\sum J(a,b_{1}c_{1})J(b_{2},c_{2})$

and

(2.2) $J(a,1)=\epsilon(a)=J(1,a),$

for all $a,b,c\in H$. If $G$ is an affine algebraic group and
$H:=\mathcal{O}(G)$ is the algebra of polynomial functions, then $J$ is often
referred to as a _Hopf $2$-cocycle for $G$_. ∎

Let $J$ and $K$ be two Hopf $2$-cocycles for $H$. Then one can construct a new
algebra ${}_{K}H_{J}$ as follows: as vector spaces, ${}_{K}H_{J}=H$, but the
new multiplication ${}_{K}m_{J}$ is given for $a,b\in H$ by

(2.3) $_{K}m_{J}(a\otimes b)=\sum
K^{-1}(a_{1},b_{1})a_{2}b_{2}J(a_{3},b_{3}).$

In particular there are special cases when $K$ or $J$ is
$\epsilon\otimes\epsilon$, denoted respectively by $H_{J}$ and by ${}_{K}H$,
with respective multiplications

(2.4) $m_{J}(a\otimes b)=\sum a_{1}b_{1}J(a_{2},b_{2})\,\,\textit{ and
}\,\,_{K}m(a\otimes b)=\sum K^{-1}(a_{1},b_{1})a_{2}b_{2}.$

In the most important case $K=J$, ${}_{J}H_{J}$ is a Hopf algebra, with
coalgebra structure unchanged from $H$, but with multiplication

(2.5) ${}_{J}m_{J}(a\otimes b)=\sum
J^{-1}(a_{1},b_{1})a_{2}b_{2}J(a_{3},b_{3}),$

and antipode

(2.6) $S^{J}(a)=\sum J^{-1}(a_{1},S(a_{2}))S(a_{3})J(S(a_{4}),a_{5}),$

for all $a,b\in H$. To avoid needless clutter we will abbreviate the notation
for the multiplication in ${}_{J}H_{J}$ to $m_{J}$ from now on, when no
confusion seems likely. We record the following basic properties, which will
be needed later.

######  Lemma 2.2.

Let $H$, $K$ and $J$ be as above, and let $G\subset(H^{*})^{\times}$ be a
subgroup.

  1. (1)

${}_{K}H_{J}$ is a right and left $G$-module under the respective actions
defined by

(2.7) $f\cdot g=\sum g(f_{1})f_{2}\,\,\textit{ and }\,\,g\cdot f=\sum
f_{1}g(f_{2});\,\,\,f\in{}_{K}H_{J},\,g\in G.$

  2. (2)

Suppose that $H=\mathcal{O}(G)$ for a group $G$. Then the first of the actions
in (1) defines an action of $G$ by algebra automorphisms on
$\mathcal{O}(G)_{J}$, and the second yields algebra automorphisms of
${}_{J}\mathcal{O}(G)$.

  3. (3)

$\Delta$ gives an algebra embedding of ${}_{K}H_{J}$ into ${}_{K}H\otimes
H_{J}$.

###### Proof.

(1) This is clear.

(2) We check the first statement, the second being similar. Use $\cdot$ to
denote multiplication in $\mathcal{O}(G)_{J}$. As in $(1)$, use $g(-)$ to
denote evaluation at $g$, and $-\cdot g$ for the right $G$-module action on
the (vector space) $\mathcal{O}(G)$. For $f,h\in\mathcal{O}(G)$ and $g\in G$,

$\displaystyle(f\cdot h)\cdot g$ $\displaystyle=\left(\sum
f_{1}h_{1}J(f_{2},h_{2})\right)\cdot g$ $\displaystyle=\sum(f_{1}h_{1})\cdot
gJ(f_{2},h_{2})=\sum(f_{1}\cdot g)(h_{1}\cdot g)J(f_{2},h_{2})$
$\displaystyle=\sum(g(f_{1})f_{2}g(h_{1})h_{2})J(f_{3},h_{3})=(f\cdot
g)\cdot(h\cdot g).$

(3) This is [G3, Lemma 2.2]. ∎

######  Remarks 2.3.

(1) If $J$ is a Hopf $2$-cocycle for $H$ and $A$ is a Hopf subalgebra of $H$,
then $J_{|A\otimes A}$ is a Hopf $2$-cocycle for $A$.

(2) By (the dualised version of) a theorem of Schauenberg [Scha2] (see also
[EG5] for an alternative proof), for all finite dimensional Hopf algebras $H$
and $A$, we have $\mathrm{Corep}(H)\cong\mathrm{Corep}(A)$ as tensor
categories if and only if $H$ and $A$ are related by a Hopf $2$-cocycle. This
extends to all Hopf algebras, with basically the same proof.

(3) Let $J$ be a Hopf $2$-cocycle for a Hopf algebra $H$. So $J\in(H\otimes
H)^{\ast}$ and there is an action of the group $(H^{*})^{\times}$ of units of
$H^{\ast}$ on the set of Hopf $2$-cocycles for $H$: for
$\chi\in(H^{*})^{\times}$ there is a Hopf $2$-cocycle $J^{\chi}$ of $H$,
defined by

$J^{\chi}(a,b):=\sum\chi(a_{1}b_{1})J(a_{2},b_{2})\chi^{-1}(a_{3})\chi^{-1}(b_{3});\,\,\,a,b\in
H.$

Two Hopf $2$-cocycles for $H$ which are in the same orbit under
$(H^{*})^{\times}$ are said to be _gauge equivalent_. If $J_{1}$ and $J_{2}$
are gauge equivalent Hopf $2$-cocycles for the finite dimensional Hopf algebra
$H$, then ${}_{J_{1}}H_{J_{1}}\cong{}_{J_{2}}H_{J_{2}}$ as Hopf algebras, by
the dualised version of [EGNO, page 115]. The same proof applies in the
infinite dimensional setting.

(4) A Hopf $2$-cocycle $J$ for $H$ is called _invariant_ if ${}_{J}H_{J}=H$ as
Hopf algebras [G3, p.2] (i.e., isomorphic via the identity map). The set of
invariant Hopf $2$-cocycles for $H$ is a subgroup of $((H\otimes
H)^{*})^{\times}$, denoted $Z^{2}_{\mathrm{inv}}(H)$; in view of the previous
remark, $Z^{2}_{\mathrm{inv}}(H)$ is normalized by $(H^{*})^{\times}$.
Invariant Hopf $2$-cocycles are studied in, for example, [BC, EG4, Scha1]. ∎

###  2.2. Cotriangular Hopf algebras

A key point for us will be the preservation of the following property under
cocycle deformation.

######  Definition 2.4.

([EGNO, Definition 8.3.19]) A _coquasitriangular_ Hopf algebra is a pair
$(H,R)$, where $H$ is a Hopf $\mathbb{C}$-algebra and $R:H\otimes
H\rightarrow\mathbb{C}$ is a convolution invertible bilinear map such that

  1. (1)

$R(h,\ell g)=\sum R(h_{1},g)R(h_{2},\ell)\,\,\textit{ and }\,\,R(gh,\ell)=\sum
R(g,\ell_{1})R(h,\ell_{2});$

  2. (2)

$\sum R(h_{1},g_{1})h_{2}g_{2}=\sum g_{1}h_{1}R(h_{2},g_{2})$

for $g,h,\ell\in H$. Such a functional $R$ is called an _$R$ -form_ for $H$.

If in addition

$\textrm{(3)}\quad\sum R(h_{1},g_{1})R(g_{2},h_{2})=\epsilon(g)\epsilon(h),$

that is, if $R^{-1}=R_{21}$, then $(H,R)$ is called _cotriangular_. ∎

There is a dual concept of a _(quasi)triangular Hopf algebra_ $(H,R)$, defined
by Drinfeld [Dr1], [EGNO, Definition 8.3.1], where now $R$, the _universal
$R$-matrix_, is an invertible element of $H\otimes H$ satisfying certain
equations [EGNO, (8.8), page 198], stemming from the braid relations. A finite
dimensional Hopf algebra $(H,R)$ is easily seen to be (quasi)triangular if and
only if $(H^{*},R)$ is co(quasi)triangular [EGNO, page 201] (noting that $R\in
H\otimes H$ can be regarded as an element of $(H^{*}\otimes H^{*})^{*}$).

The following classes of examples are key for us:

######  Example 2.5.

(1)([EGNO, Example 8.3.4]) If $H$ is a commutative Hopf algebra, then
$(H,\epsilon\otimes\epsilon)$ is cotriangular. Equivalently, any cocommutative
Hopf algebra is triangular with universal $R$-matrix $1\otimes 1$.

(2)([EGNO, Remark 8.3.24]) Let $(H,R)$ be co(quasi)triangular, and let $J$ be
a Hopf $2$-cocycle for $H$. Define $R^{J}:=J_{21}^{-1}\ast R\ast J$. Then
$R^{J}$ is an $R$-form for ${}_{J}H_{J}$, and we say that $R^{J}$ is obtained
from $R$ by _twisting by_ $J$. Thus, $({}_{J}H_{J},R^{J})$ is a
co(quasi)triangular Hopf algebra. ∎

The following properties of the $R$-forms obtained as in Example 2.5(2) will
be needed later.

######  Lemma 2.6.

Let $G$ be an affine algebraic group, so
$(\mathcal{O}(G),\epsilon\otimes\epsilon)$ is cotriangular. Let $J$ be a Hopf
$2$-cocycle for $G$, and define $R^{J}:=J_{21}^{-1}\ast J$. Then $R^{J}$ is an
$R$-form for ${}_{J}\mathcal{O}(G)_{J}$ by Example 2.5(2).

  1. (1)

For all $\beta\in\mathcal{O}(G)^{+}$, $R^{J}(\beta,1)=R^{J}(1,\beta)=0$.

  2. (2)

Let $p\in\mathcal{O}(G)$ be primitive. Then for all $\alpha\in\mathcal{O}(G)$,

$R^{J}(p,\alpha)=(J-J_{21})(p,\alpha)=(J_{21}^{-1}-J^{-1})(p,\alpha).$

###### Proof.

(1) Let $\beta\in\mathcal{O}(G)^{+}$. Then, by definition,

$R^{J}(\beta,1)=\sum J_{21}^{-1}(\beta_{1},1)J(\beta_{2},1),$

and this equals 0 by equation (2.2) in Definition 2.1, since in every summand
above, at least one of $\beta_{1},\beta_{2}$ is in $\mathcal{O}(G)^{+}$.

(2) This is [G3, Lemma 2.3]. ∎

Notice that we can reformulate Definition 2.4(2) as $m^{\mathrm{op}}=R\ast
m\ast R^{-1}$, thus exposing the relevance of $R$-forms for $H$ to the
existence of braidings on the category $\mathrm{Comod}(H)$ of right
$H$-comodules. Namely, recall the following, where (2) is the dual of [EGNO,
Proposition 8.3.14].

######  Theorem 2.7.

([EGNO, p. 200–202]) Let $H$ be a Hopf algebra, and let $\mathcal{C}$ be the
tensor category of finite dimensional right $H$-comodules.

  1. (1)

If $R$ is an $R$-form yielding a coquasitriangular structure on $H$, then the
category $\mathcal{C}$ is a braided category, with braiding

(2.8) $c_{X,Y}:=\tau\circ R^{24}\circ(\rho_{X}\otimes\rho_{Y}),$

where $X,Y\in\mathcal{C}$ with coactions $\rho_{X},\rho_{Y}$, respectively.
Conversely, if $\mathcal{C}$ is braided, then this induces a natural
coquasitriangular structure on $H$.

  2. (2)

If $(H,R)$ is a coquasitriangular Hopf algebra and $J$ is a Hopf $2$-cocycle
for $H$, then the categories of comodules over $(H,R)$ and
$({}_{J}H_{J},R^{J})$ are naturally equivalent as braided categories. There is
thus a bijection between braided structures on $\mathcal{C}$ and
coquasitriangular structures on $H$ up to twisting by a Hopf $2$-cocycle; this
bijection restricts to a bijection between symmetric structures on
$\mathcal{C}$ and cotriangular structures on $H$, up to twisting by a Hopf
$2$-cocycle. ∎

The following lemma will be important in analyzing the structure of
cotriangular Hopf algebras.

######  Lemma 2.8.

([G2, Proposition 2.1]) Let $(H,R)$ be a cotriangular Hopf algebra, and let
$I_{R}:=\\{a\in H\mid R(b,a)=0,\,\forall b\in H\\}$ be the right radical of
$R$. Then $I_{R}$ is also the left radical (with the obvious definition), and
it is a Hopf ideal of $H$. ∎

######  Definition 2.9.

A cotriangular Hopf algebra $(H,R)$ whose radical $I_{R}$ is $\\{0\\}$ is
called _minimal_ , and we say in this case that $R$ is _nondegenerate_. ∎

Clearly, by Lemma 2.8, a cotriangular Hopf algebra $(H,R)$ has a unique
minimal cotriangular Hopf quotient, which is denoted by
$(H_{\mathrm{min}},R)$.

######  Lemma 2.10.

Let $(H,R)$ be a cotriangular Hopf algebra. Then there are maps
$\Psi_{\ell},\Psi_{r}:H\to H^{\circ}$, defined respectively by
$\Psi_{\ell}(a)=R(a,-)$ and $\Psi_{r}(a)=R(-,a)$ for $a\in H$. Each of these
is a map of Hopf algebras with kernel $I_{R}$, mapping into (the isomorphic
Hopf algebras) $H^{\circ,\,{\rm cop}}$ and $H^{\circ,\,{\rm op}}$,
respectively.

###### Proof.

Clearly $\Psi_{\ell}$ and $\Psi_{r}$ are linear maps into $H^{\ast}$ with
kernel $I_{R}$. To see that their images lie in $H^{\circ}$, let $c\in H$ and
fix a finite dimensional subcoalgebra $C$ of $H$ with $c\in C$. Setting
$I_{C}:=\\{h\in H\mid R(h,d)=0,\,\forall d\in C\\}$, one sees from the
definition of $R$ that $I_{C}$ is an ideal of $H$ containing $I_{R}$, and that
$H/I_{C}$ embeds into $C^{\ast}$ via $\Psi_{\ell}$. Since
$\Psi_{r}(c)(I_{C})=0$, this proves that $\mathrm{im}(\Psi_{r})\subset
H^{\circ}$. Similarly, $\mathrm{im}(\Psi_{\ell})\subset H^{\circ}$.

Since for all $a,b,x\in H$,

$\Psi_{\ell}(ab)(x)=R(ab,x)=\sum
R(a,x_{1})R(b,x_{2})=(\Psi_{\ell}(a)\ast\Psi_{\ell}(b))(x),$

$\Psi_{\ell}$ is an algebra homomorphism from $H$ to $H^{\circ}$. Moreover,
the first part of Definition 2.4(1) shows that $\Psi_{\ell}$ is a bialgebra
homomorphism into $H^{\circ,\,{\rm cop}}$.

Recall that by [Mon, Lemma 1.5.11], $H^{\circ,\,{\rm cop}}$ has antipode
$(S^{\ast})^{-1}$, the composition inverse of the antipode $S^{\ast}$ of
$H^{\circ}$. That is,

$(S^{\ast})^{-1}f(x)=f(S^{-1}(x))$

for all $f\in H^{\circ,\,{\rm cop}}$ and $x\in H$. Thus, for $h,x\in H$,

$\Psi_{\ell}(S(h))=R(S(h),x)=R(h,S^{-1}(x))=(S^{\ast})^{-1}(\Psi_{\ell}(h)),$

proving that $\Psi_{\ell}$ is a Hopf algebra map from $H$ into
$H^{\circ,\,{\rm cop}}$.

The argument for $\Psi_{r}$ is similar, noting that $H^{\circ,\,{\rm op}}$ has
antipode $(S^{\ast})^{-1}$. Note that the two codomain Hopf algebras are
isomorphic by [K, Corollary III.3.5]. ∎

Specialise now to the case where $H$ is the coordinate ring $\mathcal{O}(G)$
of an affine algebraic group $G$, so that
$(\mathcal{O}(G),\epsilon\otimes\epsilon)$ is cotriangular by Example 2.5(1).
Let $J$ be a Hopf $2$-cocycle for $G$, so $({}_{J}\mathcal{O}(G)_{J},R^{J})$
is cotriangular by Example 2.5(2). Extending the terminology of Definition
2.9, we say that $J$ is _minimal_ if $({}_{J}\mathcal{O}(G)_{J},R^{J})$ is
minimal. We then have:

######  Proposition 2.11.

([G2, Theorem 3.1]) Let $G$ be an affine algebraic group, and let $J$ be a
Hopf $2$-cocycle for $G$. Then there is a closed subgroup $T\subseteq G$ and a
minimal Hopf $2$-cocycle $\widehat{J}$ for $T$, such that $J$ is gauge
equivalent to $\widehat{J}$ (where $\widehat{J}$ is viewed in the obvious way
as a Hopf $2$-cocycle for $G$). In particular, there is an epimorphism of
cotriangular Hopf algebras from $({}_{J}\mathcal{O}(G)_{J},R^{J})$ to
$({}_{\widehat{J}}\mathcal{O}(T)_{\widehat{J}},R^{\widehat{J}})$ with kernel
$I_{R^{J}}$, the radical of $R^{J}$. ∎

######  Definition 2.12.

The (conjugacy class of the) closed subgroup $T$ of Proposition 2.11 is called
the _support_ of $J$. ∎

###  2.3. Unipotent groups

Let $G$ be a unipotent algebraic group of dimension $n$ over $\mathbb{C}$. We
recall here some basic facts, and introduce some notation needed in the sequel
(see [G3, 2.5]). Fix a chain of closed normal subgroups $G_{i}$ of $G$, for
$0\leq i\leq n$, with $G_{0}=\\{1\\}$ and $G_{n}=G$, and with $G_{i+1}/G_{i}$
central in $G/G_{i}$ and isomorphic to $(\mathbb{C},+)$, for $i=0,\ldots,n-1$.
This corresponds to a chain of Hopf subalgebras

(2.9) $\mathbb{C}=H_{0}\subset\cdots\subset H_{i}\subset\cdots\subset
H_{n}=\mathcal{O}(G)$

of $\mathcal{O}(G)$, with $H_{i}\cong\mathcal{O}(G/G_{n-i})$ and
$\mathcal{O}(G)/H_{i}^{+}\mathcal{O}(G)\cong\mathcal{O}(G_{n-i})$ for all $i$.
We can fix elements $X_{1},\ldots,X_{n}$ of $\mathcal{O}(G)$ such that

$H_{i}=\mathbb{C}[X_{1},\ldots,X_{i}]$

for all $i$. By construction, and by the basic theory of affine algebraic
groups (or by [BOZZ] or [H]), for all $i=1,\ldots,n$, we have

(2.10) $\Delta(X_{i})=X_{i}\otimes 1+1\otimes X_{i}+q(X_{i}),$

where

(2.11) $q(X_{i}):=\sum x^{i}_{1}\otimes x^{i}_{2}\in H_{i-1}^{+}\otimes
H_{i-1}^{+}$

is a coalgebra $2$-cocycle. In $\S$3.1 we will also need the notation

$(\mathrm{id}\otimes\Delta)(q(X_{i}))=\sum x^{i}_{1}\otimes x^{i}_{21}\otimes
x^{i}_{22}.$

In studying cocycle twists of unipotent groups we will make use of the
embedding of Lemma 2.2(3) into the tensor product of one sided twisted
algebras. Here is the key result for these one-sided twists of unipotent
groups. As usual, $A_{n}(\mathbb{C})$ denotes the $n$th Weyl
$\mathbb{C}$-algebra, a simple Noetherian domain whose Gelfand-Kirillov
dimension $\mathrm{GKdim}(A_{n}(\mathbb{C}))$ is $2n$ [MR, Proposition
8.1.15(2)].

######  Theorem 2.13.

([G2, Theorem 4.7]) Let $G$ be a unipotent algebraic group of dimension $n$
with Hopf $2$-cocycle $J$, and let $T$ be the support of $J$.

  1. (1)

As algebras,

$\mathcal{O}(G)_{J}\cong\mathcal{O}(T)_{J}\otimes\mathcal{O}(G/T),$

where $G/T$ denotes the space of left cosets of $T$ in $G$, so
$\mathcal{O}(G/T)$ is the commutative polynomial algebra in
$\mathrm{dim}(G)-\mathrm{dim}(T)$ variables.

  2. (2)

$\mathcal{O}(T)_{J}\cong A_{\frac{\mathrm{dim}(T)}{2}}(\mathbb{C})$ as
algebras.

  3. (3)

$\mathcal{O}(G)_{J}$ is a noetherian domain of GK-dimension $\mathrm{dim}(G)$.

  4. (4)

$(\mathcal{O}(G)_{J})^{\mathrm{op}}\cong{}_{J}\mathcal{O}(G)$. ∎

###  2.4. Hopf $2$-cocycles for unipotent groups

A _quasi-Frobenius Lie algebra_ is a finite dimensional Lie algebra
$\mathfrak{t}$ equipped with a symplectic $2$-cocycle $\omega$. That is,
$\omega:\mathfrak{t}\times\mathfrak{t}\to\mathbb{C}$ is a non-degenerate skew-
symmetric bilinear form such that for all $x,y,z\in\mathfrak{t}$,

(2.12) $\omega([x,y],z)+\omega([z,x],y)+\omega([y,z],x)=0.$

Thus every quasi-Frobenius Lie algebra has even dimension.

######  Definition 2.14.

Given a Lie algebra $\mathfrak{g}$, $r\in\wedge^{2}\mathfrak{g}$ is a solution
to the _classical Yang-Baxter equation (CYBE)_ if
$[r_{12},r_{13}]+[r_{12},r_{23}]+[r_{13},r_{23}]=0$. ∎

The equivalence of (3) and (4) below is due to Drinfeld [Dr2] and applies to
all complex finite dimensional Lie algebras: for a pair
$(\mathfrak{t},\omega)$ as in (4) one takes
$r:=\omega^{-1}\in\wedge^{2}\mathfrak{t}\subset\wedge^{2}\mathfrak{g}$. Every
solution to the CYBE arises in this way.

The equivalence between (1) and (3) was proved in [EG6, Theorem 3.2], and
later in [G2, Theorem 5.1] by an alternative proof avoiding the use of
Etingof-Kazhdan deformation theory. Full details and references can be found
in [G2, $\S$5] (see also Remarks 2.16(2) below).

######  Theorem 2.15.

(Etingof-Gelaki, [EG6]) Let $G$ be a unipotent affine algebraic group over
$\mathbb{C}$, with $\mathfrak{g}:=\mathrm{Lie}(G)$. Then the following sets
are in bijection with each other.

  1. (1)

Gauge equivalence classes of Hopf $2$-cocycles for $G$.

  2. (2)

Equivalence classes of pairs $(T,\omega)$, where $T\subseteq G$ is a closed
subgroup and $\omega$ is a left invariant symplectic form on $T$.

  3. (3)

Equivalence classes of pairs $(\mathfrak{t},r)$, where
$\mathfrak{t}\subseteq\mathfrak{g}$ is a Lie subalgebra and
$r\in\wedge^{2}\mathfrak{t}$ is a nondegenerate solution of the CYBE.

  4. (4)

Equivalence classes of pairs $(\mathfrak{t},\omega)$, where
$\mathfrak{t}\subseteq\mathfrak{g}$ is a quasi-Frobenius subalgebra with
symplectic $2$-cocycle $\omega$. ∎

######  Remarks 2.16.

(1) Here is the recipe, given in [G2, Theorem 5.1], taking us from (1) to (4)
in Theorem 2.15.

Recall first that given a Lie algebra $\mathfrak{p}$ and a $2$-cocycle
$\omega\in\wedge^{2}\mathfrak{p}^{\ast}$, one can construct a central
extension $\widehat{\mathfrak{p}}:=\mathfrak{p}\oplus\mathbb{C}z$ of
$\mathfrak{p}$ as follows. Form the exact sequence

(2.13)
$0\rightarrow\mathbb{C}z\rightarrow\widehat{\mathfrak{p}}\rightarrow\mathfrak{p}\rightarrow
0,$

with the bracket on $\widehat{\mathfrak{p}}$ given by

$[(x,\alpha z),(y,\beta z)]:=([x,y],\omega(x,y)z)$

for $x,y\in\mathfrak{p}$ and $\alpha,\beta\in\mathbb{C}$. We define

(2.14) $U^{\omega}(\mathfrak{p}):=U(\widehat{\mathfrak{p}})/\langle
z-1\rangle,$

a _twisted enveloping algebra_ of $\mathfrak{p}$ which will feature also in
Question 5.16 below.

Now given a Hopf $2$-cocycle $J$ for $G$, let $T$ be its support and
$\mathfrak{t}:={\rm Lie}(T)$. So $J$ induces a Hopf $2$-cocycle on
$\mathcal{O}(T)$, which we also denote by $J$. By Lemma 2.2(2), the right
twisted algebra $\mathcal{O}(T)_{J}$ admits an action of $T$ by algebra
automorphisms. This yields an action of $\mathfrak{t}$ on $\mathcal{O}(T)_{J}$
by derivations. But $\mathcal{O}(T)_{J}$ is a Weyl algebra by Theorem 2.13(2),
so all its derivations are inner [Di, Lemma 4.6.8]. Thus, for each
$x\in\mathfrak{t}$, we can choose $w_{x}\in\mathcal{O}(T)_{J}$ such that $x$
acts as $[w_{x},-]$ on $\mathcal{O}(T)_{J}$. One then checks that

$w_{[x,y]}=w_{x}w_{y}-w_{y}w_{x}+\omega(x,y);\,\,x,y\in\mathfrak{t},$

where $\omega(x,y)\in\mathbb{C}$ and $\omega$ is a $2$-cocycle.

It remains to show that $(\mathfrak{t},\omega)$ is quasi-Frobenius. To this
end, form the extension $\widehat{\mathfrak{t}}=\mathbb{C}z\oplus\mathfrak{t}$
as at (2.13), and observe that the map $x\mapsto w_{x}$ yields an isomorphism
of $T$-algebras

(2.15) $U^{\omega}(\mathfrak{t})=U(\widehat{\mathfrak{t}})/\langle
z-1\rangle\cong\mathcal{O}(T)_{J}.$

Now the fact that $\mathcal{O}(T)_{J}$ is a Weyl algebra by Theorem 2.13(2)
implies that $\omega$ is nondegenerate, so that $(\mathfrak{t},\omega)$ is
quasi-Frobenius, as claimed.

(2) The recipe taking us from (3) to (1) in Theorem 2.15 is given by the
results of Etingof-Kazhdan [EK] (see, e.g., [EG1, Example 5.3, Theorem 5.3]
and [EG6, §3.1]). More precisely, there exists a twist $J_{f}(r,\hbar)\in
U(\mathfrak{t})^{\otimes 2}[[\hbar]]$ (see, e.g., [EG1, Eq. (10)]), such that
$J:=J(r)$ is given by the composition

(2.16) $J:\mathcal{O}(G)^{\otimes 2}\twoheadrightarrow\mathcal{O}(T)^{\otimes
2}\xrightarrow{J_{f}(r,\hbar)}\mathcal{O}(T)^{\otimes
2}\xrightarrow{\epsilon\otimes\epsilon}\mathbb{C}.$

(3) Let $J$ be a Hopf $2$-cocycle for $G$ with support $T$. Then $J$
corresponds to $(\mathfrak{t},r)$ as in Theorem 2.15(3). Given $t\in T$, set
$J^{t}:=(t\otimes t)*J*(t^{-1}\otimes t^{-1})$ and $r^{t}:={\rm
Ad}(t)(r)\in\wedge^{2}\mathfrak{t}$. Then it follows from [EG1, Eq. (10)] that
for any $t\in T$, $J^{t}=J(r^{t})$, so $J^{t}$ corresponds to
$(\mathfrak{t},r^{t})$. ∎

##  3\. IHOE structure of ${}_{J}\mathcal{O}(G)_{J}$ for $G$ unipotent

###  3.1. ${}_{J}\mathcal{O}(G)_{J}$ is an IHOE

The acronym IHOE stands for _iterated Hopf Ore extension_. Given a positive
integer $n$, an _$n$ -step IHOE over_ $\mathbb{C}$ is a Hopf
$\mathbb{C}$-algebra $H$ with a chain

(3.1) $\mathbb{C}=H_{0}\subset H_{1}\subset\cdots\subset H_{n}=H$

of Hopf subalgebras $H_{i}$ of $H$, such that for $i=1,\ldots,n$,

(3.2) $H_{i}=H_{i-1}[X_{i};\sigma_{i},\partial_{i}],$

with $X_{i}\in H_{i}$, $\sigma_{i}$ an algebra automorphism of $H_{i-1}$, and
$\partial_{i}$ a $\sigma_{i}$-derivation of $H_{i-1}$. For details, see
[BOZZ], where many of the basic properties of IHOEs were determined. It was
shown there that the IHOE structure imposes strong constraints on
$\Delta(X_{i})$; definitive improvements to these constraints were obtained in
[H].

The IHOEs appearing in this paper are all of the form specified in (1) of the
following definition, with the individual inclusions in the chain (3.1) as in
(2).

######  Definition 3.1.

  1. (1)

An $n$-step IHOE $H$ is _of derivation type_ if it possesses a chain (3.1) for
which each automorphism $\sigma_{i}$ is the identity map on $H_{i-1}$.

  2. (2)

A Hopf $\mathbb{C}$-algebra $H$ containing a Hopf subalgebra $A$ and element
$X$ such that $H=A[X;\partial]$ for a $\mathbb{C}$-derivation $\partial$ of
$A$ is a _Hopf Ore extension of derivation type_. ∎

As noted in $\S$2.3, the coordinate ring of a unipotent group of dimension $n$
is an $n$-step IHOE with all $\sigma_{i}$ and $\partial_{i}$ trivial. More
precisely, throughout this section we keep the following hypotheses and
notation in place:

${\bf(H)}\,\,$ $G$ is a unipotent algebraic group over $\mathbb{C}$, with
$\mathcal{O}(G)=\mathbb{C}[X_{1},\ldots,X_{n}]$ as in $\S$2.3, and with
further notation as specified there.

######  Theorem 3.2.

Let $G$ be as above, and let $J$ be a Hopf $2$-cocycle for $G$. In addition
define $Q:=J-J_{21}$ and $\overline{Q}:=J^{-1}-J^{-1}_{21}$, and set
$\mathfrak{m}:=\mathcal{O}(G)^{+}$.

  1. (1)

The chain (2.9), viewed as a chain of vector spaces, gives a chain of Hopf
subalgebras of ${}_{J}\mathcal{O}(G)_{J}$, which we write as

(3.3)
$\mathbb{C}=H_{0}\subset\cdots\subset\,_{J}(H_{i})_{J}\subset\cdots\subset{}_{J}(H_{n})_{J}={}_{J}\mathcal{O}(G)_{J}.$

  2. (2)

For all $1\leq i,j\leq n$, we have

$J(X_{i},X_{j})+J^{-1}(X_{i},X_{j})+\sum
J(x^{i}_{1},x^{j}_{1})J^{-1}(x^{i}_{2},x^{j}_{2})=0.$

  3. (3)

For all $1\leq i,j\leq n$, we have

$\displaystyle X_{i}\cdot X_{j}$ $\displaystyle=X_{i}X_{j}+\sum
J^{-1}(X_{i},x^{j}_{1})x^{j}_{2}+\sum x^{j}_{1}J(X_{i},x^{j}_{2})$
$\displaystyle+\sum x^{i}_{1}J(x^{i}_{2},X_{j})+\sum
J^{-1}(x^{i}_{1},X_{j})x^{i}_{2}+\sum
x^{i}_{1}x^{j}_{1}J(x^{i}_{2},x^{j}_{2})$
$\displaystyle+\sum_{x^{i}_{21}x^{j}_{21}\in\mathfrak{m}}J^{-1}(x^{i}_{1},x^{j}_{1})x^{i}_{21}x^{j}_{21}J(x^{i}_{22},x^{j}_{22}).$

  4. (4)

For all $1\leq i,j\leq n$, we have

$\displaystyle[X_{i},X_{j}]$ $\displaystyle=X_{i}\cdot X_{j}-X_{j}\cdot X_{i}$
$\displaystyle=\quad\sum x^{i}_{1}Q(x^{i}_{2},X_{j})+\sum
x^{j}_{1}Q(X_{i},x^{j}_{2})+\sum x^{i}_{1}x^{j}_{1}Q(x^{i}_{2},x^{j}_{2})$
$\displaystyle+\sum\left(J^{-1}(x^{i}_{1},x^{j}_{1})J(x^{i}_{22},x^{j}_{22})-J_{21}^{-1}(x^{i}_{1},x^{j}_{1})J_{21}(x^{i}_{22},x^{j}_{22})\right)x^{i}_{21}x^{j}_{21}$
$\displaystyle+\sum x^{i}_{2}\overline{Q}(x^{i}_{1},X_{j})+\sum
x^{j}_{2}\overline{Q}(X_{i},x^{j}_{1}).$

In particular, $[X_{i},X_{j}]\in H_{i-1}^{+}$ for all $1\leq j<i\leq n$.

  5. (5)

For all $1\leq i\leq n$,
${}_{J}(H_{i})_{J}\cong{}_{J}(H_{i-1})_{J}[X_{i};\partial_{i}]$ is a Hopf Ore
extension of derivation type.

  6. (6)

$X_{1}$ and $X_{2}$ are primitive elements of ${}_{J}\mathcal{O}(G)_{J}$.

  7. (7)

The space $\mathcal{P}({}_{J}\mathcal{O}(G)_{J})$ of primitive elements of
${}_{J}\mathcal{O}(G)_{J}$, which equals $\mathcal{P}(\mathcal{O}(G))$,
generates a commutative Hopf subalgebra of ${}_{J}\mathcal{O}(G)_{J}$, a
polynomial algebra of rank
$\mathrm{dim}_{k}(\mathcal{P}(\mathcal{O}(G)))=\mathrm{dim}(G/[G,G])$.

###### Proof.

(1) This is clear from (2.5) and (2.9).

(2) Omitting summation symbols for clarity where convenient, we calculate,
using (2.10) for $\Delta(X_{i})$ and $\Delta(X_{j})$, and the notation (2.11),
and applying (2.2),

$\displaystyle 0$ $\displaystyle=(\epsilon\otimes\epsilon)(X_{i}\otimes
X_{j})=(J\ast J^{-1})(X_{i}\otimes X_{j})$
$\displaystyle=J(X_{i},X_{j})J^{-1}(1,1)+J(X_{i},1)J^{-1}(1,X_{j})+J(1,X_{j})J^{-1}(X_{i},1)$
$\displaystyle+J(1,1)J^{-1}(X_{i},X_{j})+J(x^{i}_{1},X_{j})J^{-1}(x^{i}_{2},1)+J(X_{i},x^{j}_{1})J^{-1}(1,x^{j}_{2})$
$\displaystyle+J(1,x^{j}_{1})J^{-1}(X_{i},x^{j}_{2})+J(x^{i}_{1},1)J^{-1}(x^{i}_{2},X_{j})+J(x^{i}_{1},x^{j}_{1})J^{-1}(x^{i}_{2},x^{j}_{2})$
$\displaystyle=J(X_{i},X_{j})+J^{-1}(X_{i},X_{j})+J(x^{i}_{1},x^{j}_{1})J^{-1}(x^{i}_{2},x^{j}_{2}),$

where the final equality holds since the omitted terms are all $0$ by (2.2)
and (2.11).

(3) We start from the expansions

$\displaystyle(\mathrm{id}\otimes\Delta)\circ\Delta(X_{i})$
$\displaystyle=X_{i}\otimes 1\otimes 1+1\otimes X_{i}\otimes 1+1\otimes
1\otimes X_{i}$ $\displaystyle+1\otimes q(X_{i})+\sum x^{i}_{1}\otimes
x^{i}_{21}\otimes x^{i}_{22},$

and a corresponding one for $X_{j}$. Thus, from (2.5) applied to the above
expansions, and using (2.2) to see that many terms in the product are $0$, we
calculate

$\displaystyle X_{i}\cdot X_{j}$
$\displaystyle=J^{-1}(X_{i},X_{j})J(1,1)+J^{-1}(X_{i},x^{j}_{1})x^{j}_{21}J(1,x^{j}_{22})+X_{i}X_{j}+J^{-1}(1,1)J(X_{i},X_{j})$
$\displaystyle+J^{-1}(1,1)x^{j}_{1}J(X_{i},x^{j}_{2})+J^{-1}(1,1)x^{i}_{1}J(x^{i}_{2},X_{j})+J^{-1}(1,1)x^{i}_{1}x^{j}_{1}J(x^{i}_{2},x^{j}_{2})$
$\displaystyle+J^{-1}(x^{i}_{1},X_{j})x^{i}_{21}J(x^{i}_{22},1)+J^{-1}(x^{i}_{1},x^{j}_{1})x^{i}_{21}x^{j}_{21}J(x^{i}_{22},x^{j}_{22})$
$\displaystyle=J^{-1}(X_{i},X_{j})+J(X_{i},X_{j})+J^{-1}(X_{i},x^{j}_{1})x^{j}_{21}J(1,x^{j}_{22})+X_{i}X_{j}+x^{j}_{1}J(X_{i},x^{j}_{2})$
$\displaystyle+x^{i}_{1}J(x^{i}_{2},X_{j})+x^{i}_{1}x^{j}_{1}J(x^{i}_{2},x^{j}_{2})+J^{-1}(x^{i}_{1},X_{j})x^{i}_{21}J(x^{i}_{22},1)$
$\displaystyle+J^{-1}(x^{i}_{1},x^{j}_{1})x^{i}_{21}x^{j}_{21}J(x^{i}_{22},x^{j}_{22})$
$\displaystyle=J^{-1}(X_{i},X_{j})+J(X_{i},X_{j})+J^{-1}(x^{i}_{1},x^{j}_{1})J(x^{i}_{22},x^{j}_{22})+J^{-1}(X_{i},x^{j}_{1})x^{j}_{21}J(1,x^{j}_{22})$
$\displaystyle+X_{i}X_{j}+x^{j}_{1}J(X_{i},x^{j}_{2})+x^{i}_{1}J(x^{i}_{2},X_{j})+x^{i}_{1}x^{j}_{1}J(x^{i}_{2},x^{j}_{2})+J^{-1}(x^{i}_{1},X_{j})x^{i}_{21}J(x^{i}_{22},1)$
$\displaystyle+\sum_{x^{i}_{21}x^{j}_{21}\in\mathfrak{m}}J^{-1}(x^{i}_{1},x^{j}_{1})x^{i}_{21}x^{j}_{21}J(x^{i}_{22},x^{j}_{22})$
$\displaystyle=J^{-1}(X_{i},x^{j}_{1})x^{j}_{21}J(1,x^{j}_{22})+X_{i}X_{j}+x^{j}_{1}J(X_{i},x^{j}_{2})+x^{i}_{1}J(x^{i}_{2},X_{j})+x^{i}_{1}x^{j}_{1}J(x^{i}_{2},x^{j}_{2})$
$\displaystyle+J^{-1}(x^{i}_{1},X_{j})x^{i}_{21}J(x^{i}_{22},1)+\sum_{x^{i}_{21}x^{j}_{21}\in\mathfrak{m}}J^{-1}(x^{i}_{1},x^{j}_{1})x^{i}_{21}x^{j}_{21}J(x^{i}_{22},x^{j}_{22}),$

where the sum of the first three terms in the penultimate expansion equals $0$
by (2). In the final expansion above, the summands in the first term are all 0
by (2.2), except when $x^{j}_{22}=1$, which occurs when
$x^{j}_{21}=x^{j}_{2}$, so that the first term collapses to $\sum
J^{-1}(X_{i},x^{j}_{1})x^{j}_{2}$. Similarly, the penultimate term collapses
to $\sum J^{-1}(x^{i}_{1},X_{j})x^{i}_{2}$. Hence, we finally obtain

$\displaystyle X_{i}\cdot X_{j}$ $\displaystyle=X_{i}X_{j}+\sum
J^{-1}(X_{i},x^{j}_{1})x^{j}_{2}+\sum x^{j}_{1}J(X_{i},x^{j}_{2})$
$\displaystyle+\sum x^{i}_{1}J(x^{i}_{2},X_{j})+\sum
J^{-1}(x^{i}_{1},X_{j})x^{i}_{2}+\sum
x^{i}_{1}x^{j}_{1}J(x^{i}_{2},x^{j}_{2})$
$\displaystyle+\sum_{x^{i}_{21}x^{j}_{21}\in\mathfrak{m}}J^{-1}(x^{i}_{1},x^{j}_{1})x^{i}_{21}x^{j}_{21}J(x^{i}_{22},x^{j}_{22}).$

(4) Combining the calculation of $X_{i}\cdot X_{j}$ from (3) with the
corresponding one for $X_{j}\cdot X_{i}$, we arrive at the formula stated in
the proposition.

(5) This is immediate from (1) and (4) when we bear in mind that, as a vector
space, ${}_{J}(H_{i})_{J}$ has the same basis as $H_{i}$.

(6) That $X_{1}$ and $X_{2}$ are primitive follows from the fact that the
comultiplication in ${}_{J}\mathcal{O}(G)_{J}$ is unchanged from
$\mathcal{O}(G)$, since in characteristic $0$ there are no non-abelian
unipotent groups of dimension at most $2$.

(7) It is clear from (4) that primitive elements commute. The rest of (7) now
follows easily. ∎

The following key corollary is easily deduced from Theorem 3.2(5):

######  Corollary 3.3.

With $G$ and $J$ as in Theorem 3.2, ${}_{J}\mathcal{O}(G)_{J}$ is an $n$-step
IHOE of derivation type.

###### Proof.

This is essentially proved already by Theorem 3.2(5). It is perhaps worth
observing that the relations given by Theorem 3.2(4) ensure that
${}_{J}\mathcal{O}(G)_{J}$ is a _factor_ of the iterated Ore extension
$\mathbb{C}[X_{1}][X_{2};\partial_{2}]\cdots[X_{n};\partial_{n}]$. However, we
know that the coalgebra structure is preserved on the same vector space by the
twist operation, so the ideal we are factoring by is $\\{0\\}$. ∎

######  Remarks 3.4.

(1) Here is a perspective on Corollary 3.3, which is useful for instance when
discussing the finite dimensional simple ${}_{J}\mathcal{O}(G)_{J}$-modules in
$\S$6. We can view both $\mathcal{O}(G)$ and ${}_{J}\mathcal{O}(G)_{J}$ as
factor algebras of the free $\mathbb{C}$-algebra on $n$ generators,
$F:=\mathbb{C}\langle X_{1},\dots,X_{n}\rangle$, the former being the factor
of $F$ by the ideal

$D:=\langle[X_{i},X_{j}]\mid 1\leq i<j\leq n\rangle,$

and the latter the factor by the ideal

$E:=\langle[X_{i},X_{j}]-f_{ij}\mid 1\leq i<j\leq n\rangle,$

where the elements $f_{ij}\in F$ are given by Theorem 3.2(4). Let

$\mathcal{B}:=\\{X_{1}^{a_{1}}X_{2}^{a_{2}}\cdots X_{n}^{a_{n}}\mid
a_{i}\in\mathbb{Z}^{\geq 0}\\}\subset F.$

Then the cosets

$\\{b+D\mid b\in\mathcal{B}\\}\textit{ and }\\{b+E\mid b\in\mathcal{B}\\}$

provide $\mathbb{C}$-bases of $\mathcal{O}(G)$ and ${}_{J}\mathcal{O}(G)_{J}$
respectively, _with the multiplication in each case induced by the
multiplication in $F$_. In particular, one sees that the sets of powers of
each generator $X_{i}$ are the same in both $\mathcal{O}(G)$ and in
${}_{J}\mathcal{O}(G)_{J}$.

(2) Theorem 3.2(3)-(5) correct typos in [G3, Lemma 3.1]. More precisely,
Theorem 3.2(3) corrects [G3, Lemma 3.1(4)] in which the second and fifth
summands are missing. Theorem 3.2(4) corrects [G3, Lemma 3.1(5)] in which the
fifth and sixth summands are missing. Theorem 3.2(5) corrects [G3, Lemma
3.1(6)] in which it is mistakenly stated that $X_{1}$ and $X_{2}$ are central.
∎

###  3.2. Immediate consequences of IHOE structure

Here are some more or less immediate consequences of Corollary 3.3.
Definitions of the homological terms appearing in (2) can be found in [BZ],
for example. Recall for (3) that a Hopf algebra is _connected_ [Mon,
Definition 5.1.5] if its coradical is just the base field $\mathbb{C}$; note
for future use that this property passes to all factor Hopf algebras by [Mon,
Corollary 5.3.5].

######  Corollary 3.5.

Keep the notation and hypotheses of ${\bf(H)}$ from the start of $\S$3.

  1. (1)

([G3, Corollary 3.2]) ${}_{J}\mathcal{O}(G)_{J}$ is a noetherian domain of GK
dimension $n$.

  2. (2)

${}_{J}\mathcal{O}(G)_{J}$ is Artin-Schelter regular and Auslander regular
with global homological dimension $n$, and is GK-Cohen-Macaulay.

  3. (3)

${}_{J}\mathcal{O}(G)_{J}$ is a connected Hopf algebra whose associated graded
algebra with respect to its coradical filtration is the coordinate ring
$\mathcal{O}(\widehat{G})$, the associated graded ring of $\mathcal{O}(G)$,
$\widehat{G}$ being the associated graded unipotent group of $G$.

  4. (4)

Every prime ideal of ${}_{J}\mathcal{O}(G)_{J}$ is completely prime.

  5. (5)

Every finite dimensional simple ${}_{J}\mathcal{O}(G)_{J}$-module $V$ has
$\mathrm{dim}_{\mathbb{C}}(V)=1$.

###### Proof.

(1), (2) These properties are valid for every $n$-step IHOE - see [BOZZ,
Theorem 3.2(2),(5)].

(3) Since the coalgebra structure of ${}_{J}\mathcal{O}(G)_{J}$ is the same as
that of $\mathcal{O}(G)$, ${}_{J}\mathcal{O}(G)_{J}$ is a connected Hopf
algebra (which also follows from the fact that it is an IHOE, [BOZZ, Theorem
3.2(1)]). Hence its coradical filtration is an algebra filtration by [Mon,
Lemma 5.2.8]. It follows from Corollary 3.3 and [BOZZ, Theorem 3.2(4)] that
the resulting associated graded algebra of ${}_{J}\mathcal{O}(G)_{J}$ is a
coradically graded commutative polynomial Hopf algebra in $n$ variables. But
as already recalled, ${}_{J}\mathcal{O}(G)_{J}$ and $\mathcal{O}(G)$ are the
same as coalgebras, hence they have the same associated graded Hopf algebra.

(4) This is true for all iterated Ore extensions of $\mathbb{C}$ of derivation
type by [L].

(5) The annihilator $I$ of a finite dimensional simple
${}_{J}\mathcal{O}(G)_{J}$-module $V$ is a maximal ideal with
$\mathrm{dim}_{\mathbb{C}}({}_{J}\mathcal{O}(G)_{J}/I)<\infty$. So, by the
Artin-Wedderburn theorem and (4), ${}_{J}\mathcal{O}(G)_{J}/I\cong\mathbb{C}$.
Hence, $\mathrm{dim}_{\mathbb{C}}(V)=1$. ∎

Further to Corollary 3.5(2), basic properties of IHOEs allow us to say also
that ${}_{J}\mathcal{O}(G)_{J}$ is skew Calabi-Yau, but we do not include that
property here as we will later - in Corollary 5.6 \- show that
${}_{J}\mathcal{O}(G)_{J}$ is actually Calabi-Yau. The group of finite
dimensional simple ${}_{J}\mathcal{O}(G)_{J}$-modules (which exists by (5)),
is studied in $\S$6.

##  4\. Structure of twisted unipotent groups - the minimal case

Retain the notation and hypotheses (H) from the beginning of $\S$3, so
$(\mathcal{O}(G),\epsilon\otimes\epsilon)$ is cotriangular by Example 2.5(1).
Fix a Hopf $2$-cocycle $J$ for $G$, so $R^{J}:=J_{21}^{-1}\ast J$ and
$({}_{J}\mathcal{O}(G)_{J},R^{J})$ is cotriangular by Example 2.5(2). The aim
in this section is to prove Theorem 4.9 (as predicted in [G3, Theorem 3.4];
see Remark 4.10 below), describing the algebra structure of
${}_{J}\mathcal{O}(G)_{J}$ in the case where
$({}_{J}\mathcal{O}(G)_{J},R^{J})$ is minimal (Definition 2.9).

###  4.1. The minimal case - inclusion of $\mathrm{im}(\Psi)$

We assume throughout this subsection that $R^{J}$ is nondegenerate for
${}_{J}\mathcal{O}(G)_{J}$, as in Definition 2.9. By Lemma 2.10, there is an
embedding of Hopf algebras

(4.1)
$\Psi:{}_{J}\mathcal{O}(G)_{J}\xrightarrow{1:1}({}_{J}\mathcal{O}(G)_{J})^{\circ,{\rm
op}},\,\,\,a\mapsto R^{J}(-,a).$

Observe that $({}_{J}\mathcal{O}(G)_{J})^{\circ,{\rm op}}$ is a subalgebra of
$({}_{J}\mathcal{O}(G)_{J})^{*,{\rm op}}$ (where
$(_{J}\mathcal{O}(G)_{J})^{*}$ is the full vector space dual of
${}_{J}\mathcal{O}(G)_{J}$). Since $\mathcal{O}(G)={}_{J}\mathcal{O}(G)_{J}$
as coalgebras, their vector space duals are the same as algebras. Namely, via
the identity map, as algebras,

(4.2) $({}_{J}\mathcal{O}(G)_{J})^{*,{\rm op}}=(\mathcal{O}(G)^{*})^{{\rm
op}}.$

In particular, $({}_{J}\mathcal{O}(G)_{J})^{*,{\rm op}}$ contains as a
subalgebra the (opposite algebra of the) finite dual $\mathcal{O}(G)^{\circ}$
of $\mathcal{O}(G)$. Since $\mathcal{O}(G)$ is commutative, its finite dual is
cocommutative, so by the Cartier-Gabriel-Kostant structure theorem [Mon,
Corollary 5.6.4(3) and Theorem 5.6.5] and (4.2),
$({}_{J}\mathcal{O}(G)_{J})^{*,{\rm op}}$ contains the subalgebra
$(U(\mathfrak{g})\\#\mathbb{C}[G])^{{\rm op}}$, where $\mathfrak{g}:={\rm
Lie}(G)$. Thus, $({}_{J}\mathcal{O}(G)_{J})^{*,{\rm op}}$ contains the
subalgebra $U(\mathfrak{g})^{{\rm op}}\cong U(\mathfrak{g})$, consisting of
the distributions supported (set-theoretically) at $1$, that is vanishing on
some power of the augmentation ideal $\mathfrak{m}$ of $\mathcal{O}(G)$. We
shall show that

(4.3) $\mathrm{im}(\Psi)=U(\mathfrak{g})^{{\rm op}}.$

First we use the work of Etingof and Kazhdan [EK] to prove

######  Proposition 4.1.

With the above hypotheses and notation,

$\mathrm{im}(\Psi)\subseteq U(\mathfrak{g})^{{\rm op}}.$

###### Proof.

Let $J_{f}(r,\hbar)\in U(\mathfrak{g})^{\otimes 2}[[\hbar]]$ be such that $J$
is given by the composition

(4.4) $J:\mathcal{O}(G)^{\otimes
2}\xrightarrow{J_{f}(r,\hbar)}\mathcal{O}(G)^{\otimes
2}\xrightarrow{\epsilon\otimes\epsilon}\mathbb{C}$

(see Remarks 2.16(2)). Then setting

$R_{f}(r,\hbar):=J_{f}(r,\hbar)_{21}^{-1}J_{f}(r,\hbar)\in
U(\mathfrak{g})^{\otimes 2}[[\hbar]],$

we have that $R^{J}$ is given by the composition

$R^{J}:\mathcal{O}(G)^{\otimes
2}\xrightarrow{R_{f}(r,\hbar)}\mathcal{O}(G)^{\otimes
2}\xrightarrow{\epsilon\otimes\epsilon}\mathbb{C}.$

Namely, writing

$R_{f}(r,\hbar)=1+\sum_{i=1}^{\infty}(r_{i1}\otimes r_{i2})\hbar^{i}\in
U(\mathfrak{g})^{\otimes 2}[[\hbar]],$

the bilinear form $R^{J}$ is given by

$R^{J}(y,z)=\epsilon(yz)+\sum_{i=1}^{\infty}r_{i1}(y)r_{i2}(z);\,\,\,y,z\in\mathcal{O}(G),$

where the sum is well defined since it is actually finite (as we see below).

Let $z\in\mathfrak{m}:=\mathcal{O}(G)^{+}$. Pick $n$ such that $r_{i2}(z)=0$
for every $i>n$, which is possible since $z$ vanishes on some power of
$U(\mathfrak{g})^{+}$, and $r_{i2}$ lies in $(U(\mathfrak{g})^{+})^{2i}$ for
every $i$ [EG1, Example 5.3]. Next fix $N$ such that $r_{11},\dots,r_{1n}$
vanish on $\mathfrak{m}^{N}$. Then for every $y\in\mathfrak{m}^{N}$, we have

$\Psi(z)(y)=R^{J}(y,z)=\sum_{i=1}^{\infty}r_{i1}(y)r_{i2}(z)=\sum_{i=1}^{N}r_{i1}(y)r_{i2}(z)=0.$

Thus, $\Psi(z)$ vanishes on $\mathfrak{m}^{N}$, so $\Psi$ maps
${}_{J}\mathcal{O}(G)_{J}$ into $U(\mathfrak{g})$, as claimed. ∎

###  4.2. The minimal case - surjectivity of $\Psi$

We continue to assume throughout this subsection that $R^{J}$ is nondegenerate
for ${}_{J}\mathcal{O}(G)_{J}$, and we retain all the notation introduced so
far in $\S$4. The proof that $\Psi$ maps ${}_{J}\mathcal{O}(G)_{J}$ onto
$U(\mathfrak{g})$ is by induction on $\mathrm{dim}(G)$, so we start by
introducing the induction setup.

Recall from $\S$2.4 that $n:=\dim(G)$ is even. Given elements $x,y$ in (the
vector space) $\mathcal{O}(G)$, when confusion seems possible we will use the
notation $x\cdot y$ to denote multiplication in ${}_{J}\mathcal{O}(G)_{J}$ and
$xy$ to denote multiplication in $\mathcal{O}(G)$.

Choose a $1$-dimensional closed central subgroup $C$ of $G$, and let $K/C$ be
the support of $J$ on $\mathcal{O}(G/C)\subset\mathcal{O}(G)$. By [G2,
Proposition 4.6], $K/C$ has codimension $1$ in $G/C$, so $K$ has codimension
$1$ in $G$, and hence is normal. Denote the Lie algebras of $K$ [resp. $G$] by
$\mathfrak{k}$ [resp. $\mathfrak{g}$]. Clearly, the exact sequence
$0\rightarrow\mathfrak{k}\rightarrow\mathfrak{g}\rightarrow\mathfrak{g}/\mathfrak{k}\rightarrow
0$ splits, with $\mathfrak{g}=\mathfrak{k}\oplus\mathbb{C}x$ for any element
$x\in\mathfrak{g}\setminus\mathfrak{k}$. Hence, the same is true for the
corresponding groups, namely, we have

(4.5) $G=K\rtimes Y\cong K\rtimes(G/K),$

where the subgroup $Y$ of $G$ is isomorphic to $(\mathbb{C},+)$. Note that
$C\subseteq K$ and $C$ is central in $G$, so that

$C\rtimes Y=C\times Y\subseteq G.$

By (4.5), we can pick a primitive element $y_{1}\in\mathcal{O}(G)$ such that
$y_{1}(K)=0$ and $\mathcal{O}(Y)\cong\mathcal{O}(G/K)=\mathbb{C}[y_{1}]$ as
Hopf algebras. Consider the chain of group epimorphisms

$G\twoheadrightarrow G/C\twoheadrightarrow G/K.$

By the discussion in $\S$2.3, this yields the chain of cotriangular Hopf
algebras

(4.6)
$\mathbb{C}[y_{1}]={}_{J}\mathcal{O}(G/K)_{J}\subseteq{}_{J}\mathcal{O}(G/C)_{J}\subset{}_{J}\mathcal{O}(G)_{J},$

where we will elaborate on the first equality in Lemma 4.3 below. Pick
$y_{n}\in\mathcal{O}(G)$ whose restriction to $C$ generates $\mathcal{O}(C)$.
Then

${}_{J}\mathcal{O}(G)_{J}={}_{J}\mathcal{O}(G/C)_{J}[y_{n};\partial_{n}]$

is a Hopf Ore extension, and by (2.11) we see that

(4.7) $\Delta(y_{n})-y_{n}\otimes 1-1\otimes y_{n}:=\sum Y_{n1}\otimes
Y_{n2}\in\mathcal{O}(G/C)^{+}\otimes\mathcal{O}(G/C)^{+}.$

It follows from (4.6), and (again) will be further explicated in Lemma 4.3,
that $y_{1}$ generates the kernel of the surjective cotriangular Hopf algebra
map

(4.8)
${}_{J}\mathcal{O}(G/C)_{J}\twoheadrightarrow{}_{J}\mathcal{O}(K/C)_{J}.$

We need the following easy lemma.

######  Lemma 4.2.

Let $(G,J)$ be a unipotent group with a Hopf $2$-cocycle, and let $p$ be a
primitive element of $\mathcal{O}(G)$. Let $R^{J}$ be the $R$-form
$J_{21}^{-1}\ast J$ as in Example 2.5(2). Define

(4.9) $I_{p}:=\\{\alpha\in(_{J}\mathcal{O}(G)_{J})^{+}\mid
R^{J}(\alpha,p)=0\\}.$

  1. (1)

$I_{p}$ is an ideal of ${}_{J}\mathcal{O}(G)_{J}$.

  2. (2)

$I_{p}=\\{\alpha\in(_{J}\mathcal{O}(G)_{J})^{+}\mid R^{J}(p,\alpha)=0\\}.$

###### Proof.

$(1)$ Clearly $I_{p}$ is a vector subspace. Let $\alpha\in I_{p}$ and
$\beta\in{}_{J}\mathcal{O}(G)_{J}$. Then

$R^{J}(\alpha\cdot\beta,p)=R^{J}(\alpha,p)R^{J}(\beta,1)+R^{J}(\alpha,1)R^{J}(\beta,p)=0+0=0,$

where the first term is $0$ since $\alpha\in I_{p}$ and the second is $0$ by
Lemma 2.6(1). A similar calculation deals with $\beta\cdot\alpha$.

$(2)$ Since $({}_{J}\mathcal{O}(G)_{J},R^{J})$ is cotriangular, and using
[Maj, Lemma 2.2.2],

$R^{J}(x,p)=(R^{J})^{-1}(x,S(p))=R^{J}(S(p),x)$

for all $x\in{}_{J}\mathcal{O}(G)_{J}$. So (2) follows from the fact that
$S(p)=-p$. ∎

Returning to our induction setup, we prove the following basic facts.

######  Lemma 4.3.

The following hold:

  1. (1)

For every $x\in\mathcal{O}(G/C)$, we have $R^{J}(x,y_{1})=R^{J}(y_{1},x)=0$.

  2. (2)

$y_{1}$ is a central primitive element of ${}_{J}\mathcal{O}(G)_{J}$.

  3. (3)

For all $x\in\mathcal{O}(G)^{+}$, $R^{J}(y_{1}\cdot
x,y_{n})=R^{J}(y_{n},y_{1}\cdot x)=0$.

  4. (4)

Using the notation (4.9),

$\displaystyle I_{y_{1}}$
$\displaystyle=({}_{J}\mathcal{O}(G/C)_{J})^{+}{}_{J}\mathcal{O}(G)_{J}+y_{n}^{2}{}_{J}\mathcal{O}(G)_{J}$
$\displaystyle=({}_{J}\mathcal{O}(G/C)_{J})^{+}{}_{J}\mathcal{O}(G)_{J}+y_{n}\cdot
y_{n}{}_{J}\mathcal{O}(G)_{J}.$

  5. (5)

We may assume that $R^{J}(y_{n},y_{1})=1$.

###### Proof.

(1) By definition of $K$ and of $y_{1}$, $y_{1}$ lies in the radical of
$R^{J}$ restricted to ${}_{J}\mathcal{O}(G/C)_{J}$.

(2) By (1) and Lemma 2.6(2), for all $x\in\mathcal{O}(G/C)$, we have

(4.10) $0=R^{J}(y_{1},x)=(J-J_{21})(y_{1},x)=(J_{21}^{-1}-J^{-1})(y_{1},x).$

The chain $1\subset C\subseteq K\subset G$ of normal subgroups of $G$ can be
refined to a chain of normal subgroups with successive subfactors all
isomorphic to $(\mathbb{C},+)$. This in turn yields a chain of Hopf Ore
extensions

$\\{0\\}\subset\mathcal{O}(G/K)=\mathbb{C}[y_{1}]\subseteq\cdots\subseteq\mathcal{O}(G/C)\subset\mathcal{O}(G/C)[y_{n}]=\mathcal{O}(G),$

with a similar chain for ${}_{J}\mathcal{O}(G)_{J}$ formed from identical
subspaces, as explained in $\S$2.3 and Theorem 3.2. Fix generators
$X_{2},\ldots,X_{n-1}$ for the Hopf Ore extensions from
${}_{J}\mathcal{O}(G/K)_{J}$ to ${}_{J}\mathcal{O}(G/C)_{J}$, and define
$Q:=J-J_{21}$ and $\overline{Q}:=J^{-1}-J^{-1}_{21}$. Recall that we write,
for each $i$,

(4.11) $\Delta(X_{i})-X_{i}\otimes 1-1\otimes X_{i}=\sum x^{i}_{1}\otimes
x^{i}_{2},$

where, as in (4.7), the elements $x^{i}_{1},x^{i}_{2}$ lie in lower Hopf
algebras, and in particular are in ${}_{J}\mathcal{O}(G/C)_{J}$. Hence, by
Theorem 3.2(4), taking commutators in ${}_{J}\mathcal{O}(G/C)_{J}$, we have
for $i=2,\ldots,n-1$,

(4.12) $[X_{i},y_{1}]=\sum x^{i}_{1}Q(x^{i}_{2},y_{1})+\sum
x^{i}_{2}\overline{Q}(x^{i}_{1},y_{1})=0,$

where the second equality follows from (4.10).

An exactly similar argument shows that $[y_{n},y_{1}]=0$, in view of (4.7).
Since ${}_{J}\mathcal{O}(G)_{J}$ is generated by
$\\{y_{1},X_{2},\ldots,X_{n-1},y_{n}\\}$, (2) is proved.

(3) By (4.7) and by parts (1) and (2) of the lemma, we have

$\displaystyle R^{J}(y_{1}\cdot x,y_{n})$
$\displaystyle=R^{J}(y_{1},y_{n})R^{J}(x,1)+R^{J}(y_{1},1)R^{J}(x,y_{n})$
$\displaystyle+\sum R^{J}(y_{1},Y_{n_{1}})R^{J}(x,Y_{n_{2}})=\sum
R^{J}(y_{1},Y_{n_{1}})R^{J}(x,Y_{n_{2}})=0,$

where the second equality follows from Lemma 2.6(1) and the third from (4.7)
and (1). The argument for $R^{J}(y_{n},y_{1}\cdot x)$ is exactly similar.

(4), (5) Note first that $y_{n}^{2}=y_{n}\cdot y_{n}+z$ for some $z$ in
$\mathcal{O}(G/C)^{+}$ by Theorem 3.2(3). Thus, the ideals appearing in the
second and third entries of the identity in (4) are equal. Moreover,
$({}_{J}\mathcal{O}(G/C)_{J})^{+}{}_{J}\mathcal{O}(G)_{J}\subset I_{y_{1}}$ by
Lemmas 4.2(1) and 4.3(1). So to prove that the ideal on the right is contained
in $I_{y_{1}}$ it remains only to show that $y_{n}\cdot y_{n}\in I_{y_{1}}$.
But this is immediate from Lemma 2.6(1), since $y_{1}$ is primitive.

For the reverse inclusion, note that $R^{J}(-,y_{1})$ is a nonzero functional
on ${}_{J}\mathcal{O}(G)_{J}$ by minimality of $J$. However, by the inclusion
proved above and Lemma 2.6(1),

$R^{J}\left(\left({}_{J}\mathcal{O}(G/C)_{J}\right)^{+}{}_{J}\mathcal{O}(G)_{J}+y_{n}\cdot
y_{n}{}_{J}\mathcal{O}(G)_{J}+\mathbb{C},y_{1}\right)=0.$

Hence we must have $R^{J}(y_{n},y_{1})\neq 0$, which proves the reverse
inclusion and also shows that we can rescale so that $R^{J}(y_{n},y_{1})=1$,
proving (5). ∎

We proceed now to prove Theorem 4.4, that $\mathrm{Im}(\Psi)$ is
$U(\mathfrak{g})^{{\rm op}}$, thus completing the proof of (4.3). Recall that
$\mathfrak{g}$ and $\mathfrak{k}$ are the Lie algebras of $G$ and $K$,
respectively, and let $\mathfrak{c}$ denote the Lie algebra of $C$. By (4.5),
we have a chain of ideals
$\mathfrak{c}\subseteq\mathfrak{k}\subset\mathfrak{g}$, such that
$\mathfrak{c}\cong\mathbb{C}$ is central in $\mathfrak{g}$, and

$\mathfrak{g}=\mathfrak{k}\rtimes(\mathfrak{g}/\mathfrak{k})\cong\;\mathfrak{k}\rtimes\mathbb{C}.$

It follows (see, e.g., [MR, §1.7.10, §1.7.12]) that we have algebra
isomorphisms

(4.13) $U(\mathfrak{g})\cong
U(\mathfrak{k})\\#U(\mathfrak{g}/\mathfrak{k})\cong\left(U(\mathfrak{c})*U(\mathfrak{k}/\mathfrak{c})\right)\\#U(\mathfrak{g}/\mathfrak{k}),$

where $*$ denotes a crossed product and $\\#$ a smash product. Thus,
$U(\mathfrak{g})$ is generated by its subspaces $U(\mathfrak{c})$,
$U(\mathfrak{k}/\mathfrak{c})$, and $U(\mathfrak{g}/\mathfrak{k})$. Bearing in
mind that $\mathfrak{g}$ constitutes precisely the $\mathbb{C}$-derivations of
$\mathcal{O}(G)$, we can specify a basis $u_{1},\ldots,u_{n}$ of
$\mathfrak{g}$ as follows:

$\displaystyle u_{1}(y_{1})$ $\displaystyle=u_{1}(X_{j})=0,\;2\leq j\leq
n-1,\;u_{1}(y_{n})=1;$ $\displaystyle u_{i}(y_{1})$
$\displaystyle=u_{i}(y_{n})=0,\;u_{i}(X_{j})=\delta_{i,n+1-j},\;2\leq i,j\leq
n-1;$ $\displaystyle u_{n}(X_{j})$ $\displaystyle=u_{n}(y_{n})=0,\;2\leq j\leq
n-1,\;u_{n}(y_{1})=1.$

Set $\mathfrak{m}:=\mathcal{O}(G)^{+}$. Since the above maps are derivations,
$u_{i}(\mathfrak{m}^{2})=u_{i}(1)=0$ for all $i=1,\ldots,n$, noting that
$\mathfrak{m}^{2}$ is the product $\mathfrak{m}\mathfrak{m}$ using the
_commutative_ multiplication. Recall that $U(\mathfrak{g})$ is the subalgebra
of $\mathcal{O}(G)^{*}$ consisting of distributions supported (set-
theoretically) at $1$, that is, vanishing on some power of $\mathfrak{m}$.

######  Theorem 4.4.

Keep all the notation introduced above, and in the rest of $\S$4. The image of
the Hopf algebra monomorphism $\Psi$ from ${}_{J}\mathcal{O}(G)_{J}$ defined
at (5.15) is $U(\mathfrak{g})^{{\rm op}}$.

###### Proof.

Since $\Psi$ is an algebra map, it is enough to show that

(4.14) $u_{i}\in\mathrm{im}(\Psi),\;i=1,\ldots,n.$

The proof of (4.14) is quite involved, requiring four lemmas, the first being:

######  Lemma 4.5.

After multiplying if necessary by a non-zero scalar, we have

(4.15) $u_{1}=\Psi(y_{1})\in\mathrm{im}(\Psi).$

###### Proof.

This follows from Lemma 4.3(1),(4),(5) and the definition of $u_{1}$. ∎

We now proceed by induction on even $n\geq 0$, the case $n=0$ being trivial.
So we assume that the result is true for all minimal $2$-cocycle deformations
of unipotent groups whose dimension is less than $n$. In particular, by
definition of $K$, the restriction of $J$ to $\mathcal{O}(G/C)$ induces a
minimal Hopf $2$-cocycle $\overline{J}$ for $K/C$, with corresponding $R$-form
$\overline{R}^{\overline{J}}$, and injective algebra homomorphism

$\overline{\Psi}:{}_{\overline{J}}\mathcal{O}(K/C)_{\overline{J}}\xrightarrow{1:1}U(\mathfrak{k}/\mathfrak{c}).$

Thus, our induction hypothesis ensures that $\overline{\Psi}$ maps onto
$U(\mathfrak{k}/\mathfrak{c})$, so there exist elements
$\overline{z}_{i}\in{}_{\overline{J}}\mathcal{O}(K/C)_{\overline{J}}$, $2\leq
i\leq n-1$, such that we have

(4.16) $\overline{\Psi}(\overline{z_{i}})=\overline{u}_{i},\quad\textit{for
}i=2,\ldots,n-1,$

where
$\overline{\Psi}(\overline{a}):=\overline{R}^{\overline{J}}(-,\overline{a})$
for $\overline{a}\in{}_{\overline{J}}\mathcal{O}(K/C)_{\overline{J}}$, and
$\overline{u}_{i}\in\mathcal{O}(K/C)^{*}$ is the derivation mapping
$\overline{X}_{j}$ to $\delta_{n+1-i,j}$, $2\leq i,j\leq n-1$. (Note that it
makes sense to use the notation $\overline{u}_{i}$ since $u_{i}(y_{1})=0$ for
$i=2,\ldots n-1$.)

Recall from Corollary 3.5(3), and the remarks preceding it, that
${}_{J}\mathcal{O}(G/C)_{J}$ is connected. This implies by a result of
Schneider [Schn] \- see [Mon, Theorem 8.4.8 and Example (2) in remarks
following] \- that the exact sequence of Hopf algebras

$0\rightarrow\mathcal{O}(G/K)\rightarrow{}_{J}\mathcal{O}(G/C)_{J}\rightarrow{}_{\overline{J}}\mathcal{O}(K/C)_{\overline{J}}\rightarrow
0$

is cleft. By definition, this means that there is an invertible cleaving map
$\gamma$ of ${}_{\overline{J}}\mathcal{O}(K/C)_{\overline{J}}-$comodules,

$\gamma:{}_{\overline{J}}\mathcal{O}(K/C)_{\overline{J}}\xrightarrow{1:1}{}_{J}\mathcal{O}(G/C)_{J}.$

Thus, we can use $\gamma$ to view
${}_{\overline{J}}\mathcal{O}(K/C)_{\overline{J}}$ as a subspace of
${}_{J}\mathcal{O}(G/C)_{J}$, so that there is a crossed product decomposition

${}_{J}\mathcal{O}(G/C)_{J}=\mathcal{O}(G/K)\\#_{\sigma}{}_{\overline{J}}\mathcal{O}(K/C)_{\overline{J}}\subset{}_{J}\mathcal{O}(G)_{J}$

by [Mon, Theorem 7.2.2].

Let $\iota:K/C\hookrightarrow G/C$ denote the inclusion morphism, so we have a
surjective Hopf algebra map
$\iota^{*}:\mathcal{O}(G/C)\twoheadrightarrow\mathcal{O}(K/C)$, and

$\iota^{*}\circ\gamma={\rm
id}_{|{}_{\overline{J}}\mathcal{O}(K/C)_{\overline{J}}}.$

Denote the canonical surjection from $U(\mathfrak{k})$ onto
$U(\mathfrak{k}/\mathfrak{c})$ by $\pi$. Then, by definition, for every
$\overline{x}\in{}_{\overline{J}}\mathcal{O}(K/C)_{\overline{J}}$, we have

$\pi\circ\Psi(\gamma(\overline{x}))=\overline{\Psi}(\iota^{*}\gamma(\overline{x}))=\overline{\Psi}(\overline{x}).$

Thus, lifting back along $\gamma$ we obtain elements

$z_{i}:=\gamma(\overline{z}_{i})\in{}_{J}\mathcal{O}(G/C)_{J},\,\,\,2\leq
i\leq n-1,$

such that

$\Psi(z_{i})\left(y_{1}{}_{J}\mathcal{O}(G/C)_{J}\right)=0,\,\,\,2\leq i\leq
n-1,$

since $y_{1}$ is in the radical of $R^{J}_{|{}_{J}\mathcal{O}(G/C)_{J}}$. Then
the induced map on ${}_{J}\mathcal{O}(G/C)_{J}$ is

(4.17)
$\Psi(z_{i})_{|{}_{J}\mathcal{O}(G/C)_{J}}=\overline{\Psi}(\iota^{*}\circ\gamma(\overline{z}_{i}))\circ\iota^{*}=\overline{\Psi}(\overline{z}_{i})\circ\iota^{*}=\overline{u}_{i}\circ\iota^{*}=u_{i\,|{}_{J}\mathcal{O}(G/C)_{J}}.$

The second and third lemmas together will prove that $u_{2},\ldots,u_{n-1}$
are in $\mathrm{im}(\Psi)$.

######  Lemma 4.6.

For each $i=2,\ldots,n-1$, after replacing $z_{i}$ by
$\widehat{z_{i}}\in{}_{J}\mathcal{O}(G/C)_{J}$,

$\Psi(\widehat{z_{i}})=u_{i}\in\mathrm{im}(\Psi),\,\,\,2\leq i\leq n-1.$

###### Proof.

It suffices to prove that after suitable adjustments changing the elements
$z_{i}$ to $\widehat{z_{i}}\in{}_{J}\mathcal{O}(G/C)_{J}$ for every $2\leq
i\leq n-1$, we have

$\displaystyle\Psi(\widehat{z_{2}}),\ldots,$
$\displaystyle\Psi(\widehat{z_{n-1}})\textit{ are linearly independent
elements of }\mathfrak{g},$ $\displaystyle\textit{still satisfying
(\ref{capture}), and with }\Psi(\widehat{z_{i}})(y_{n})=0.$

First, for $i=2,\ldots,n-1$, define
$\tau_{i}:=R^{J}(y_{n},z_{i})\in\mathbb{C}$, and set

$\widetilde{z_{i}}:=z_{i}-\tau_{i}y_{1}.$

Then for every $w\in{}_{J}\mathcal{O}(G/C)_{J}$, and $2\leq i\leq n-1$, we
have

(4.18) $\Psi(\widetilde{z_{i}})(w)=R^{J}(w,\widetilde{z_{i}})=R^{J}(w,z_{i})$

since $y_{1}$ lies in the radical of $({}_{J}\mathcal{O}(G/C)_{J},R^{J})$ by
Lemma 4.3(1). Moreover,

(4.19)
$R^{J}(y_{n},\widetilde{z_{i}})=R^{J}(y_{n},z_{i})-\tau_{i}R^{J}(y_{n},y_{1})=\tau_{i}-\tau_{i}=0$

for every $i=2,\ldots,n-1$, by Lemma 4.3(5).

To complete the proof it remains to show that for every $2\leq i\leq n-1$ we
can adjust $\widetilde{z_{i}}$ to an element
$\widehat{z_{i}}\in{}_{J}\mathcal{O}(G/C)_{J}$, without damaging the versions
for $\widehat{z_{i}}$ of (4.18) and (4.19), but such that
$\Psi(\widehat{z_{i}})\in\mathfrak{g}$; that is, that

(4.20) $\Psi(\widehat{z_{i}})(\mathfrak{m}^{2})=0.$

Fix $2\leq i\leq n-1$. We already know by Proposition 4.1, that there exists
$\ell>1$ such that

(4.21) $\Psi(\widetilde{z_{i}})(\mathfrak{m}^{\ell})=0.$

Also, by (4.17) and (4.18),
$\Psi(\widetilde{z_{i}})_{|{}_{J}\mathcal{O}(G/C)_{J}}=u_{i\,|{}_{J}\mathcal{O}(G/C)_{J}}$,
and $u_{i\,|{}_{J}\mathcal{O}(G/C)_{J}}$ is a derivation of
$\mathcal{O}(G/C)$, hence we have

(4.22)
$\Psi(\widetilde{z_{i}})\left(\mathfrak{m}^{2}\cap{}_{J}\mathcal{O}(G/C)_{J}\right)=0.$

For each $n$-tuple ${\bf b}=(b_{1},b_{2},\dots,b_{n})\in(\mathbb{Z}_{\geq
0})^{n}$ define, using the usual commutative product in $\mathcal{O}(G)$,

(4.23) ${\bf y}^{{\bf b}}:=y_{1}^{b_{1}}X_{2}^{b_{2}}\cdots
X_{n-1}^{b_{n-1}}y_{n}^{b_{n}}.$

In view of (4.21) and (4.22) we must show that for each $i=2,\ldots,n-1$,
$\widetilde{z_{i}}$ can be modified to $\widehat{z_{i}}$ so that
$\Psi(\widehat{z_{i}})$ maps to $0$ every monomial in the set

(4.24) $\mathcal{M}:=\left\\{{\bf y}^{{\bf
b}}\,\bigg{|}\,2\leq\sum_{i=1}^{n}b_{i}<\ell,\,b_{n}\neq
0\right\\}\subset\mathfrak{m}^{2}.$

Denote by $\mathcal{N}$ the set of $(n-1)$-tuples ${\bf
a}=(a_{2},\dots,a_{n})\in(\mathbb{Z}_{\geq 0})^{n-1}$ such that

$2\leq\sum_{i=2}^{n}a_{i}<\ell\,\,\,{\rm and}\,\,\,a_{n}\neq 0.$

Using the usual product in $\mathcal{O}(G)$, for each ${\bf a}\in\mathcal{N}$
define the element

${\bf z}_{{\bf a}}:=\frac{1}{a_{n}!\cdots
a_{2}!}y_{1}^{a_{n}}\widetilde{z_{2}}^{a_{n-1}}\cdots\widetilde{z_{n-1}}^{a_{2}}\in{}_{J}\mathcal{O}(G)_{J}.$

Set $\mathcal{A}:=\\{{\bf z}_{\bf a}\mid{\bf a}\in\mathcal{N}\\}.$

To complete the proof of Lemma 4.6, we therefore need:

######  Lemma 4.7.

  1. (1)

Let ${\bf b}=(b_{1},\ldots,b_{n})\in(\mathbb{Z}_{\geq 0})^{n}$ with $b_{1}>0$.
For every ${\bf z}_{{\bf a}}\in\mathcal{A}$,

$R^{J}({\bf y}^{{\bf b}},{\bf z}_{{\bf a}})=0.$

  2. (2)

For every ${\bf a}\in\mathcal{A}$, and every $n$-tuple ${\bf
b}=(0,b_{2},\ldots,b_{n})\in(\mathbb{Z}_{\geq 0})^{n}$, we have $R^{J}({\bf
y}^{{\bf b}},{\bf z}_{{\bf a}})=1$ when ${\bf b}=(0,{\bf a})$ and $0$
otherwise.

###### Proof.

(1) By hypothesis, ${\bf y}^{{\bf b}}=y_{1}\cdot{\bf z}$ for some monomial
${\bf z}\in{}_{J}\mathcal{O}(G)_{J}$. Thus

$R^{J}({\bf y}^{{\bf b}},{\bf z}_{{\bf a}})=R^{J}(y_{1}\cdot{\bf z},{\bf
z}_{{\bf a}})=\sum R^{J}(y_{1},\left({\bf z}_{{\bf a}}\right)_{1})R^{J}({\bf
z},\left({\bf z}_{{\bf a}}\right)_{2})=0,$

where the second equality follows from Definition 2.4(2), and the final
equality holds because all the terms $R^{J}(y_{1},({\bf z}_{{\bf a}})_{1})$
are $0$ by Lemma 4.3(1).

(2) Fix ${\bf a}\in\mathcal{A}$. Then for any ${\bf
b}=(0,b_{2},\ldots,b_{n})\in(\mathbb{Z}_{\geq 0})^{n}$, using the centrality
of $y_{1}$ in ${}_{J}\mathcal{O}(G)_{J}$ (Lemma 4.3(2)) for the second
equality, and Definition 2.4(1) for the third, we have (omitting the summation
symbol)

$\displaystyle a_{n}!\cdots a_{2}!R^{J}({\bf y}^{{\bf b}},{\bf z}_{{\bf a}})$
$\displaystyle=$ $\displaystyle R^{J}\left(X_{2}^{b_{2}}\cdots
X_{n-1}^{b_{n-1}}y_{n}^{b_{n}},y_{1}^{a_{n}}\widetilde{z_{2}}^{a_{n-1}}\cdots\widetilde{z_{n-1}}^{a_{2}}\right)$
$\displaystyle=$ $\displaystyle R^{J}\left(X_{2}^{b_{2}}\cdots
X_{n-1}^{b_{n-1}}y_{n}^{b_{n}},\widetilde{z_{2}}^{a_{n-1}}\cdots\widetilde{z_{n-1}}^{a_{2}}\cdot
y_{1}^{a_{n}}\right)$ $\displaystyle=$ $\displaystyle
R^{J}\left(\left(X_{2}^{b_{2}}\cdots
X_{n-1}^{b_{n-1}}y_{n}^{b_{n}}\right)_{1},y_{1}^{a_{n}}\right)R^{J}\left(\left(X_{2}^{b_{2}}\cdots
X_{n-1}^{b_{n-1}}y_{n}^{b_{n}}\right)_{2},\widetilde{z_{2}}^{a_{n-1}}\cdots\widetilde{z_{n-1}}^{a_{2}}\right)$
$\displaystyle=$
$\displaystyle\Psi\left(y_{1}\right)^{a_{n}}\left(\left(X_{2}^{b_{2}}\cdots
X_{n-1}^{b_{n-1}}y_{n}^{b_{n}}\right)_{1}\right)\Psi\left(\widetilde{z_{2}}^{a_{n-1}}\cdots\widetilde{z_{n-1}}^{a_{2}}\right)\left(\left(X_{2}^{b_{2}}\cdots
X_{n-1}^{b_{n-1}}y_{n}^{b_{n}}\right)_{2}\right)$ $\displaystyle=$
$\displaystyle u_{1}^{a_{n}}\left(\left(X_{2}^{b_{2}}\cdots
X_{n-1}^{b_{n-1}}y_{n}^{b_{n}}\right)_{1}\right)\Psi\left(\widetilde{z_{2}}^{a_{n-1}}\cdots\widetilde{z_{n-1}}^{a_{2}}\right)\left(\left(X_{2}^{b_{2}}\cdots
X_{n-1}^{b_{n-1}}y_{n}^{b_{n}}\right)_{2}\right),$

where the last equality follows from Lemma 4.5. Using (4.11) and the
definition of $u_{1}$ to calculate the terms
$u_{1}^{a_{n}}\left(\left(X_{2}^{b_{2}}\cdots
X_{n-1}^{b_{n-1}}y_{n}^{b_{n}}\right)_{1}\right)$, it is straightforward to
verify that if $R^{J}({\bf y}^{{\bf b}},{\bf z}_{{\bf a}})\neq 0$, then
$b_{n}=a_{n}$ and

$a_{n}!\cdots a_{2}!R^{J}({\bf y}^{{\bf b}},{\bf z}_{{\bf
a}})=a_{n}!\Psi\left(\widetilde{z_{2}}^{a_{n-1}}\cdots\widetilde{z_{n-1}}^{a_{2}}\right)\left(X_{2}^{b_{2}}\cdots
X_{n-1}^{b_{n-1}}\right)\neq 0.$

Therefore, by (4.17) and (4.18), we have

$a_{n}!\cdots a_{2}!R^{J}({\bf y}^{{\bf b}},{\bf z}_{{\bf
a}})=a_{n}!u_{2}^{a_{n-1}}\cdots u_{n-1}^{a_{2}}\left(X_{2}^{b_{2}}\cdots
X_{n-1}^{b_{n-1}}\right)\neq 0.$

This implies that if $R^{J}({\bf y}^{{\bf b}},{\bf z}_{{\bf a}})\neq 0$, then
${\bf a}={\bf b}$ and $R^{J}({\bf y}^{{\bf b}},{\bf z}_{{\bf a}})=1$, as
claimed. ∎

Return now to the proof of Lemma 4.6. For $i=2,\ldots,n-1$, set

$\widetilde{z_{i}}^{\prime}:=\sum_{{\bf a}\in\mathcal{A}}R^{J}({\bf y}^{{\bf
a}},\widetilde{z_{i}}){\bf z}_{{\bf a}}\in{}_{J}\mathcal{O}(G/C)_{J}.$

By Lemma 4.7, for any monomial ${\bf y}^{{\bf b}}$ in the set $\mathcal{M}$
defined in (4.24), we have

$\displaystyle\Psi(\widetilde{z_{i}}-\widetilde{z_{i}}^{\prime})({\bf y}^{{\bf
b}})=\Psi(\widetilde{z_{i}})({\bf y}^{{\bf
b}})-\Psi(\widetilde{z_{i}}^{\prime})({\bf y}^{{\bf b}})$ $\displaystyle=$
$\displaystyle R^{J}({\bf y}^{{\bf b}},\widetilde{z_{i}})-\sum_{{\bf
a}\in\mathcal{A}}R^{J}({\bf y}^{{\bf a}},\widetilde{z_{i}})R^{J}({\bf y}^{{\bf
b}},{\bf z}_{{\bf a}})=R^{J}({\bf y}^{{\bf b}},\widetilde{z_{i}})-R^{J}({\bf
y}^{{\bf b}},\widetilde{z_{i}})=0.$

Thus, replacing $\widetilde{z_{i}}$ by
$\widehat{z_{i}}:=\widetilde{z_{i}}-\widetilde{z_{i}}^{\prime}$ for each
$i=2,\ldots,n-1$ completes the proof of Lemma 4.6, since now
$\Psi(\widehat{z_{i}})$ coincides with $u_{i}$ on every monomial in
$\mathcal{O}(G)$. ∎

For Theorem 4.4 it thus remains only to show that

######  Lemma 4.8.

$u_{n}\in\mathrm{im}(\Psi)$.

###### Proof.

By Proposition 4.1, $\Psi(y_{n})$ lies in $U(\mathfrak{g})$, so we can fix an
integer ${\ell}\geq 2$ such that $\Psi(y_{n})$ vanishes on
$\mathfrak{m}^{\ell}$.

Retain the notation (4.23), and for each ${\bf b}\in(\mathbb{Z}_{\geq
0})^{n}$, let ${\bf y}^{{\bf b*}}$ denote the linear form on $\mathcal{O}(G)$
which is $1$ on ${\bf y}^{{\bf b}}$ and $0$ on every monomial ${\bf y}^{{\bf
i}}$ such that ${\bf i}\neq{\bf b}$. Let ${\bf
b}=(0,b_{2},\dots,b_{n})\in(\mathbb{Z}_{\geq 0})^{n}$. Then ${\bf y}^{{\bf
b*}}$ vanishes on some power of the ideal generated by
$y_{1},X_{2},\ldots,X_{n-1},y_{n}$, so that

(4.25) ${\bf y}^{{\bf b*}}\in U(\mathfrak{k})\subset U(\mathfrak{g}).$

Hence, by Lemmas 4.5 and 4.6, for each ${\bf
b}=(0,i_{2},\dots,i_{n})\in(\mathbb{Z}_{\geq 0})^{n}$, there exists an element
${\bf z}_{{\bf b}}\in{}_{J}\mathcal{O}(G/C)_{J}$ such that

(4.26) ${\bf y}^{{\bf b*}}=\Psi({\bf z}_{{\bf b}}).$

For each ${\bf b}=(b_{1},\ldots,b_{n})\in(\mathbb{Z}_{\geq 0})^{n}$, define
$|{\bf b}|:=\sum_{i=1}^{n}b_{i}$. Set

$z:=\sum_{{\bf b}=(0,b_{2},\dots,b_{n})\in(\mathbb{Z}_{\geq
0})^{n},\,2\leq|{\bf b}|<{\ell}}R^{J}({\bf y}^{{\bf b}},y_{n}){\bf z}_{{\bf
b}}\in{}_{J}\mathcal{O}(G/C)_{J}.$

Then by (4.25) and (4.26), we have

$\Psi(z)=\sum_{{\bf b}=(0,b_{2},\dots,b_{n})\in(\mathbb{Z}_{\geq
0})^{n},\,2\leq|{\bf b}|<{\ell}}R^{J}({\bf y}^{{\bf b}},y_{n})\Psi({\bf
z}_{{\bf b}})\in U(\mathfrak{k}).$

We claim that $\Psi(z-y_{n})$ vanishes on $\mathfrak{m}^{2}$; that is,

$\Psi(z-y_{n})({\bf y}^{{\bf d}})=0$

for every ${\bf d}\in(\mathbb{Z}_{\geq 0})^{n}$ with $|{\bf d}|\geq 2$.
Indeed, if ${\bf d}=(0,d_{2},\dots,d_{n})\in(\mathbb{Z}_{\geq 0})^{n}$ such
that $2\leq|{\bf d}|<\ell$, then

$\displaystyle\Psi(z)({\bf y}^{{\bf d}})=\sum_{{\bf
b}=(0,b_{2},\dots,b_{n})\in(\mathbb{Z}_{\geq 0})^{n},\,2\leq|{\bf
b}|<{\ell}}R^{J}({\bf y}^{{\bf b}},y_{n})\Psi({\bf z}_{{\bf b}})({\bf y}^{{\bf
d}})$ $\displaystyle=$ $\displaystyle\sum_{{\bf
b}=(0,b_{2},\dots,b_{n})\in(\mathbb{Z}_{\geq 0})^{n},\,2\leq|{\bf
b}|<{\ell}}R^{J}({\bf y}^{{\bf b}},y_{n})\langle{\bf y}^{{\bf b*}},{\bf
y}^{{\bf d}}\rangle=R^{J}({\bf y}^{{\bf d}},y_{n})=\Psi(y_{n})(y^{{\bf d}}).$

Next if ${\bf b}=(b_{1},\ldots,b_{n})\in(\mathbb{Z}_{\geq 0})^{n}$ such that
$2\leq|{\bf b}|<\ell$ and $b_{1}\geq 1$, then by Lemma 4.3(3),

$\Psi(z)({\bf y}^{{\bf b}})=0=\Psi(y_{n})({\bf y}^{{\bf b}}).$

Finally, it is clear that if $|{\bf b}|\geq\ell$, then

$\Psi(z)({\bf y}^{{\bf b}})=0=\Psi(y_{n})({\bf y}^{{\bf b}}).$

Thus, $\Psi(z-y_{n})$ vanishes on $\mathfrak{m}^{2}$, as claimed.

Finally, since ${}_{J}\mathcal{O}(G)_{J}$ is cotriangular and $y_{1}$ is
primitive, [Maj, Lemma 2.2.2] shows that

$R^{J}(y_{n},y_{1})=(R^{J})^{-1}(y_{1},y_{n})=R^{J}(S(y_{1}),y_{n})=-R^{J}(y_{1},y_{n}).$

Hence, by Lemma 4.3(5),

(4.27) $R^{J}(y_{1},y_{n})=-1.$

Thus, $\Psi(z-y_{n})(y_{1})=R^{J}(y_{1},z)-R^{J}(y_{1},y_{n})=1$, so
$\Psi(z-y_{n})=u_{n}$, as desired. ∎

The theorem follows now from Lemmas 4.5, 4.6, and 4.8. ∎

Combining Proposition 4.1 and Theorem 4.4 yields

######  Theorem 4.9.

Let $G$ be a unipotent affine algebraic group, and let $J$ be a Hopf
$2$-cocycle for $G$ such that the $R$-form $R^{J}:=J_{21}^{-1}\ast J$ is
nondegenerate. Then the map $\Psi$ defined at (5.15) yields an isomorphism of
Hopf algebras

${}_{J}\mathcal{O}(G)_{J}\cong\left(U(\mathfrak{g})^{J}\right)^{{\rm cop}},$

where $U(\mathfrak{g})^{J}=U(\mathfrak{g})$ as algebras, with coproduct given
by $\Delta^{J}(\cdot)=J^{-1}*\Delta(\cdot)*J$, where $\Delta$ is the standard
cocommutative coproduct of $U(\mathfrak{g})$. ∎

######  Remark 4.10.

Theorem 4.9 is [G3, Theorem 3.4]. However, the proof given in [G3] is
incomplete. More precisely, the proof in [G3] assumes a positive answer to
Question 4.11 below, which we are unable to prove. ∎

######  Question 4.11.

Set $\mathfrak{m}:=\mathcal{O}(G)^{+}$. Is it true that every power of
$\mathfrak{m}$ in the algebra ${}_{J}\mathcal{O}(G)_{J}$ contains a power of
$\mathfrak{m}$ in the algebra $\mathcal{O}(G)$? ∎

######  Corollary 4.12.

Under the hypotheses of Theorem 4.9, there are equivalences of tensor
categories

${\rm Rep}({}_{J}\mathcal{O}(G)_{J})\xrightarrow{\cong}{\rm
Rep}(U(\mathfrak{g}))\xrightarrow{\cong}{\rm Rep}(\mathfrak{g}).$

###### Proof.

By Theorem 4.9, it suffices to prove that ${\rm Rep}(U(\mathfrak{g}))\cong{\rm
Rep}(U(\mathfrak{g})^{J})$ as tensor categories. To this end, note that the
functor

$\left({\rm Id},{\rm J}\right):{\rm
Rep}(U(\mathfrak{g}))\xrightarrow{\cong}{\rm Rep}(U(\mathfrak{g})^{J}),$

where ${\rm J}_{V,W}:=J_{f}(r,\hbar)_{\mid V\otimes W}:V\otimes
W\xrightarrow{\cong}V\otimes W$ (4.4), is a well defined tensor equivalence. ∎

##  5\. Structure of twisted unipotent groups - the general case

We assume throughout this section that $G$ is a unipotent algebraic group over
$\mathbb{C}$ of dimension $n$, and that $J$ is a Hopf $2$-cocycle for $G$ with
corresponding $R$-form $R^{J}:=J_{21}^{-1}*J$ for ${}_{J}\mathcal{O}(G)_{J}$.
Let $T$ be a chosen representative of the conjugacy class of supports of $J$,
and $(\mathfrak{t},\omega)$ the Lie algebra of $T$, so $(\mathfrak{t},\omega)$
is a quasi-Frobenius Lie algebra as explained in $\S$2.4.

###  5.1. Crossed product decomposition

Recall that since $T$ is a closed subgroup of $G$, $\mathcal{O}(G/T)$ and
$\mathcal{O}(T\backslash G)$ are left and right coideal subalgebras of
$\mathcal{O}(G)$ respectively, and hence are polynomial subalgebras in
$\mathrm{dim}(G)-\mathrm{dim}(T)$ variables; see for example [BG].

Following Proposition 2.11, there is an epimorphism $\pi$ of Hopf algebras
from ${}_{J}\mathcal{O}(G)_{J}$ onto ${}_{J}\mathcal{O}(T)_{J}$ with kernel
$I_{R^{J}}$, the radical of $R^{J}$. The notation (2.4) is used below for the
algebras ${}_{J}\mathcal{O}(G)$ and $\mathcal{O}(G)_{J}$, and extended in the
obvious way to subalgebras of these algebras. Finally, recall that $S^{J}$
denotes the antipode of ${}_{J}\mathcal{O}(G)_{J}$.

######  Lemma 5.1.

With the above notation, we have the following:

  1. (1)

The algebra ${}_{J}\mathcal{O}(G/T)_{J}={}_{J}\mathcal{O}(G/T)$ is a left
coideal subalgebra of ${}_{J}\mathcal{O}(G)_{J}$, and a subalgebra of
${}_{J}\mathcal{O}(G)$. Correspondingly, ${}_{J}\mathcal{O}(T\backslash
G)_{J}=\mathcal{O}(T\backslash G)_{J}$ is a right coideal subalgebra of
${}_{J}\mathcal{O}(G)_{J}$, and a subalgebra of $\mathcal{O}(G)_{J}$.

  2. (2)

$I_{R^{J}}=(\mathcal{O}(T\backslash
G)_{J})^{+}{}_{J}\mathcal{O}(G)_{J}=({}_{J}\mathcal{O}(G/T))^{+}{}_{J}\mathcal{O}(G)_{J}.$

  3. (3)

$S^{J}(\mathcal{O}(T\backslash G)_{J})={}_{J}\mathcal{O}(G/T)$, and vice
versa.

###### Proof.

(1) The maps $\pi:\mathcal{O}(G)\twoheadrightarrow\mathcal{O}(T)$ and $\Delta$
are the same when applied to ${}_{J}\mathcal{O}(G)$. Therefore, since
${}_{J}\mathcal{O}(G/T)_{J}$ is defined using the coalgebra structure, we see
that, as vector subspaces of $\mathcal{O}(G)$, we have

(5.1)
${}_{J}\mathcal{O}(G/T)_{J}=\mathcal{O}(G/T)=\\{f\in\mathcal{O}(G)\mid(\mathrm{id}\otimes\pi)\circ\Delta(f)=f\otimes\overline{1}\\}.$

In particular, ${}_{J}\mathcal{O}(G/T)_{J}$ is a left coideal subalgebra of
${}_{J}\mathcal{O}(G)_{J}$ since $\mathcal{O}(G/T)$ is a left coideal
subalgebra of $\mathcal{O}(G)$. Hence, for $a,b\in{}_{J}\mathcal{O}(G/T)_{J}$,
we have

$\displaystyle m_{J}(a\otimes b)$ $\displaystyle=\sum
J^{-1}(a_{1},b_{1})a_{2}b_{2}J(a_{3},b_{3})$ $\displaystyle=\sum
J^{-1}(a_{1},b_{1})a_{2}b_{2}\epsilon(a_{3})\epsilon(b_{3})$
$\displaystyle=\sum J^{-1}(a_{1},b_{1})a_{2}b_{2}.$

That is, ${}_{J}\mathcal{O}(G/T)_{J}={}_{J}\mathcal{O}(G/T)$. This proves the
first half of (1); the second half is exactly similar.

(2), (3) These follow from (5.1) by standard properties of Hopf ideals in
connected Hopf algebras due to Masuoka, see e.g. [BG, Proposition 2.3]. ∎

The Hopf algebra $U(\mathfrak{t})^{J}$ featuring in the next theorem is
$U(\mathfrak{t})$ with twisted coproduct, as defined in Theorem 4.9.

######  Theorem 5.2.

([G3, Theorem 3.8]) With the notation and hypotheses given at the start of
$\S$5, ${}_{J}\mathcal{O}(G)_{J}$ decomposes as a crossed product,

$\displaystyle{}_{J}\mathcal{O}(G)_{J}$
$\displaystyle\cong{}_{J}\mathcal{O}(G/T)\\#_{\sigma}{}_{J}\mathcal{O}(T)_{J}\cong\mathcal{O}(T\backslash
G)_{J}\\#_{\sigma^{\prime}}{}_{J}\mathcal{O}(T)_{J}$
$\displaystyle\cong{}_{J}\mathcal{O}(G/T)\\#_{\sigma}U(\mathfrak{t})^{J}\cong\mathcal{O}(T\backslash
G)_{J}\\#_{\sigma^{\prime}}U(\mathfrak{t})^{J},$

where $\sigma$ and $\sigma^{\prime}$ are invertible $2$-cocycles from
$({}_{J}\mathcal{O}(T)_{J})^{\otimes 2}$ to ${}_{J}\mathcal{O}(G/T)$ and
$\mathcal{O}(T\backslash G)_{J}$, respectively. ∎

######  Remarks 5.3.

(1) Suppose in the above that $T$ is normal in $G$. Then of course
$\mathcal{O}(G/T)$ is a Hopf subalgebra of $\mathcal{O}(G)$, and from the
proof of Lemma 5.1 we see that ${}_{J}\mathcal{O}(G/T)_{J}=\mathcal{O}(G/T)$;
in fact by Proposition 5.4 below, it is a central (polynomial) Hopf subalgebra
of ${}_{J}\mathcal{O}(G)_{J}$. But in general, ${}_{J}\mathcal{O}(G/T)$ is not
commutative, as will be seen in Example 7.7.

(2) We do not know whether the $2$-cocycle $\sigma$ in Theorem 5.2 can be non-
trivial. But the action of ${}_{J}\mathcal{O}(T)_{J}$ on
${}_{J}\mathcal{O}(G/T)$ can be non-trivial, as is shown by Examples 7.7 and
7.8 below. ∎

###  5.2. Consequences of Theorem 5.2

Although we are unable to give a precise description of the centre of
${}_{J}\mathcal{O}(G)_{J}$ for unipotent affine algebraic groups $G$, we can
at least identify an important central subalgebra:

######  Proposition 5.4.

([G3, Lemma 3.7]) If $G$ is a unipotent affine algebraic group with a Hopf
$2$-cocycle $J$ whose support is $T$, then the algebra
$\mathcal{O}(T\backslash G/T)$ of functions which are constant on the double
cosets of $T$ in $G$ is contained in the centre of ${}_{J}\mathcal{O}(G)_{J}$.

###### Proof.

By [G2, Theorem 3.5] (without needing the hypothesis of unipotence), there are
algebra embeddings

$\mathcal{O}(G/T)=\mathcal{O}(G/T)_{J}\subseteq
Z(\mathcal{O}(G)_{J})\,\,\,{\rm and}\,\,\,\mathcal{O}(T\backslash
G)={}_{J}\mathcal{O}(T\backslash G)\subseteq Z({}_{J}\mathcal{O}(G)),$

where the equalities emphasise that the multiplication is the same in the
linked algebras. Hence, we have

(5.2) $\mathcal{O}(T\backslash
G/T)\subseteq\mathcal{O}(G/T)=\mathcal{O}(G/T)_{J}\subseteq
Z(\mathcal{O}(G)_{J})$

and

(5.3) $\mathcal{O}(T\backslash G/T)\subseteq\mathcal{O}(T\backslash
G)={}_{J}\mathcal{O}(T\backslash G)\subseteq Z(_{J}\mathcal{O}(G)).$

Moreover (5.2) and (5.3) imply that

(5.4) $\Delta(\mathcal{O}(T\backslash G/T))\subseteq\mathcal{O}(T\backslash
G)\otimes\mathcal{O}(G/T)\subseteq
Z({}_{J}\mathcal{O}(G)\otimes\mathcal{O}(G)_{J})),$

since $\mathcal{O}(T\backslash G)$ and $\mathcal{O}(G/T)$ are right and left
coideal subalgebras of $\mathcal{O}(G)$, respectively. Thus, the result
follows from (5.4), and the fact that by Lemma 2.2(3), $\Delta$ embeds
${}_{J}\mathcal{O}(G)_{J}$ in ${}_{J}\mathcal{O}(G)\otimes\mathcal{O}(G)_{J}$.
∎

Recall that an ideal $I$ of a ring $A$ is called _polycentral_ if there is a
finite subset $\\{y_{1},\ldots,y_{t}\\}\subset I$, which generate $I$ as a
right ideal, such that for all $i=1,\ldots,t$,

$y_{i}+\left(\sum_{j<i}y_{j}A\right)\textit{ is central in
}A/\left(\sum_{j<i}y_{j}A\right).$

The ideal $I$ and its corresponding generating set are called _regular
polycentral_ if in addition each $y_{i}$ is not a zero divisor in
$A/(\sum_{j<i}y_{j}A)$.

######  Theorem 5.5.

Let $G$ be a unipotent affine algebraic group of dimension $n$, and $J$ a Hopf
$2$-cocycle for $G$. Then the augmentation ideal
${}_{J}\mathcal{O}(G)_{J}^{+}$ of ${}_{J}\mathcal{O}(G)_{J}$ has a regular
polycentral generating set of $n$ elements.

###### Proof.

By Theorem 5.2, with $T$ as usual denoting the support of $J$, and
$\mathfrak{t}$ the Lie algebra of $T$, we have

(5.5)
${}_{J}\mathcal{O}(G)_{J}\cong{}_{J}\mathcal{O}(G/T)_{J}\\#_{\sigma}U(\mathfrak{t})^{J}.$

If $T=G$ then ${}_{J}\mathcal{O}(G)_{J}\cong U(\mathfrak{t})$ as an algebra,
and the result follows by taking as generators of
${}_{J}\mathcal{O}(G)_{J}^{+}$ a basis of $\mathfrak{t}$ obtained from its
upper central series. So we may suppose without loss of generality that
$T\subsetneq G$. By standard properties of unipotent groups in characteristic
$0$, the normaliser $N_{G}(T)$ is a closed subgroup of $G$ strictly containing
$T$. Thus, $N_{G}(T)/T$ is a non-trivial unipotent group, so we can find a
chain of subgroups

$1\subset T_{1}/T\subset T_{2}/T\subset\cdots\subset T_{\ell}/T=N_{G}(T)/T,$

with each subgroup normal in the next, and successive factors isomorphic to
$(\mathbb{C},+)$. Repeating with $N_{G}(T)$ in place of $T$, we eventually
obtain a normal subgroup $N$ of $G$ with $T\subseteq N$ and
$G/N\cong(\mathbb{C},+)$.

By the discussion at the start of $\S$4.2, the factor $G/N$ of $G$ corresponds
to a Hopf subalgebra $H_{1}=\mathbb{C}[X]$ of $\mathcal{O}(G)$. Here $X$ is
primitive, and for all $g\in G$, $X$ is a constant function on the coset
$gN=Ng$. Moreover, for all $g\in G$, we have

$TgT\subset NgN=gN,$

so that $X$ is constant on the double coset $TgT$. In particular, by
Proposition 5.4, $X$ is central in ${}_{J}\mathcal{O}(G)_{J}$. Since
$X\in{}_{J}\mathcal{O}(G/T)^{+}$, we can factor by the Hopf ideal
$X{}_{J}\mathcal{O}(G)_{J}$ of ${}_{J}\mathcal{O}(G)_{J}$. The ideal
$X{}_{J}\mathcal{O}(G)_{J}$ is, by construction, contained in
$({}_{J}\mathcal{O}(G/T))^{+}_{J}\mathcal{O}(G)_{J}$. Hence, Lemma 5.1(2)
applies, and we obtain a quotient Hopf algebra of GK-dimension $n-1$, namely
${}_{J}\mathcal{O}(N)_{J}$. Now one inducts on $n$ to obtain the theorem. ∎

######  Corollary 5.6.

Let $G$ be a unipotent affine algebraic group, and $J$ a Hopf $2$-cocycle for
$G$.

  1. (1)

$(S^{J})^{2}=\mathrm{id}$.

  2. (2)

${}_{J}\mathcal{O}(G)_{J}$ is Calabi-Yau; that is, its Nakayama automorphism
is trivial.

###### Proof.

Suppose that $\mathrm{dim}(G)=n$, and for brevity, denote
${}_{J}\mathcal{O}(G)_{J}$ by $H$. Since $H^{+}$ has a regular polycentral
sequence of $n$ generators by Theorem 5.5, it follows from the change of rings
theorem [BZ, Lemma, $\S$6.5] that, with $\mathbb{C}$ denoting the trivial
$H$-module, we have

(5.6) $\mathrm{Ext}^{i}_{H}(\mathbb{C},H)\cong\delta_{in}\mathbb{C},$

where the isomorphism is of $(H,H)$-bimodules.

Therefore, since ${}_{J}\mathcal{O}(G)_{J}$ is Artin-Schelter regular by
Corollary 3.5(2), it follows from [BZ, Proposition 4.5] that the Nakayama
automorphism $\nu$ of $H$ is given by

(5.7) $\nu=(S^{J})^{2}.$

On the other hand, since $H$ is an IHOE by Proposition 3.2(5), its only units
are the nonzero scalars $\mathbb{C}^{\times}$, so by (5.6) and [BZ, Corollary
4.6], we have

(5.8) $(S^{J})^{4}=\mathrm{id}.$

However, we know from [BOZZ, Theorem 3.2(viii)] that the antipode of an IHOE
over $\mathbb{C}$ is either of infinite order or has order $2$. The first
option is ruled out by (5.8), so (1) is proved, and then (2) follows from
(5.7). ∎

######  Remark 5.7.

If $G$ is not unipotent, $S^{J}$ may not be involutive, as demonstrated in
[EG1, Example 5.2]. ∎

###  5.3. Stratification of representations by double cosets

We keep the notation as specified at the start of $\S$5. When $J$ is trivial,
so that ${}_{J}\mathcal{O}(G)_{J}=\mathcal{O}(G)$ is commutative and
$T=\\{1\\}$, the simple $\mathcal{O}(G)$-modules are of course parametrized by
the points of $G$. We generalize this to arbitrary $J$ here.

The starting point is Proposition 5.4. By [MR, Proposition 9.1.7], the
endomorphism algebra of each simple ${}_{J}\mathcal{O}(G)_{J}$-module $V$ is
$\mathbb{C}$, so $V$ is killed by a maximal ideal of the central subalgebra
${}_{J}\mathcal{O}(T\backslash G/T)_{J}$. Our target is to make this
observation more precise. For $g\in G$, let $Z_{g}:=TgT$ be the double coset
of $T$ in $G$ containing $g$, so that $Z_{g}$ is the orbit containing $g$
under the left action of $T\times T$ on $G$ given by

(5.9) $(r,s)\cdot g:=rgs^{-1};\,\,\,r,s\in T.$

######  Lemma 5.8.

([G3, §4]) Fix $g\in G$, and set $T_{g}:=T\cap gTg^{-1}$.

  1. (1)

$Z_{g}$ is an irreducible closed subset of $G$ of dimension

$2\mathrm{dim}(T)-\mathrm{dim}(T_{g}).$

  2. (2)

Define the embedding $\theta:T_{g}\rightarrow T\times
T,\,\,\,a\mapsto(a,g^{-1}ag).$ Then $\theta(T_{g})$ acts on $T\times T$ on the
right, and

$\mathcal{O}(Z_{g})\cong(\mathcal{O}(T)\otimes\mathcal{O}(T))^{\theta(T_{g})}.$

  3. (3)

Let $\mathcal{I}(Z_{g})$ be the defining ideal of $Z_{g}$ in $\mathcal{O}(G)$,
and let $i:T\hookrightarrow G$ be the inclusion morphism. Then there is an
algebra epimorphism

$m_{g}^{\ast}:=(i^{\ast}\otimes
i^{\ast})\circ(\mathrm{id}\otimes\lambda_{g^{-1}})\circ\Delta:\mathcal{O}(G)\twoheadrightarrow\left(\mathcal{O}(T)\otimes\mathcal{O}(T)\right)^{\theta(T_{g})}$

with kernel $\mathcal{I}(Z_{g})$. ∎

###### Proof.

(1), (2) Since $T\times T$ is unipotent its orbits on $G$ under the action
(5.9) are closed by the theorem of Kostant, Rosenberg and Rosenlicht, [Mi,
Theorem 17.64], so $Z_{g}$ is an irreducible closed subset of $G$. The
stabilizer of $g$ in $T\times T$ is $\\{(a,g^{-1}ag)\mid a\in T\cap
gTg^{-1}\\}$, so that $Z_{g}\cong(T\times T)/(T\cap gTg^{-1})$. More
precisely, the closed subgroup $\theta(T_{g})$ acts on $T\times T$ on the
right by

$(x,y)\cdot\theta(a)=(xa,g^{-1}a^{-1}gy),$

for $x,y\in T$ and $a\in T_{g}$, yielding an isomorphism of affine varieties

$(T\times T)/\theta(T_{g})\rightarrow Z_{g},\quad\overline{(x,y)}\mapsto xgy.$

This proves (1) and (2).

(3) The above right action of $\theta(T_{g})$ induces a left action of
$\theta(T_{g})$ on $\mathcal{O}(T)\otimes\mathcal{O}(T)$, given, for $a\in
T_{g}$, $x,y\in T$, and $\alpha,\beta\in\mathcal{O}(T)$, by

$\theta(a)(\alpha\otimes\beta)(x,y)=\alpha(xa)\beta(g^{-1}a^{-1}gy).$

That is, $\theta(a)$ acts via $\rho_{a}\otimes\lambda_{g^{-1}ag}$, where
$\rho$ and $\lambda$ are the right and left regular actions of $G$ on
$\mathcal{O}(G)$ as noted in (2.7). So there is an algebra isomorphism

$\mathcal{O}(Z_{g})\cong(\mathcal{O}(T)\otimes\mathcal{O}(T))^{\theta(T_{g})}.$

Now (3) is just a reformulation of the above. ∎

The value of Lemma 5.8 lies in the following result:

######  Proposition 5.9.

([G3, Proposition 4.1]) Retain the above notation. The map $m_{g}^{\ast}$ is
also a surjective algebra homomorphism from ${}_{J}\mathcal{O}(G)_{J}$ onto
$({}_{J}\mathcal{O}(T)\otimes\mathcal{O}(T)_{J})^{\theta(T_{g})}$. Hence,
$\mathcal{I}(Z_{g})$ is also a two-sided ideal of ${}_{J}\mathcal{O}(G)_{J}$.
∎

###### Proof.

Since $i^{\ast}:{}_{J}\mathcal{O}(G)_{J}\rightarrow{}_{J}\mathcal{O}(T)_{J}$
is an algebra epimorphism, by Proposition 2.11, it is enough by Lemma 5.8(3)
to show that $(\mathrm{id}\otimes\lambda_{g^{-1}})\circ\Delta$ is an algebra
homomorphism from ${}_{J}\mathcal{O}(G)_{J}$ to
${}_{J}\mathcal{O}(G)_{J}\otimes\,{}_{J}\mathcal{O}(G)_{J}$. This is a routine
check. ∎

In view of Proposition 5.9, we follow [G3] and make the following

######  Definition 5.10.

For $g\in G$, set
${}_{J}\mathcal{O}(Z_{g})_{J}:={}_{J}\mathcal{O}(G)_{J}/\mathcal{I}(Z_{g})$. ∎

######  Corollary 5.11.

[G3, Corollary 4.2] For any $g\in G$, ${}_{J}\mathcal{O}(Z_{g})_{J}$ is a
noetherian domain, with
$\mathrm{GKdim}({}_{J}\mathcal{O}(Z_{g})_{J})=2\mathrm{dim}(T)-\mathrm{dim}(T_{g})$.

###### Proof.

Since ${}_{J}\mathcal{O}(Z_{g})_{J}$ is a factor of
${}_{J}\mathcal{O}(G)_{J}$, it is affine Noetherian by Corollary 3.5(1). It is
a domain because Proposition 5.9 shows that it is a subalgebra of
${}_{J}\mathcal{O}(T)\otimes\mathcal{O}(T)_{J}$, which is a Weyl algebra by
Theorem 2.13. The associated (commutative) graded algebras of $\mathcal{O}(G)$
and of ${}_{J}\mathcal{O}(G)_{J}$ with respect to the filtrations induced by
their coradical filtrations are identical, since the coradical filtration of
these algebras is the same. The same is thus true of the factor algebras
$\mathcal{O}(G)/\mathcal{I}(Z_{g})$ and
${}_{J}\mathcal{O}(G)_{J}/\mathcal{I}(Z_{g})$. Denoting this commutative
affine algebra by $A$, we have

$\mathrm{GKdim}(\mathcal{O}(G)/\mathcal{I}(Z_{g}))=\mathrm{GKdim}(A)=\mathrm{GKdim}({}_{J}\mathcal{O}(G)_{J}/\mathcal{I}(Z_{g}))$

by two applications of [KL, Proposition 6.6]. The value of this common GK-
dimension is given by Lemma 5.8(1). ∎

Keep the notation from the start of $\S$5 and from earlier in $\S$5.3. We now
provide an elaborated version of the proof of [G3, Theorem 5.1] for the
reader’s convenience.

######  Theorem 5.12.

([G3, Theorem 5.1]) Every simple ${}_{J}\mathcal{O}(G)_{J}$-module factors
through a unique quotient ${}_{J}\mathcal{O}(Z_{g})_{J}$.

###### Proof.

Let $V$ be a simple (left) ${}_{J}\mathcal{O}(G)_{J}$-module. By Proposition
5.4 and Schur’s lemma [MR, Proposition 9.1.7], $V$ is annihilated by
$K{}_{J}\mathcal{O}(G)_{J}$ for a maximal ideal $K$ of
$\mathcal{O}(T\backslash G/T)$. Now,
$K{}_{J}\mathcal{O}(G)_{J}=K\mathcal{O}(G)$ is also an ideal of
$\mathcal{O}(G)$, and is clearly $(T\times T)$-invariant under the action of
the left and right winding automorphisms of $T$ on $\mathcal{O}(G)$ (5.18).
Therefore, the same invariance is true of $\sqrt{K\mathcal{O}(G)}$, where
$\sqrt{K\mathcal{O}(G)}/K\mathcal{O}(G)$ denotes the nilradical of
$\mathcal{O}(G)/K\mathcal{O}(G)$. Thus, the variety
$\mathcal{V}(\sqrt{K\mathcal{O}(G)})$ is a disjoint union of orbits $TgT$, all
of them closed by Lemma 5.8(1), as $g$ runs through those elements of $G$
whose maximal ideals $\mathfrak{m}_{g}$ contain $K\mathcal{O}(G)$. Choose a
set of representatives $\mathcal{S}\subset G$ for these orbits. Thus, by the
commutative Nullstellensatz, we have

(5.10) $\bigcap_{g\in\mathcal{S}}\mathcal{I}(Z_{g})=\sqrt{K\mathcal{O}(G)}.$

Note that (5.10) is valid whether viewed in $\mathcal{O}(G)$ or in
${}_{J}\mathcal{O}(G)_{J}$, and that for $g\in\mathcal{S}$, the ideals
$\mathcal{I}(Z_{g})$ of ${}_{J}\mathcal{O}(G)_{J}$ are comaximal, because they
are by definition comaximal as ideals of $\mathcal{O}(G)$, and since addition
of ideals involves only the vector space structure. Hence, $V$ is annihilated
by at most one of the ideals $\mathcal{I}(Z_{g})$ for $g\in\mathcal{S}$. The
theorem is thus equivalent to the claim that

(5.11) $V\textit{ is killed by }\mathcal{I}(Z_{g})\textit{ for some
}g\in\mathcal{S}.$

To prove (5.11), we first determine the $(T\times T)$-invariants of the
strata: namely,

(5.12)
${}^{T}\left({}_{J}\mathcal{O}(Z_{g})_{J}\right)^{T}=\,{}^{T}\left({}\mathcal{O}(Z_{g})\right)^{T}=\mathbb{C}.$

The first equality in (5.12) holds because the space of invariants is
independent of the multiplication on the algebra, and the second equality
holds because the maximal ideals of $\mathcal{O}(Z_{g})$ form a single
$(T\times T)$-orbit intersecting in ${0}$, so that every nonzero $(T\times
T)$-invariant in $\mathcal{O}(Z_{g})$ must be a unit, while the only units of
$\mathcal{O}(Z_{g})$ are the scalars $\mathbb{C}^{\times}$ by Lemma 5.8(2),
noting that $\mathcal{O}(T)^{\otimes 2}$ is a polynomial algebra.

Now observe that by (5.10), there is a canonical embedding of algebras

(5.13)
$\varphi:{}_{J}\mathcal{O}(G)_{J}/\sqrt{K\mathcal{O}(G)}\hookrightarrow\Pi_{s\in\mathcal{S}}\,{}_{J}\mathcal{O}(Z_{s})_{J}.$

Moreover, $\varphi$ is clearly $(T\times T)$-equivariant, so the restriction
of $\varphi$ gives an embedding

(5.14)
$\varphi:{}^{T}\left({}_{J}\mathcal{O}(G)_{J}/\sqrt{K\mathcal{O}(G)}\right)^{T}\hookrightarrow\Pi_{s\in\mathcal{S}}\,{}^{T}({}_{J}\mathcal{O}(Z_{s})_{J})^{T}=\Pi_{s\in\mathcal{S}}\mathbb{C},$

where the final equality follows from (5.12). From (5.13) and (5.14) we see
that

(5.15)
${}^{T}\left({}_{J}\mathcal{O}(G)_{J}/\sqrt{K\mathcal{O}(G)}\right)^{T}\subset
Z\left({}_{J}\mathcal{O}(G)_{J}/\sqrt{K\mathcal{O}(G)}\right).$

If $|\mathcal{S}|=1$, then
${}_{J}\mathcal{O}(Z_{g})_{J}={}_{J}\mathcal{O}(G)_{J}/\sqrt{K\mathcal{O}(G)}$,
and (5.11) is immediate by definition of $K$. So suppose that
$|\mathcal{S}|>1$, and let $TxT$ be a double coset with $x\in\mathcal{S}$.
Then $TxT$ is preserved by $T\times T$, and so therefore is
$\mathcal{I}(Z_{x})$. Therefore, $T\times T$ acts rationally on the non-zero
space $\mathcal{I}(Z_{x})/\sqrt{K\mathcal{O}(G)}$. Since $T\times T$ is
unipotent, its only rational irreducible module is the trivial module, hence
there exists $0\neq\alpha\in\mathcal{I}(Z_{x})\setminus\sqrt{K\mathcal{O}(G)}$
with $\alpha$ being $(T\times T)$-invariant.

By (5.15), $\alpha\in
Z\left({}_{J}\mathcal{O}(G)_{J}/\sqrt{K\mathcal{O}(G)}\right)$, but $\alpha$
is not constant on $\mathcal{V}(\sqrt{K\mathcal{O}(G)})$ since $\alpha(TxT)=0$
whereas $TxT\subsetneq\mathcal{V}(\sqrt{K\mathcal{O}(G)})$ and
$\alpha\notin\sqrt{K\mathcal{O}(G)}$. Hence, by the noncommutative
Nullstellensatz [MR, Proposition 9.1.7], there exists $\lambda\in\mathbb{C}$
such that $(\alpha-\lambda)V=0$, and
$\alpha-\lambda\notin\sqrt{K\mathcal{O}(G)}$. Define

$I_{1}:=\sqrt{K\mathcal{O}(G)}+(\alpha-\lambda)\mathcal{O}(G),$

so that $I_{1}$ is a $(T\times T)$-invariant ideal of both $\mathcal{O}(G)$
and of ${}_{J}\mathcal{O}(G)_{J}$ with

$\sqrt{K\mathcal{O}(G)}\subsetneq I_{1}\textit{ and }I_{1}V=0.$

Now $\mathcal{V}_{1}:=\mathcal{V}(\sqrt{I_{1}})$ is a proper subvariety of
$\mathcal{V}(\sqrt{K\mathcal{O}(G)})$ and is a union of $(T\times T)$-orbits
labelled by a proper subset $\mathcal{S}_{1}$ of $\mathcal{S}$. If
$|\mathcal{S}_{1}|=1$, say $\mathcal{S}_{1}=\\{y\\}$, then $\sqrt{I_{1}}V=0$
with $\sqrt{I_{1}}=\mathcal{I}(Z_{y})$, proving (5.11).

If on the other hand $|\mathcal{S}_{1}|>1$, then we can repeat the above
argument with $\sqrt{I_{1}}$ replacing $\sqrt{K\mathcal{O}(G)}$. Since
$\mathcal{O}(G)$, or equivalently, ${}_{J}\mathcal{O}(G)_{J}$ are noetherian,
this process must terminate after finitely many repeats, so that (5.11)
follows, completing the proof of the theorem. ∎

###  5.4. On the structure of the algebras ${}_{J}\mathcal{O}(Z_{g})_{J}$

Retain the notation and hypotheses from the start of $\S$5, the definitions of
the algebras ${}_{J}\mathcal{O}(Z_{g})_{J}$ from Definition 5.10, and the
ideals $\mathcal{I}(Z_{g})$ (of $\mathcal{O}(G)$ and of
${}_{J}\mathcal{O}(G)_{J}$) from Lemma 5.8(3). Let $g\in G$, and recall from
Lemma 5.8 that we denote $T\cap gTg^{-1}$ by $T_{g}$. We divide the discussion
of ${}_{J}\mathcal{O}(Z_{g})_{J}$ into the following three cases, recalling
that this algebra depends only on the double coset $TgT$ of $T$ in $G$ to
which $g$ belongs. Here, and in $\S$6, we denote the normaliser $N_{G}(T)$ of
$T$ in $G$ by $N$.

Case I: $T_{g}=\\{1\\}$.

Case II: $T_{g}=T$; that is, $g\in N$.

Case III: $\\{1\\}\subsetneq T_{g}\subsetneq T$.

We treat Cases I and II separately in the following results, keeping the above
notation $G$, $J$ and $T$ in force throughout.

######  Theorem 5.13.

([G3, Remark 4.4]) Let $g\in G$ be such that Case I holds; that is
$T_{g}=\\{1\\}$. Then ${}_{J}\mathcal{O}(Z_{g})_{J}$ is a Weyl algebra;
specifically,

${}_{J}\mathcal{O}(Z_{g})_{J}\cong A_{\mathrm{dim}(T)}(\mathbb{C}).$

###### Proof.

In this case $\theta(T_{g})=\\{1\\}$, so
${}_{J}\mathcal{O}(Z_{g})_{J}\cong{}_{J}\mathcal{O}(T)\otimes\mathcal{O}(T)_{J}$
by Proposition 5.9. The result follows from Theorem 2.13(2),(4). ∎

In contrast to Case I, the Case II strata are no longer in general mutually
isomorphic algebras. The following is a refinement and elaboration of Theorem
5.2.

######  Theorem 5.14.

The following hold:

  1. (1)

The Hopf ideal $\mathcal{O}(G/N)^{+}\mathcal{O}(G)$ of $\mathcal{O}(G)$ is
also a Hopf ideal of ${}_{J}\mathcal{O}(G)_{J}$, so
$\mathcal{O}(G/N)^{+}\mathcal{O}(G)=\mathcal{O}(G/N)^{+}{}_{J}\mathcal{O}(G)_{J}=({}_{J}\mathcal{O}(G/N)_{J})^{+}{}_{J}\mathcal{O}(G)_{J}$,
and

${}_{J}\mathcal{O}(G)_{J}/({}_{J}\mathcal{O}(G/N)_{J})^{+}{}_{J}\mathcal{O}(G)_{J}\cong{}_{J}\mathcal{O}(N)_{J}$

as Hopf algebras.

  2. (2)

${}_{J}\mathcal{O}(G/T)_{J}/({}_{J}\mathcal{O}(G/N)_{J})^{+}{}_{J}\mathcal{O}(G/T)_{J}$
is isomorphic to the central (polynomial) Hopf subalgebra $\mathcal{O}(N/T)$
of the quotient Hopf algebra ${}_{J}\mathcal{O}(N)_{J}$ of
${}_{J}\mathcal{O}(G)_{J}$, with

(5.16)
${}_{J}\mathcal{O}(N)_{J}\cong\mathcal{O}(N/T)\\#_{\sigma}\left(U(\mathfrak{t})^{J}\right),$

where $\sigma$ is an invertible 2-cocycle from
$\left(U(\mathfrak{t})^{J}\right)\otimes\left(U(\mathfrak{t})^{J}\right)$ to
$\mathcal{O}(N/T)$.

  3. (3)

Let $g$ be such that Case II holds; that is $g\in N$. Then
$\mathcal{I}(Z_{g})$ is generated by its intersection with
${}_{J}\mathcal{O}(G/T)_{J}$. That is, we have

$\mathcal{I}(Z_{g})\cap{}_{J}\mathcal{O}(G/T)_{J}=\mathfrak{m}_{gT},$

where $\mathfrak{m}_{gT}$ denotes the functions in $\mathcal{O}(G/T)$ which
are 0 on $gT$. Thus, $\mathfrak{m}_{gT}$ is an ideal of
${}_{J}\mathcal{O}(G/T)_{J}$, which contains
$({}_{J}\mathcal{O}(G/N)_{J})^{+}{}_{J}\mathcal{O}(G/T)_{J}$, with
${}_{J}\mathcal{O}(G/T)_{J}/\mathfrak{m}_{gT}\cong\mathbb{C}$. Moreover,
$\mathcal{I}(Z_{g})=\mathfrak{m}_{gT}{}_{J}\mathcal{O}(G)_{J}$.

###### Proof.

(1) Working first in $\mathcal{O}(G)$, observe that

$\mathcal{O}(G/N)^{+}\mathcal{O}(G)=\bigcap_{x\in N}\mathfrak{m}_{x},$

where as usual, for $x\in G$, $\mathfrak{m}_{x}=\\{f\in\mathcal{O}(G)\mid
f(x)=0\\}$. Since for each $g\in N$,

$\mathcal{I}(Z_{g})=\bigcap_{x\in gT}\mathfrak{m}_{x},$

by the definition of $\mathcal{I}(Z_{g})$ in Lemma 5.8(3) and its proof, it
follows that

(5.17) $\mathcal{O}(G/N)^{+}\mathcal{O}(G)=\bigcap_{g\in
N}\mathcal{I}(Z_{g}).$

Since each $\mathcal{I}(Z_{g})$ is an ideal of ${}_{J}\mathcal{O}(G)_{J}$ by
Proposition 5.9, we see from (5.17) that $\mathcal{O}(G/N)^{+}\mathcal{O}(G)$
is an ideal of ${}_{J}\mathcal{O}(G)_{J}$. Since the comultiplication is
unchanged in the switch from $\mathcal{O}(G)$ to ${}_{J}\mathcal{O}(G)_{J}$,
$\mathcal{O}(G/N)^{+}\mathcal{O}(G)$ is a Hopf ideal of
${}_{J}\mathcal{O}(G)_{J}$. The displayed isomorphism now follows because
$\mathcal{O}(G/N)^{+}\mathcal{O}(G)$ is contained in the radical of $R^{J}$ by
Lemma 5.1(2), so that ${}_{J}\mathcal{O}(N)_{J}$ is well-defined, with an
epimorphism to it from ${}_{J}\mathcal{O}(G)_{J}$ as in Proposition 2.11.

(2) Working in ${}_{J}\mathcal{O}(N)_{J}$, we see that since $T\triangleleft
N$, the first claim is a special case of Proposition 5.4. Moreover,
$\mathcal{O}(N/T)$ is a polynomial algebra because $N/T$ is a unipotent group.
Finally, (5.16) is a special case of Theorem 5.2, where in this case the
action is trivial thanks to the centrality of $\mathcal{O}(N/T)$.

(3) The displayed equality is proved by first noting that the equivalent
statement is obviously true in $\mathcal{O}(N)$, hence also true in
${}_{J}\mathcal{O}(N)_{J}$ since $\mathcal{O}(N/T)$ is central in
${}_{J}\mathcal{O}(N)_{J}$. Finally, we can lift the statements back to
${}_{J}\mathcal{O}(G)_{J}$ using the displayed isomorphism in (1). ∎

For any $g\in G$, let

(5.18)
$\begin{split}\tau^{\ell}_{g}:\mathcal{O}(G)\rightarrow\mathcal{O}(G),\,\,\,\alpha\mapsto\sum\alpha_{1}(g)\alpha_{2},\\\
\tau^{r}_{g}:\mathcal{O}(G)\rightarrow\mathcal{O}(G),\,\,\,\alpha\mapsto\sum\alpha_{1}\alpha_{2}(g),\end{split}$

be the associated winding automorphisms of $\mathcal{O}(G)$.

######  Lemma 5.15.

Let $g\in G$. The following hold:

  1. (1)

([G3, Lemma 3.16])
$\tau^{\ell}_{g}:{}_{J}\mathcal{O}(G)_{J}\rightarrow{}_{J^{g}}\mathcal{O}(G)_{J}$
is an algebra isomorphism, where ${}_{J^{g}}\mathcal{O}(G)_{J}$ is defined at
(2.3).

  2. (2)

Assume that $g\in N$. With ${}_{J^{g}}\mathcal{O}(T)_{J}$ defined as at (2.3),
there is an algebra isomorphism
${}_{J}\mathcal{O}(Z_{g})_{J}\cong{}_{J^{g}}\mathcal{O}(T)_{J}$.

###### Proof.

(1) Note that $\tau^{\ell}_{g}$ is at least a linear isomorphism of the
underlying vector space, and let $\alpha,\beta\in{}_{J}\mathcal{O}(G)_{J}$.
Use $\cdot$ [resp.$\ast$] to denote multiplication in
${}_{J}\mathcal{O}(G)_{J}$ [resp. in ${}_{J^{g}}\mathcal{O}(G)_{J}$], with
juxtaposition for multiplication in $\mathcal{O}(G)$. Then

$\displaystyle\tau^{\ell}_{g}(\alpha\cdot\beta)$
$\displaystyle=\sum\tau^{\ell}_{g}\left(J^{-1}(\alpha_{1},\beta_{1})\alpha_{2}\beta_{2}J(\alpha_{3},\beta_{3})\right)$
$\displaystyle=\sum
J^{-1}(\alpha_{1},\beta_{1})\tau^{\ell}_{g}(\alpha_{2})\tau^{\ell}_{g}(\beta_{2})J(\alpha_{3},\beta_{3})$
$\displaystyle=\sum
J^{-1}(\alpha_{1},\beta_{1})\alpha_{2}(g)\beta_{2}(g)\alpha_{3}\beta_{3}J(\alpha_{4},\beta_{4}).$

On the other hand,

$\displaystyle\tau^{\ell}_{g}(\alpha)\ast\tau^{\ell}_{g}(\beta)$
$\displaystyle=\left(\sum\alpha_{1}(g)\alpha_{2}\right)\ast\left(\sum\beta_{1}(g)\beta_{2}\right)=\sum\alpha_{1}(g)\beta_{1}(g)\alpha_{2}\ast\beta_{2}$
$\displaystyle=\sum\alpha_{1}(g)\beta_{1}(g)J^{-1}\left(\alpha_{2}(g^{-1})\alpha_{3}\alpha_{4}(g),\beta_{2}(g^{-1})\beta_{3}\beta_{4}(g)\right)\alpha_{5}\beta_{5}J(\alpha_{6},\beta_{6})$
$\displaystyle=\sum\epsilon(g)\epsilon(g)J^{-1}(\alpha_{1},\beta_{1})\alpha_{2}(g)\beta_{2}(g)\alpha_{3}\beta_{3}J(\alpha_{4},\beta_{4})$
$\displaystyle=\sum
J^{-1}(\alpha_{1},\beta_{1})\alpha_{2}(g)\beta_{2}(g)\alpha_{3}\beta_{3}J(\alpha_{4},\beta_{4}).$

This proves (1).

(2) The isomorphism $\tau^{\ell}_{g}$ from (1) maps
$\mathfrak{m}_{gT}\mathcal{O}(G)$ onto $\mathfrak{m}_{T}\mathcal{O}(G)$. Since
$g\in N$, Theorem 5.14(3) shows that

${}_{J}\mathcal{O}(Z_{g})_{J}={}_{J}\mathcal{O}(G)_{J}/\mathcal{I}(gT)\cong{}_{J^{g}}\mathcal{O}(G)_{J}/\mathcal{I}(T)={}_{J^{g}}\mathcal{O}(T)_{J},$

as claimed. ∎

######  Question 5.16.

Let $g\in N$. Is it true that ${}_{J}\mathcal{O}(Z_{g})_{J}\cong
U^{\omega_{g}}(\mathfrak{t})$ for a $2$-cocycle $\omega_{g}$ of
$\mathfrak{t}$? If yes, is it true that
$\omega_{g}=\omega-\omega\circ(\mathrm{Ad}g\otimes\mathrm{Ad}g)$, where
$\omega$ is the $2$-cocycle of $\mathfrak{t}$ corresponding to $J$? (In [G3,
Theorem 4.5] it is claimed without a proof that the answer to the second
question is positive.) ∎

Although we do not currently have detailed information on the structure of the
Case III strata, we note here that if $g\in G$ is such that Case III or Case I
holds that is, if $g\notin N$, then ${}_{J}\mathcal{O}(Z_{g})_{J}$ has no
nonzero finite dimensional modules, by Theorem 5.13, and Theorem 6.8 below.

##  6\. Finite dimensional simple ${}_{J}\mathcal{O}(G)_{J}$-modules

In this section we study the group of finite dimensional simple
${}_{J}\mathcal{O}(G)_{J}$-modules. (In particular, we correct [G3, Theorem
5.2].) Since the finite dimensional simple ${}_{J}\mathcal{O}(G)_{J}$-modules
are $1$-dimensional by Corollary 3.5(5), the set of these modules forms a
group, which we denote in this section by $\Gamma$. We show in $\S$6.1 that
$\Gamma$ is a closed subgroup of $G$, and then we give a precise description
of $\Gamma$ in $\S$6.2, proving Theorem 6.8 (= Theorem 1.6).

###  6.1. $\Gamma$ is a subgroup of $G$

We start with the following elementary observation.

######  Lemma 6.1.

Let $k$ be a field, $n\geq 1$, and $F=k\langle X_{1},\dots,X_{n}\rangle$ the
free $k$-algebra of rank $n$. Let $D:=\langle[X_{i},X_{j}]\mid 1\leq i<j\leq
n\rangle$ be the commutator ideal of $F$, so $F/D\cong k[X_{1},\dots,X_{n}]$.
Let $S:=F/E$ be any factor algebra of $F$. Then the commutator ideal
$\langle[S,S]\rangle$ of $S$ is $I:=(D+E)/E$.

###### Proof.

Since $D\subseteq D+E$, $S/I$ is commutative, so $\langle[S,S]\rangle\subseteq
I$. On the other hand, the image in $S$ of every generator $[X_{i},X_{j}]$ of
$D$ lies in $\langle[S,S]\rangle$, so $I\subseteq\langle[S,S]\rangle$. ∎

Now we apply Lemma 6.1 in the setting of Remarks 3.4(1). That is, in the lemma
we take $k=\mathbb{C}$ and $n=\mathrm{dim}(G)$, so
$\mathcal{O}(G)\xrightarrow{\pi_{D}}F/D$, $X_{i}\mapsto X_{i}+D$, is an
algebra isomorphism, and we define the ideal $E$ of $F$ to be the ideal $E$ of
Remarks 3.4(1), so that ${}_{J}\mathcal{O}(G)_{J}\xrightarrow{\pi_{E}}F/E$,
$X_{i}\mapsto X_{i}+E$, is an algebra isomorphism. Then by Lemma 6.1,
$(D+E)/E$ is the commutator ideal of $F/E$.

Now let $I:=\langle[{}_{J}\mathcal{O}(G)_{J},{}_{J}\mathcal{O}(G)_{J}]\rangle$
be the commutator ideal of ${}_{J}\mathcal{O}(G)_{J}$. Since the commutator
ideal of any Hopf algebra is a Hopf ideal, $I$ is a Hopf ideal of
${}_{J}\mathcal{O}(G)_{J}$, and $\pi_{E}$ induces a canonical Hopf algebra
isomorphism

${}_{J}\mathcal{O}(G)_{J}/I\xrightarrow{\overline{\pi_{E}}}(F/E)/((D+E)/E),\,\,\,X_{i}+I\mapsto\pi_{E}(X_{i})+(D+E)/E.$

In particular, since the algebra $(F/E)/((D+E)/E)$ is canonically isomorphic
to the algebra $F/(D+E)$, we see that the algebra $F/(D+E)$ is a factor Hopf
algebra of ${}_{J}\mathcal{O}(G)_{J}$ in a canonical way.

Now set $\mathscr{I}:=\pi_{D}^{-1}\left((D+E)/D\right)$; it is an ideal of
$\mathcal{O}(G)$, and $\pi_{D}$ induces a canonical algebra isomorphism

$\mathcal{O}(G)/\mathscr{I}\xrightarrow{\overline{\pi_{D}}}(F/D)/((D+E)/D),\,\,\,X_{i}+\mathscr{I}\mapsto\pi_{D}(X_{i})+(D+E)/D.$

Since the algebra $(F/D)/((D+E)/D)$ is canonically isomorphic to the algebra
$F/(D+E)$, it follows from the above that $\mathscr{I}$ is a Hopf ideal of
$\mathcal{O}(G)$, so the algebra $F/(D+E)$ is also a factor Hopf algebra of
$\mathcal{O}(G)$ in a canonical way.

It follows that the Hopf algebras $\mathcal{O}(G)/\mathscr{I}$ and
${}_{J}\mathcal{O}(G)_{J}/I$ are canonically isomorphic via the “identity map”

$\mathcal{O}(G)/\mathscr{I}\xrightarrow{{\rm
id}}{}_{J}\mathcal{O}(G)_{J}/I,\,\,\,X_{i}+\mathscr{I}\mapsto X_{i}+I,$

hence $I=\mathscr{I}$, so $I$ is also a Hopf ideal of $\mathcal{O}(G)$.

Thus, combining the above with Remarks 3.4(1), we have obtained the following.

######  Theorem 6.2.

Let $G$ be a unipotent affine algebraic group, $J$ a Hopf $2$-cocycle for $G$,
$I:=\langle[{}_{J}\mathcal{O}(G)_{J},{}_{J}\mathcal{O}(G)_{J}]\rangle$ the
commutator ideal of ${}_{J}\mathcal{O}(G)_{J}$, and $\Gamma$ the group of
finite dimensional simple ${}_{J}\mathcal{O}(G)_{J}$-modules. The following
hold:

  1. (1)

${}_{J}\mathcal{O}(G)_{J}/I\cong\mathcal{O}(\Gamma)$, so $\Gamma$ is a
unipotent affine algebraic group.

  2. (2)

$I$ is a Hopf ideal of both ${}_{J}\mathcal{O}(G)_{J}$ and $\mathcal{O}(G)$.

  3. (3)

The identity map $\mathcal{O}(G)/I\xrightarrow{{\rm
id}}{}_{J}\mathcal{O}(G)_{J}/I$ is a Hopf algebra isomorphism.

  4. (4)

Every maximal ideal of ${}_{J}\mathcal{O}(G)_{J}$ of codimension $1$ is also a
maximal ideal of $\mathcal{O}(G)$.

  5. (5)

$\Gamma$ is the closed subgroup of $G$ whose defining ideal is
$I\subseteq\mathcal{O}(G)$. ∎

###  6.2. The determination of $\Gamma$

In this subsection we obtain an explicit description of the subgroup $\Gamma$
of $G$. We deal first in Theorem 6.3 with the case where $J$ is minimal, then
address the general case from Lemma 6.5 onwards. The main result of this
subsection is Theorem 6.8.

Retain the notation of $\S$5.4 and Lemma 5.15. By [ES, Proposition 3.3 and
Definition 3.2], $(\mathfrak{t},[\,,\,],r,\delta)$ is a _nondegenerate
coboundary Lie bialgebra_ with cobracket

(6.1) $\delta:\mathfrak{t}\to\wedge^{2}\mathfrak{t},\,\,x\mapsto[x\otimes
1+1\otimes x,r];$

here, as in Theorem 2.15, $r:=\omega^{-1}\in\wedge^{2}\mathfrak{t}$ is the
associated nondegenerate solution to the classical Yang-Baxter equation, and
we can restate (6.1) as $\delta=\partial_{1}(r)$. By [ES, Definitions 3.4],
this means that $(\mathfrak{t},[\,,\,],r,\partial_{1}(r))$ is a _triangular_
Lie bialgebra.

Recall also from [ES, Proposition 2.2] that $\mathfrak{t}^{*}$ is a Lie
bialgebra with bracket $\delta^{*}$ and cobracket $[\,,\,]^{*}$, and the map

(6.2)
$F_{r}:\mathfrak{t}^{*}\xrightarrow{\cong}\mathfrak{t},\,\,\alpha\mapsto(\alpha\otimes{\rm
id})(r),$

is an isomorphism of Lie bialgebras, where [CP, Proposition 2.1.4] shows that
the map is a homomorphism of Lie bialgebras, with bijectivity following from
nondegeneracy combined with [ES, Proposition 3.2]. We consider first the group
$\Gamma\cap T$ of finite dimensional simple representations of
${}_{J}\mathcal{O}(T)_{J}$. We show in (1) and (2) of the following result
that $\Gamma\cap T$ is isomorphic to a particular closed abelian subgroup $F$
of $T$. Later, in Theorem 6.8(3), we shall see that this isomorphism is in
fact an equality.

######  Theorem 6.3.

Retain the above notation. The following hold:

  1. (1)

There is a group isomorphism $\Gamma\cap T\cong T/[T,T]$, so in particular
$\Gamma\cap T$ is a unipotent abelian affine algebraic group.

  2. (2)

$\Gamma\cap T$ is canonically isomorphic to a closed abelian subgroup $F$ of
$T$.

  3. (3)

$\mathrm{Lie}(F)=\mathrm{ker}(\delta)=\\{x\in\mathfrak{t}\mid[x\otimes
1+1\otimes x,r]=0\\}$.

  4. (4)

$F=\\{t\in T\mid r^{t}=r\\}=\\{t\in T\mid J^{t}=J\\}$.

###### Proof.

(1) This follows from Corollary 4.12.

(2) It is sufficient to show that ${\rm
Lie}(T/[T,T])=\mathfrak{t}/[\mathfrak{t},\mathfrak{t}]$ is isomorphic to a Lie
subalgebra of $\mathfrak{t}$. To this end, using the discussion and notation
at the start of this subsection, note that the canonical projection
$\pi:\mathfrak{t}\twoheadrightarrow\mathfrak{t}/[\mathfrak{t},\mathfrak{t}]$
is a Lie bialgebra map by (6.1). Since
$\mathfrak{t}/[\mathfrak{t},\mathfrak{t}]\xrightarrow{\cong}\left(\mathfrak{t}/[\mathfrak{t},\mathfrak{t}]\right)^{*}$
as Lie bialgebras, we obtain an injective map of Lie bialgebras

$\mathfrak{t}/[\mathfrak{t},\mathfrak{t}]\xrightarrow{\cong}\left(\mathfrak{t}/[\mathfrak{t},\mathfrak{t}]\right)^{*}\xrightarrow{\pi^{*}}\mathfrak{t}^{*}\xrightarrow{F_{r}}\mathfrak{t},$

as desired. Denote the Lie subalgebra
$F_{r}\circ\pi^{*}((\mathfrak{t}/[\mathfrak{t},\mathfrak{t}])^{*})$ of
$\mathfrak{t}$ by $\mathfrak{f}$. Thus, $\mathfrak{f}$ is an abelian Lie
subalgebra of $\mathfrak{t}$ of dimension $\mathrm{dim}(T/[T,T])$, and (2)
follows with $F$ being the closed abelian subgroup of $T$ with
$\mathrm{Lie}(F)=\mathfrak{f}$.

(3) Since $\delta$ is a $1$-cocycle, $\mathrm{ker}(\delta)$ is a Lie
subalgebra of $\mathfrak{t}$. By the proof of Theorem 6.3,
$\delta(F_{r}(\pi^{*}(\alpha)))=0$ for every
$\alpha\in\left(\mathfrak{t}/[\mathfrak{t},\mathfrak{t}]\right)^{*}$. Namely,
$r\in\left(\wedge^{2}\mathfrak{t}\right)^{\mathfrak{f}}$ is invariant under
the adjoint action of $\mathfrak{f}$ on $\wedge^{2}\mathfrak{t}$. Thus,
$\mathfrak{f}\subseteq\mathrm{ker}(\delta)$.

Now since $\mathfrak{t}^{*}\xrightarrow{\cong}\mathfrak{t}$ as Lie bialgebras,
it follows that ${\rm dim}(\mathrm{ker}(\delta))={\rm
dim}(\mathrm{ker}([\,,\,]^{*})$. But clearly,
$\mathrm{ker}([\,,\,]^{*})=\left(\mathfrak{t}/[\mathfrak{t},\mathfrak{t}]\right)^{*}$.
Hence, by Theorem 6.3, we have

${\rm dim}(\mathrm{ker}(\delta))={\rm
dim}(\left(\mathfrak{t}/[\mathfrak{t},\mathfrak{t}]\right)^{*})={\rm
dim}(\mathfrak{f}),$

which implies the statement.

(4) By the well known correspondence between closed subgroups of $T$ and Lie
subalgebras of $\mathfrak{t}$, we have $F=\\{t\in T\mid r^{t}=r\\}$, since by
(3), both closed subgroups have the same Lie algebra. Also, by Remarks
2.16(3), $J^{t}$ corresponds to $r^{t}$ for any $t\in T$. Thus, for any $t\in
T$, $J^{t}=J$ if and only if $r^{t}=r$. This implies that $\\{t\in T\mid
r^{t}=r\\}=\\{t\in T\mid J^{t}=J\\}$, as claimed. ∎

######  Remark 6.4.

Let $\mathfrak{t}$ have basis $\\{a,b,c,d\\}$ with nonzero brackets $[a,b]=c$,
$[c,b]=d$, and let $r:=a\wedge c+d\wedge b$ (see Example 7.6 below). Then
$\mathfrak{f}={\rm sp}_{\mathbb{C}}\\{c,d\\}$ is an abelian ideal of
$\mathfrak{t}$, which is strictly contained in the abelian ideal ${\rm
sp}_{\mathbb{C}}\\{a,c,d\\}$. Thus, $\mathfrak{f}$ is not always a maximal
abelian subalgebra of $\mathfrak{t}$. ∎

By Theorem 5.12, every simple ${}_{J}\mathcal{O}(G)_{J}$-module is a simple
${}_{J}\mathcal{O}(Z_{g})_{J}$-module for a unique stratum $Z_{g}$, where for
$g,h\in G$, ${}_{J}\mathcal{O}(Z_{g})_{J}={}_{J}\mathcal{O}(Z_{h})_{J}$ if and
only if $TgT=ThT$. Thus, we can analyse the group $\Gamma$ of finite
dimensional simple ${}_{J}\mathcal{O}(G)_{J}$-modules by considering the
finite dimensional simple ${}_{J}\mathcal{O}(Z_{g})_{J}$-modules for each
$g\in G$.

Let $(\mathcal{O}(G)^{*})_{1}^{\times}$ denote the group of invertible
elements $\chi\in\mathcal{O}(G)^{*}$ with $\epsilon(\chi)=1$, and note that
$G\subseteq(\mathcal{O}(G)^{*})_{1}^{\times}$. Recall from Remark 2.3(3) that
$(\mathcal{O}(G)^{*})_{1}^{\times}$ acts on the set of Hopf $2$-cocycles $J$
for $G$ via

$J^{\chi}:=(\chi\circ
m)*J*\left(\chi^{-1}\otimes\chi^{-1}\right);\,\,\,\chi\in\left(\mathcal{O}(G)^{*}\right)_{1}^{\times},$

where $m$ is the multiplication map. Let $[J]$ denote the orbit of $J$ under
the action of the subgroup $(\mathcal{O}(T)^{*})_{1}^{\times}$ of
$(\mathcal{O}(G)^{*})_{1}^{\times}$.

######  Lemma 6.5.

Let $L$ and $J$ be Hopf $2$-cocycles for $G$, and let
$\chi\in(\mathcal{O}(G)^{*})_{1}^{\times}$.

  1. (1)

$\chi:{}_{L}\mathcal{O}(G)_{J}\to\mathbb{C}$ is an algebra homomorphism if and
only if $L=J^{\chi}$.

  2. (2)

$\chi$ yields a representation of ${}_{J}\mathcal{O}(G)_{J}$ if and only if
$J=J^{\chi}$.

  3. (3)

Suppose that $L$ and $J$ both have support $T$. Then the algebra
${}_{L}\mathcal{O}(T)_{J}$ has $1$-dimensional representations if and only if
$[L]=[J]$.

###### Proof.

(1) Let $\cdot$ denote the multiplication in ${}_{L}\mathcal{O}(G)_{J}$, as in
(2.3). Then a linear map $\chi:{}_{L}\mathcal{O}(G)_{J}\to\mathbb{C}$ is an
algebra homomorphism if and only if we have

$\displaystyle(\chi\otimes\chi)(x\otimes y)=\chi(x)\chi(y)=\chi(x\cdot y)$
$\displaystyle=$ $\displaystyle L^{-1}(x_{1}\otimes
y_{1})\chi(x_{2}y_{2})J(x_{3}\otimes y_{3})=L^{-1}(x_{1}\otimes
y_{1})(\chi\circ m)(x_{2}\otimes y_{2})J(x_{3}\otimes y_{3})$ $\displaystyle=$
$\displaystyle\left(L^{-1}*(\chi\circ m)*J\right)(x\otimes y)$

for all $x,y\in{}_{L}\mathcal{O}(G)_{J}$. But this is equivalent to
$\chi\otimes\chi=L^{-1}*(\chi\circ m)*J$, that is to

$L=(\chi\circ m)*J*(\chi^{-1}\otimes\chi^{-1})=J^{\chi},$

as claimed.

(2) This follows immediately from (1).

(3) Apply the special case $G=T$ of (1). ∎

######  Notation 6.6.

Viewing $J$ as a Hopf $2$-cocycle for $T$, define the following closed
subgroups of $N=N_{G}(T)$, where as above, $[J]$ denotes the orbit of $J$
under the action of $(\mathcal{O}(T)^{*})_{1}^{\times}$:

(6.3) $C_{0}:=\\{g\in N\mid J^{g}=J\\}\,\,\,{\rm and}\,\,\,C:=\\{g\in
N\mid[J^{g}]=[J]\\}.$

######  Remarks 6.7.

(1) Clearly, $C_{0}\subseteq C$ is a closed subgroup, and $T\subseteq C$ is a
closed normal subgroup (since $C\subseteq N=N_{G}(T)$ and, by definition,
$[J^{t}]=[J]$ for $t\in T$).

(2) $\Gamma$ is the group of grouplike elements of
$\left({}_{J}\mathcal{O}(G)_{J}\right)^{\circ}$, so is a subgroup of
$\left(\mathcal{O}(G)^{*}\right)_{1}^{\times}$. More precisely, by Theorem 6.2
and Lemma 6.5(2),

(6.4) $\Gamma=\\{\chi\in(\mathcal{O}(G)^{*})_{1}^{\times}\mid
J^{\chi}=J\\}\subseteq G.$

(3) In view of (6.4, $Z(G)\subseteq\Gamma$. ∎

We can now state and prove our main result on the group $\Gamma$ of
1-dimensional representations of ${}_{J}\mathcal{O}(G)_{J}$.

######  Theorem 6.8.

Retain the notation used throughout this section, so $G$ is an affine
algebraic unipotent group, $J$ is a Hopf $2$-cocycle for $G$ with support $T$,
$N=N_{G}(T)$, $C_{0}\subseteq C$ are the subgroups of $N$ defined at (6.3),
and $F$ is the closed subgroup of $T$ defined at Theorem 6.3(3),(4).

  1. (1)

$\Gamma=C_{0},$ and $C_{0}=\\{g\in G\mid J^{g}=J\\}$.

  2. (2)

$C=C_{0}T.$

  3. (3)

$F=C_{0}\cap T$ is a closed abelian subgroup of $T$, normal in $C_{0}$, with

$\mathrm{dim}(F)=\mathrm{dim}(T/[T,T]).$

  4. (4)

$C_{0}/F\cong C/T$, so that there is an exact sequence of algebraic groups

$1\rightarrow F\rightarrow\Gamma\rightarrow C/T\rightarrow 1.$

  5. (5)

$\mathrm{dim}(\Gamma)=\mathrm{dim}(C)-\mathrm{dim}([T,T]).$

  6. (6)

The algebra ${}_{J}\mathcal{O}(Z_{g})_{J}$ has nonzero finite dimensional
representations if and only if $g\in C$. The distinct strata admitting nonzero
finite dimensional modules are $\\{{}_{J}\mathcal{O}(Z_{g})_{J}\mid
g\in\mathcal{C}\\},$ where $\mathcal{C}$ is a set of coset representatives for
$F=C_{0}\cap T$ in $C_{0}$, or equivalently for $T$ in $C$.

  7. (7)

For any $g\in\mathcal{C}$, the set of finite dimensional simple
${}_{J}\mathcal{O}(Z_{g})_{J}$-modules is parametrized by $gF=g(C_{0}\cap T)$.
Specifically, the corresponding functionals have kernels the maximal ideals
$\\{\mathfrak{m}_{gx}\mid x\in C_{0}\cap T\\}$ of $\mathcal{O}(G)$.

###### Proof.

(1) Let $\chi\in\Gamma$. By (6.4), $\chi$ is a grouplike element of
$\left({}_{J}\mathcal{O}(G)_{J}\right)^{\circ}$, with

(6.5) $\chi\in G\textit{ and }J=J^{\chi}=(\chi\circ
m)*J*(\chi^{-1}\otimes\chi^{-1}).$

It follows that
$R^{J}=(\chi\otimes\chi)*R^{J}*(\chi^{-1}\otimes\chi^{-1})\in\left(\mathcal{O}(T)\otimes\mathcal{O}(T)\right)^{*}$.
Note that $\left({}_{J}\mathcal{O}(T)_{J},R^{J}\right)$ is the minimal
cotriangular quotient Hopf algebra of
$\left({}_{J}\mathcal{O}(G)_{J},R^{J}\right)$ by Lemma 2.8 (quoted from [G2,
Section 2.2]), and that since $J^{\chi}=J$, the map

${}_{J}\mathcal{O}(G)_{J}\to{}_{J}\mathcal{O}(G)_{J},\,\,x\mapsto\sum\chi(x_{1})x_{2}\chi^{-1}(x_{3}),$

is a Hopf algebra automorphism by Lemma 6.5. Therefore, it follows that

(6.6)
$(\chi\otimes\chi)*\left(\mathcal{O}(T)\otimes\mathcal{O}(T)\right)^{*}*(\chi^{-1}\otimes\chi^{-1})=\left(\mathcal{O}(T)\otimes\mathcal{O}(T)\right)^{*}$

inside the algebra $\left(\mathcal{O}(G)\otimes\mathcal{O}(G)\right)^{*}$.
Hence, $\chi\in N=N_{G}(T)$ and so $\Gamma\subseteq C_{0}$ by definition of
the latter group. Conversely, if $g\in C_{0}$ then $g\in\Gamma$ by Lemma
6.5(2).

The second equality in (1) is also proved in the argument given above.

(2), (6) Suppose that $g\in G$ is such that the stratum
${}_{J}\mathcal{O}(Z_{g})_{J}$ admits a $1$-dimensional simple module $\chi$.
Then, by (1), $\chi\in C_{0}\subseteq N$. Hence, by definition of $Z_{g}$,
$\chi=gt$ for some element $t\in T$. Thus $g=\chi t^{-1}\in C_{0}T$.

Conversely, suppose $g\in C$, so that $[J^{g}]=[J]$ by definition of $C$. That
is, there exists $t_{0}\in T$ such that $J^{g}=J^{t_{0}}$. Hence

$gt_{0}^{-1}\in\\{x\in N\mid J^{x}=J\\}=C_{0}.$

Therefore $C=C_{0}T$ and $\chi:=gt_{0}^{-1}$ is a 1-dimensional representation
of ${}_{J}\mathcal{O}(Z_{g})_{J}$.

The final sentence of (6) is a consequence of the first together with Theorem
5.12, since the latter guarantees that every finite dimensional simple
${}_{J}\mathcal{O}(G)_{J}$-module factors through
${}_{J}\mathcal{O}(Z_{g})_{J}$ for a unique $g\in\mathcal{C}$.

(3) That $F=C_{0}\cap T$ is immediate from Theorem 6.3(4) and the definition
of $C_{0}$ at (6.3). The dimension of $F$ was also determined at Theorem 6.3.
Since $C_{0}\subseteq N=N_{G}(T)$ it is clear that $C_{0}\cap T$ is normal in
$C_{0}$.

(4) The isomorphism holds because $C_{0}\cap T=F\triangleleft C_{0}$ by (1),
and $C_{0}T=C$ by (2). The exact sequence follows at once, since
$C_{0}=\Gamma$ by (1).

(5) This is a straightforward consequence of (3) and (4).

(7) Let $g\in C_{0}$. There is an algebra isomorphism from
${}_{J}\mathcal{O}(T)_{J}={}_{J}\mathcal{O}(Z_{1})_{J}$ to
${}_{J}\mathcal{O}(Z_{g})_{J}$ given by the automorphism $\tau^{\ell}_{g}$ of
${}_{J}\mathcal{O}(G)_{J}$, as shown by Lemma 5.15. Clearly, this map carries
the group $F=C_{0}\cap T$ of 1-dimensional representations of
${}_{J}\mathcal{O}(T)_{J}$ to the set $gF=g(C_{0}\cap T)$ of all such
representations of ${}_{J}\mathcal{O}(Z_{g})_{J}$. ∎

##  7\. Special cases and examples

###  7.1. Abelian support