# Classification of Hopf superalgebras associated with quantum special linear
superalgebra at roots of unity using Weyl groupoid

Alexander Mazurenko 111Corresponding Author: Alexander Mazurenko,
<EMAIL_ADDRESS><EMAIL_ADDRESS>Vladimir A. Stukopin
<EMAIL_ADDRESS>MCCME (Moscow Center for Continuous Mathematical Education)
MIPT (Moscow Institute of Physics and Technology)
SMI of VSC RAS (South Mathematical Institute of Vladikavkaz Scientific Center
of Russian Academy of Sciences)
MCCME (Moscow Center for Continuous Mathematical Education)

###### Abstract

We summarize the definition of the Weyl groupoid using supercategory approach
in order to investigate quantum superalgebras at roots of unity. We show how
the structure of a Hopf superalgebra on a quantum superalgebra is determined
by the quantum Weyl groupoid. The Weyl groupoid of $\mathfrak{sl}(m|n)$ is
constructed to this end as some supercategory. We prove that in this case
quantum superalgebras associated with Dynkin diagrams are isomorphic as
superalgebras. It is shown how these quantum superalgebras considered as Hopf
superalgebras are connected via twists and isomorphisms. We explicitly
construct these twists using the Lusztig isomorphisms considered as elements
of the Weyl quantum groupoid. We build a PBW basis for each quantum
superalgebra, and investigate how quantum superalgebras are connected with
their classical limits, i. e. Lie superbialgebras. We find explicit
multiplicative formulas for universal $R$-matrices, describe relations between
them for each realization and classify Hopf superalgebras and triangular
structures for the quantum superalgebra $U_{q}(\mathfrak{sl}(m|n))$.

###### keywords:

Dynkin diagram , Lie superalgebras , Lie superbialgebras , Lusztig
isomorphisms , PBW basis , quantum superalgebras , universal R-matrix ,
quantum Weyl groupoid , supercategory MSC-class: 16W35 (Primary) 16W55, 17B37,
81R50, 16W30 (Secondary)

##  1 Introduction

In this paper we investigate quantum deformation [13] of the Lie superalgebra
at roots of unity in the case of quantum superalgebra $sl(m|n)$, where $m\neq
n$ and $m,n>0$. Our considerations are based on a Weyl groupoid, which we
define as some supercategory, see Definition 3.2. We show how to associate
quantum superalgebras at roots of unity to Dynkin diagrams in Section 4.2.
Although the definition of the Weyl groupoid used by us is a slight
modification of the definition given in [21], it seems to us to be more
suitable for solving further problems of geometrization and categorification
of the main objects of the theory of quantum superalgebras. One of our main
results is Theorem 4.2 where we show that the two realizations are isomorphic
as superalgebras. We investigate how to build a PBW basis for each realization
in Theorem 4.4. In Theorem 4.5 we show how the two realizations are connected
as Hopf superalgebras. We also compute universal $R$-matrices and describe
relations between them for each realization using the formulas for the twists
corresponding to reflections in the Weyl groupoid.

Our work is motivated by results obtained in [21] and reformulated in [12]. In
these papers is defined a Weyl groupoid. In [36] is investigated a Weyl
groupoid related to the Lie superalgebras. The case of quantum superalgebras
is considered in [23]. We were inspired also by results obtained in [30], [31]
and [32]. We also note that this paper generalizes and refines results
obtained in our previous article [35].

We study the structures of Hopf superalgebras and triangular structures
defined by universal R-matrices and describe the connection between them. This
connection is described by twists in terms of the elements of the Weyl quantum
groupoid. Thus, the Weyl groupoid plays a central and connecting role in our
work. We define the Weyl groupoid in a form convenient for the purpose of our
work. Nonetheless, our definition is essentially equivalent to the generally
accepted one.

It should be emphasized that the central role in our reasoning is played by
the construction of Weyl quantum groupoid. Following V. Serganova, H. Yamane,
I. Heckenberger ideas, we define the Weyl groupoid as a supercategory in
generality sufficient for our purposes of studying quantum superalgebras.
Based on this abstract definition, we give an explicit realization of the Weyl
quantum groupoid in terms of isomorphisms of quantum superalgebras generated
by isomorphisms, which are induced by reflections (relatively both even and
odd roots). We interpret the elements of the quantum Weyl groupoid as adjoint
maps given by the elements of the quantum superalgebra $U_{q}(sl(m|n)$ (or
some its extension). Thus, we associate the elements of the quantum Weyl
groupoid with the group-like elements of the quantum superalgebra. Among other
things, we note that these elements satisfy some analogue of the braid group
relations and define braid groupoid structure. It should be noted that the
classical Weyl groupoid, acting by reflections with respect to odd roots,
permutes non-conjugate Borel subalgebras. Thus, it permutes the corresponding
non-isomorphic Manin triples defined by these Borel subalgebras. This gives a
natural description and, moreover, a classification of the structures of Lie
bisuperalgebras on superalgebra $sl(m|n)$. We confine ourselves here to a
detailed consideration of the case $sl(2|1)$. Similarly, the action of the
quantum Weyl groupoid allows one to classify the structures of the Hopf
superalgebra on the quantized universal enveloping superalgebra
$U_{q}(sl(m|n))$. We describe a connection between the structures of the Hopf
superalgebras and give explicit formulas for both twists which intertwines
different comultiplication operations and for universal $R$-matrices of
various Hopf superalgebra structures on $U_{q}(sl(m|n))$. We consider in
detail the case $U_{q}(sl(2|1))$ We explicitly describe the connection of
these constructions with the Weyl quantum groupoid.

We consider the above constructions by specializing the quantization parameter
$q$ with a value equal to the odd root of unity $q^{n}=1,n=2m+1$.

To make the basic constructions as explicit as possible, we restrict ourselves
to considering a particular case and investigate only the Weyl groupoid of the
Lie superalgebra $sl(m|n)$, where $m\neq n$ and $m,n>0$. Nonetheless, all our
considerations can be adopt to the more general case of an arbitrary basic Lie
superalgebra. Thus, our definition is based on the definition of the Weyl
groupoid given in [21], [12], [36] and contains the classical and quantum
versions of the Weyl groupoid. We also give an explicit construction of the
Weyl quantum groupoid using Lustig isomorphisms in the spirit of the [31],
[32] and [33] (see also [20], [5]). Using this explicit description of the
Weyl quantum groupoid we investigate Hopf superalgebras structures and
triangular structures associated with Dynkin diagrams and show how they are
connected via twists and isomorphisms.

Note that our initial goal, partially motivated by applications, was to study
the Hopf structures of quantum affine superalgebras. The second goal is to try
to partially clarify the categorical and geometric nature of quantum
superalgebras and to show how these structures are determined by simple
combinatorial data. But we decided to introduce these important and general
constructions based on the Weyl quantum groupoid in the simplest possible
form, so that technical difficulties would not impede the understanding of
simple basic ideas.

It is worth mentioning that we use the traditional language of the theory of
quantum groups (which are deformations of Lie bialgebras), but our
constructions are equivalent to the constructions used by algebraists in the
abstract theory of Hopf and Nichols algebras (see [5], [6], [4], [20]).

We will now give an outline of this paper. In Section 2 we recall basic facts
about Lie superalgebras, and remind some categorical definitions about
supercategories. Next we describe Lie superalgebra $sl(2|1)$ and show how to
endow it with the Lie superbialgebra structure.

Section 3 is divided in three parts. In Subsection 3.1 we give the definition
of Cartan scheme, use it to construct a category called Weyl groupoid and show
how to build $\mathcal{W}(\mathcal{R})$ the Weyl groupoid of the Lie
superalgebra $sl(2|1)$ in Subsection 3.2. Next in Subsection 3.3 we construct
a faithful covariant functor from the $\mathcal{W}(\mathcal{R})$ to the
category of Lie superalgebras associated with Dynkin diagrams. Moreover, we
show how to endow these Lie superalgebras with the structure of Lie
superbialgebras and investigate how they are related to each other.

Section 4 is divided in six parts. In Subsection 4.1 we recall the definition
of the quantized universal enveloping superalgebras. Next in Subsection 4.2 it
is shown how to associate with Dynkin diagram quantum superalgebra at roots of
unity. Subsection 4.3 contains auxiliary categorical definitions and results
about Hopf superalgebras. In Subsection 4.4 we construct a faithful covariant
functor from the $\mathcal{W}(\mathcal{R})$ to the category of superalgebras
associated with Dynkin diagrams and prove that these superalgebras are
isomorphic. In Subsection 4.5 we show how to build a PBW basis for these
superalgebras in cas of $sl(2|1)$ (see, also [22]). In Subsection 4.6 we
investigate braided Hopf superalgebras associated with Dynkin diagrams and
show how they are connected via twists and isomorphisms. The central result is
Theorem 4.6 which allows us to classify all Hopf superalgebras which we
investigate.

In this paper we use the following notation. Let $\mathbb{N}$, $\mathbb{Z}$
and $\mathbb{Q}$ denote the sets of natural numbers, integers and rational
numbers, respectively. Let $\Bbbk$ be an algebraically closed field of
characteristic zero. We also use Iverson bracket defined by
$[P]=\begin{cases}1\text{ if }P\text{ is true;}\\\ 0\text{
otherwise},\end{cases}$ where $P$ is a statement that can be true or false.

##  2 Special Lie superalgebra $sl(2|1)$

As for the terminology concerning Lie superalgebras, we refer to [26], [15].

A super vector space (superspace) $V$ over field $\Bbbk$ is a $\Bbbk$-vector
space endowed with a $\mathbb{Z}_{2}$-grading, in other words, it writes as a
direct sum of two vector spaces $V=V_{\bar{0}}\oplus V_{\bar{1}}$ such as
$V_{\bar{0}}$ is the even part and $V_{\bar{1}}$ is the odd part. Define a
parity function $\lvert\cdot\rvert:V\to\mathbb{Z}_{2}$ for a homogeneous
element $x$ in a superspace by $|x|=\bar{a}$, where $v\in V_{\bar{a}}$ and
$\bar{a}\in\mathbb{Z}_{2}$. A superalgebra $A$ over the field $\Bbbk$ is a
$\mathbb{Z}_{2}$-graded algebra $A=A_{\bar{0}}\oplus A_{\bar{1}}$ over
$\Bbbk$. A Lie superalgebra is a superalgerba
$\mathfrak{g}=\mathfrak{g}_{\bar{0}}\oplus\mathfrak{g}_{\bar{1}}$ with the
bilinear bracket (the super Lie bracket)
$[\cdot,\cdot]:\mathfrak{g}\times\mathfrak{g}\to\mathfrak{g}$ which satisfies
the following axioms, with homogeneous $x,y,z\in\mathfrak{g}$:

$[x,y]=-(-1)^{|x||y|}[y,x],$ $[x,[y,z]]=[[x,y],z]+(-1)^{|x||y|}[y,[x,z]].$

A Lie superbialgebra $(\mathfrak{g},[\cdot,\cdot],\delta)$ (see [19], [28]) is
a Lie superalgebra $(\mathfrak{g},[\cdot,\cdot])$ with a skew-symmetric linear
map $\delta:\mathfrak{g}\to\mathfrak{g}\otimes\mathfrak{g}$ that preserves the
$\mathbb{Z}_{2}$-grading and satisfies the following conditions:

$(\delta\otimes
id_{\mathfrak{g}})\circ\delta-(id_{\mathfrak{g}}\otimes\delta)\circ\delta=(id_{\mathfrak{g}}\otimes\tau_{\mathfrak{g},\mathfrak{g}})\circ(\delta\otimes
id)\circ\delta,$ (2.1) $\delta([x,y])=(ad_{x}\otimes
id_{\mathfrak{g}}+id_{\mathfrak{g}}\otimes ad_{x})\delta(y)-(ad_{y}\otimes
id_{\mathfrak{g}}+id_{\mathfrak{g}}\otimes ad_{y})\delta(x),$ (2.2)

where $x,y\in\mathfrak{g}$, $id_{\mathfrak{g}}$ is the identity map on
$\mathfrak{g}$, $ad_{x}y=[x,y]$ is the adjoint and $\tau_{V,W}:V\otimes W\to
W\otimes V$ is the linear function given by

$\tau_{V,W}(v\otimes w)=(-1)^{|v||w|}w\otimes v$ (2.3)

for homogeneous $v\in V$ and $w\in W$.

We use the well-known result (for more detail see [40], [41]).

######  Proposition 2.1.

Let $\mathfrak{g}$ be a Lie superalgebra of type A with associated Cartan
matrix $(A=(a_{ij})_{i,j\in I},\tau)$, where $\tau$ is a subset of
$I=\\{1,2,...,n\\}$. Then $\mathfrak{g}$ is generated by $h_{i}$, $e_{i}$ and
$f_{i}$ for $i\in I$ (whose parities are all even except for $e_{t}$ and
$f_{t}$, $t\in\tau$, which are odd), where the generators satisfy the
relations

$[h_{i},h_{j}]=0,\;[h_{i},e_{j}]=a_{ij}e_{j},\;[h_{i},f_{j}]=-a_{ij}f_{j},\;[e_{i},f_{j}]=\delta_{ij}h_{i}$

and the ”super classical Serre-type” relations

$[e_{i},e_{i}]=[f_{i},f_{i}]=0,\;\text{if}\;i\in\tau,$
$(ad_{e_{i}})^{1+|a_{ij}|}e_{j}=(ad_{f_{i}})^{1+|a_{ij}|}f_{j}=0,\;\text{if}\;i\neq
j,\;\text{and}\;i\notin\tau,$
$[[[e_{m-1},e_{m}],e_{m+1}],e_{m}]=[[[f_{m-1},f_{m}],f_{m+1}],f_{m}]=0,\;\text{if}\;m-1,m,m+1\in
I\;\text{and}\;m\in\tau,$

where for $x\in\mathfrak{g}$ the linear mapping
$ad_{x}:\mathfrak{g}\to\mathfrak{g}$ is defined by $ad_{x}(y)=[x,y]$ for all
$y\in\mathfrak{g}$.

Denote by $\mathfrak{n}^{+}$ (resp. $\mathfrak{n}^{-}$) and $\mathfrak{h}$ the
subalgebra of $\mathfrak{g}(A,\tau)$ generated by $e_{1}$, $...$, $e_{n}$
(resp. $f_{1}$, $...$, $f_{n}$) and $h_{1}$, $...$, $h_{n}$. Then define by
$\mathfrak{b}^{+}=\mathfrak{h}\oplus\mathfrak{n}^{+}$ (resp.
$\mathfrak{b}^{-}=\mathfrak{h}\oplus\mathfrak{n}^{-}$) the positive Borel
subalgebra (resp. the negative Borel subalgebra) of $\mathfrak{g}(A,\tau)$.

We remind some categorical definitions. Our notations here follow [9] (see
also [10]). Let $\mathcal{SV}ec$ denote the category of superspaces and all
(not necessarily homogeneous) linear maps. Set $\underline{\mathcal{SV}ec}$ to
be the subcategory of $\mathcal{SV}ec$ consisting of all superspaces but only
the even linear maps (superspace morphisms). The tensor product equips
$\underline{\mathcal{SV}ec}$ with a monoidal structure, and the map $u\otimes
v\to(-1)^{|u||v|}v\otimes u$ makes $\underline{\mathcal{SV}ec}$ into a strict
symmetric monoidal category.

######  Definition 2.1.

  1. 1.

A supercategory means a category enriched in $\underline{\mathcal{SV}ec}$, i.
e. each morphism space is a superspace and composition induces an even linear
map. A superfunctor between categories is a
$\underline{\mathcal{SV}ec}$-enriched functor, i. e. a functor
$F:\mathcal{A}\to\mathcal{B}$ such that the function
$\text{Hom}_{\mathcal{A}}(\lambda,\mu)\to\text{Hom}_{\mathcal{B}}(F\lambda,F\mu)$,
$f\to Ff$ is an even linear map for all
$\lambda,\mu\in\text{Obj}(\mathcal{B})$.

  2. 2.

For any supercategory $\mathcal{A}$, the underlying category
$\underline{\mathcal{A}}$ is the category with the same objects as
$\mathcal{A}$ but only its even morphisms.

  3. 3.

Let sLieAlg be the supercategory which objects are Lie superalgebras over
field $\Bbbk$. A morphism $f\in\text{Hom}_{\text{sLieAlg}}(V,W)$ between Lie
superalgebras $(V,[\cdot,\cdot]_{V})$ and $(W,[\cdot,\cdot]_{W})$ is a linear
map of the underlying vector spaces such that $f([x,y]_{V})=[f(x),f(y)]_{W}$
for all $x,y\in V$.

  4. 4.

Let sBiLieAlg be the supercategory which objects are Lie superbialgebras over
field $\Bbbk$. A morphism $f\in\text{Hom}_{\text{sLieAlg}}(V,W)$ between Lie
superbialgebras $(V,[\cdot,\cdot]_{V},\delta_{V})$ and
$(W,[\cdot,\cdot]_{W},\delta_{W})$ is a linear map of the underlying vector
spaces such that $f([x,y]_{V})=[f(x),f(y)]_{W}$ and $(f\otimes
f)\circ\delta_{V}(x)=\delta_{W}\circ f(x)$ for all $x,y\in V$.

We also need the following general result.

######  Proposition 2.2.

Let
$f\in\text{Hom}_{\underline{\text{sLieAlg}}}(\mathfrak{g_{1}},\mathfrak{g_{2}})$
be an isomorphism. Suppose that $\mathfrak{g_{1}}$ is a Lie superbialgebra
with a skew-symmetric even linear map
$\delta_{\mathfrak{g_{1}}}:\mathfrak{g_{1}}\to\mathfrak{g_{1}}\otimes\mathfrak{g_{1}}$
which satisfies (2.1) - (2.2). Then $f$ induces a Lie superbialgebra structure
on $\mathfrak{g_{2}}$, where a skew-symmetric even linear map
$\delta_{\mathfrak{g_{2}}}:\mathfrak{g_{2}}\to\mathfrak{g_{2}}\otimes\mathfrak{g_{2}}$
which satisfies (2.1) - (2.2) is defined by

$\delta_{\mathfrak{g_{2}}}:=(f\otimes f)\circ\delta_{\mathfrak{g_{1}}}\circ
f^{-1}.$

The special Lie superalgebra $sl(2|1)$ over $\Bbbk$ is the algebra
$M_{3,3}(\Bbbk)$ of $3\times 3$ matrices over $\Bbbk$, $\mathbb{Z}_{2}$-graded
as $sl(2|1)_{\bar{0}}\oplus sl(2|1)_{\bar{1}}$, where

$sl(2|1)_{\bar{0}}=\\{X=diag(A,D)|Str(X):=tr(A)-tr(D)=0,\;A\in
M_{2,2}(\Bbbk),\;D\in M_{1,1}(\Bbbk)\\},$

and

$sl(2|1)_{\bar{1}}=\\{\left(\begin{smallmatrix}0&B\\\
C&0\end{smallmatrix}\right)|B\in M_{2,1}(\Bbbk),\;C\in M_{1,2}(\Bbbk)\\},$

with the bilinear super bracket $[x,y]=xy-(-1)^{ab}yx$ for $x\in
sl(2|1)_{\bar{a}},\;y\in
sl(2|1)_{\bar{b}},\;\bar{a},\bar{b}\in\mathbb{Z}_{2}$, on $sl(2|1)$.

We choose for basis of Lie superalgebra $sl(2|1)$ over $\Bbbk$ the following
elements: $h_{1}=e_{1,1}-e_{2,2},\quad h_{2}=e_{2,2}+e_{3,3},$
$e_{1}=e_{1,2},\quad f_{1}=e_{2,1},$ $e_{2}=e_{2,3},\quad f_{2}=e_{3,2},$
$e_{3}=[e_{1},e_{2}]=e_{1,3},\quad f_{3}=[f_{1},f_{2}]=-e_{3,1},$ where
$e_{i,j}\in M_{3,3}(\Bbbk)$ denotes matrix with $1$ at $(i,j)$-position and
zeros elsewhere. The elements $h_{1},h_{2},e_{1},f_{1}$ are even and
$e_{2},f_{2},e_{3},f_{3}$ are odd. We have $[h_{i},h_{j}]=0$,
$[h_{i},e_{j}]=a_{ij}e_{j}$, $[h_{i},f_{j}]=-a_{ij}f_{j}$,
$[e_{i},f_{j}]=\delta_{ij}h_{i}$, $[e_{2},e_{2}]=[f_{2},f_{2}]=0$,
$[e_{1},[e_{1},e_{2}]]=[f_{1},[f_{1},f_{2}]]=0$ with $(a_{ij})$ the matrix

$A=\left(\begin{array}[]{cc}2&-1\\\ -1&0\\\ \end{array}\right).$

The Cartan subalgebra of $sl(2|1)$ is the $\Bbbk$-span $\mathfrak{h}=\langle
h_{1},h_{2}\rangle$. Denote by $\mathfrak{h}^{*}$ the dual space of
$\mathfrak{h}$. $sl(2|1)$ decomposes as a direct sum of root spaces
$\mathfrak{h}\oplus\bigoplus_{\alpha\in\mathfrak{h}^{*}}sl(2|1)_{\alpha}$,
where

$sl(2|1)_{\alpha}=\\{X\;|\;[h,X]=\alpha(h)X,\;\forall h\in\mathfrak{h}\\}.$

An $\alpha\in\mathfrak{h}^{*}-\\{0\\}$ is called a root if the root space
$sl(2|1)_{\alpha}$ is not zero. The root system for $sl(2|1)$ is defined to be
$\Delta=\\{\alpha\in\mathfrak{h}^{*}\;|\;sl(2|1)_{\alpha}\neq 0,\alpha\neq
0\\}$. Define sets of even and odd roots, respectively, to be
$\Delta_{\bar{0}}=\\{\alpha\in\Delta\;|\;sl(2|1)_{\alpha}\cap
sl(2|1)_{\bar{0}}\neq 0\\}$,
$\Delta_{\bar{1}}=\\{\alpha\in\Delta\;|\;sl(2|1)_{\alpha}\cap
sl(2|1)_{\bar{1}}\neq 0\\}$. Thus we can define a parity function
$\lvert\cdot\rvert_{\Delta}:\Delta\to\mathbb{Z}_{2}$ by $|x|_{\Delta}=\bar{a}$
if $x\in\Delta_{\bar{a}}$, where $\bar{a}\in\mathbb{Z}_{2}$.

Consider the $\Bbbk$-span $\mathfrak{d}=\langle e_{11},e_{22},e_{33}\rangle$
and it’s dual space
$\mathfrak{d}^{*}=\langle\epsilon_{1},\epsilon_{2},\delta_{1}\rangle$. We
define a non-degenerate symmetric bilinear form
$(\cdot,\cdot):\mathfrak{d}^{*}\times\mathfrak{d}^{*}\to\Bbbk$ by

$(\epsilon_{i},\epsilon_{j})=\delta_{ij},\;(\epsilon_{i},\delta_{1})=0,\;(\delta_{1},\delta_{1})=-1$

for all $i,j\in I$, where $I:=\\{1,2\\}$.

Notice that $\mathfrak{h}^{*}\subset\mathfrak{d}^{*}$. Then the root system
$\Delta\subseteq\mathfrak{h}^{*}$ has the form
$\Delta=\Delta_{\bar{0}}\oplus\Delta_{\bar{1}}$, where
$\Delta_{\bar{0}}=\\{\pm(\epsilon_{1}-\epsilon_{2})\\},$
$\Delta_{\bar{1}}=\\{\pm(\epsilon_{1}-\delta_{1}),\pm(\epsilon_{2}-\delta_{1})\\}$.
Accordingly, we also have the decomposition $\Delta=\Delta^{+}\cup\Delta^{-}$,
where
$\Delta^{+}=\\{\epsilon_{1}-\epsilon_{2},\epsilon_{1}-\delta_{1},\epsilon_{2}-\delta_{1}\\}$
and
$\Delta^{-}=\\{\epsilon_{2}-\epsilon_{1},\delta_{1}-\epsilon_{1},\delta_{1}-\epsilon_{2}\\}$.
We choose the basis
$\tau=\\{\alpha_{1}:=\epsilon_{1}-\epsilon_{2},\;\alpha_{2}:=\epsilon_{2}-\delta_{1}\\}$.
The form $(\cdot,\cdot)$ on $\mathfrak{d}^{*}$ induces a non-degenerate
symmetric bilinear form on $\mathfrak{h}^{*}$, which will be denoted by
$(\cdot,\cdot)$ as well. We define a natural pairing
$\langle\cdot,\cdot\rangle:\mathfrak{h}\times\mathfrak{h}^{*}\to\Bbbk$ by
linearity with $\langle h_{i},\alpha\rangle=\alpha(h_{i})$ for all $i\in
I,\;\alpha\in\tau$.

The Cartan matrix is $A=(a_{ij}=\alpha_{j}(h_{i});\;\alpha_{j}\in\tau,i,j\in
I)$. One could describe $A$ by the corresponding Dynkin diagram. Join vertex
$i$ with vertex $j$ if $a_{ij}\neq 0$. We need two types of vertices: $\circ$
if $a_{ii}=2$ and $|\alpha_{i}|_{\Delta}=0$ (it is called white dot);
$\otimes$ if $a_{ii}=0$ and $|\alpha_{i}|_{\Delta}=1$ (it is called grey dot),
where $i\in I$. We say that two Dynkin diagrams or even graphs that correspond
to them are isomorphic if they have the equal number of white and grey dots.

Define the linear function $\delta_{sl(2|1)}:sl(2|1)\to sl(2|1)\otimes
sl(2|1)$ on the generators by

$\delta_{sl(2|1)}(h_{i})=0,\;\delta_{sl(2|1)}(e_{i})=\frac{1}{2}(h_{i}\otimes
e_{i}-e_{i}\otimes h_{i}),\;\delta_{sl(2|1)}(f_{i})=\frac{1}{2}(h_{i}\otimes
f_{i}-f_{i}\otimes h_{i})$ (2.4)

for $i\in I$, and extend it to all the elements of $sl(2|1)$ using equation
(2.2) and by linearity. Then $sl(2|1)$ becomes a Lie superbialgebra.

##  3 Weyl groupoid

We give a categorical definition of a Weyl groupoid. This enable us to
describe the Weyl groupoid of $sl(2|1)$ by generators and relations. We
mention how it is connected with the supercategory of Lie superalgebras.

###  3.1 Cartan schemes and definition of Weyl groupoid

We adopt to our purposes the definition of a Weyl groupoid which was
introduced in [21] and reformulated in [12] (see also [36], [24], [31], [27]).
Thus we define Weyl groupoid as a supercategory. In Section 3.2 we give the
example how Lie superalgebra $sl(2|1)$ fits in our definition.

In order to define Weyl groupoid we need auxiliary data. In this way we
associate with an object (Dynkin diagram) of the Weyl groupoid an element of a
set $D$ which labels it, root basis $\tau$, maps $\rho$ which indicate the
direction of the action and special matrices $C$ used to define reflections.

Let $V$ be a fixed $(N_{\bar{0}},N_{\bar{1}})$-dimensional super vector space
and $N=N_{\bar{0}}+N_{\bar{1}}$.

######  Definition 3.1.

Let $A$ and $D$ be non-empty sets, where $A=(a_{d})_{d\in D}$, $\tau^{d}$ be a
basis of $V$ for all ${d\in D}$, $\rho_{\alpha}:S_{d}\rightarrow
S_{d^{{}^{\prime}}}$ bijective mappings for all $\alpha\in\tau^{d}$, $d\in D$
and for some non-empty $S_{d},S_{d^{{}^{\prime}}}\subseteq A$,
$d^{{}^{\prime}}\in D$, and $N\times N$ matrices
$C^{d}=(c_{\alpha,\beta}^{d}\in\mathbb{Z})_{\alpha,\beta\in\tau^{d}}$ for all
$d\in D$. The tuple

$\mathcal{C}=\mathcal{C}(A,D,V,(\tau^{d})_{d\in
D},(\rho^{d}_{\alpha})_{\alpha\in\tau^{d},d\in D},(C^{d})_{d\in D})$

is called a Cartan scheme if for all $d\in D$

  1. 1.

if $\rho_{\alpha}:S_{d}\to S_{d^{{}^{\prime}}}$ then $\exists\beta\in
S_{d^{{}^{\prime}}}$: $\rho_{\beta}\rho_{\alpha}=id_{S_{d}}$ and
$\rho_{\alpha}\rho_{\beta}=id_{S_{d^{{}^{\prime}}}}$ for all
$\alpha\in\tau_{d}$, $d\in D$ and for some $d^{{}^{\prime}}\in D$,

  2. 2.

$c_{\alpha,\alpha}^{d}=2$ and $c_{\alpha,\beta}^{d}\leq 0$, where
$\alpha,\beta\in\tau^{d}$ with $\alpha\neq\beta$,

  3. 3.

if $c_{\alpha,\beta}^{d}=0$, then $c_{\beta,\alpha}^{d}=0$, where
$\alpha,\beta\in\tau^{d}$,

  4. 4.

if $\rho_{\alpha}:S_{d}\to S_{d^{{}^{\prime}}}$ then there exists bijective
mapping $i:\tau^{d}\to\tau^{d^{\prime}}$ such that
$c^{d}_{\alpha,\beta}=c^{d^{\prime}}_{i(\alpha),i(\beta)}$ for all
$\alpha,\beta\in\tau_{d}$, $d\in D$ and for some $d^{{}^{\prime}}\in D$.

Now we are able to formulate the definition of a Weyl groupoid where morphisms
are generalizations of compositions of reflections.

######  Definition 3.2.

Let $\mathcal{C}=\mathcal{C}(A,D,V,(\tau^{d})_{d\in
D},(\rho_{\alpha})_{\alpha\in\tau^{d},d\in D},(C^{d})_{d\in D})$ be a Cartan
scheme. For all $d\in D$ and $\alpha,\beta\in\tau^{d}$ define the function
$\sigma_{\alpha}^{d}\in GL_{N}(V)$ by

$\sigma_{\alpha}^{d}(\beta)=\beta-c_{\alpha,\beta}^{d}\alpha.$ (3.1)

The Weyl groupoid of $\mathcal{C}$ is the supercategory
$\mathcal{W}(\mathcal{C})$ such that $\text{Obj}(\mathcal{W}(\mathcal{C}))=A$
and the morphisms are compositions of maps $\sigma_{\alpha}^{d}$ with $d\in D$
and $\alpha\in\tau^{d}$, where $\sigma_{\alpha}^{d}$ is considered as an
element of super vector space
$\text{Hom}_{\mathcal{W}(\mathcal{C})}(a_{d},\rho_{\alpha}(a_{d}))$. The
cardinality of $D$ is the rank of $\mathcal{W}(\mathcal{C})$.

It is easy to see that the above definition is correct and the Weyl groupoid
is in fact a supercategory, where the parity of morphism $\sigma_{\alpha}^{d}$
is determined by parity of $\alpha$ as an element of a basis $\tau^{d}$, where
$d\in D$. We reformulate the condition 4 in definition 3.1 in the following
way
$c^{d}_{\alpha,\beta}=c^{d^{{}^{\prime}}}_{\sigma_{\alpha}^{d}(\alpha),\sigma_{\alpha}^{d}(\beta)}$,
where $\rho_{\alpha}(a_{d})=a_{d^{{}^{\prime}}}$ ($d^{{}^{\prime}}\in D$) and
for all $\alpha,\beta\in\tau_{d}$, $d\in D$. Thus a morhpism
$\sigma_{\alpha}^{d}$ (3.1) becomes invertible and the inverse is
$\sigma_{\sigma_{\alpha}^{d}(\alpha)}^{d^{{}^{\prime}}}$.

######  Definition 3.3.

A Cartan scheme is called connected if its Weyl groupoid is connected, that
is, if for all $a,b\in A$ there exists
$w\in\text{Hom}_{\mathcal{W}(\mathcal{C})}(a,b)$. The Cartan scheme is called
simply connected, if it is connected and

$\text{Hom}_{\mathcal{W}(\mathcal{C})}(a,a)=\\{id_{a}\\}$

for all $a\in A$.

We characterize root systems in axiomatic way and also add explicit conditions
that are imposed on reflections.

######  Definition 3.4.

Let $\mathcal{C}=\mathcal{C}(A,D,V,(\tau^{d})_{d\in
D},(\rho^{d}_{\alpha})_{\alpha\in\tau^{d},d\in D},(C^{d})_{d\in D})$ be a
Cartan scheme. For all $a_{d}\in A$ let $R^{a_{d}}\subseteq V$, and define
$m_{\alpha,\beta}^{a_{d}}=|R^{a_{d}}\cap(\mathbb{N}_{0}\alpha+\mathbb{N}_{0}\beta)|$
for all $\alpha,\beta\in\tau^{d}$ and $d\in D$. We say that

$\mathcal{R}=\mathcal{R}(\mathcal{C},(R^{a})_{a\in A})$

is a root system of type $\mathcal{C}$, if it satisfies the following axioms:

  1. 1.

exists decomposition $R^{a}=R_{+}^{a}\cup-R_{+}^{a}$, for all $a\in A$;

  2. 2.

$R^{a_{d}}\cap\mathbb{Z}\alpha=\\{\alpha,-\alpha\\}$ for all
$\alpha\in\tau^{d}$ and $d\in D$;

  3. 3.

$\sigma_{\alpha}^{d}(R^{a_{d}})=R^{\rho_{\alpha}^{d}(a_{d})}$ for all
$\alpha\in\tau^{d}$ and $d\in D$;

  4. 4.

for $a,b\in A$, if $id_{V}\in\text{Hom}_{\mathcal{W}(\mathcal{C})}(a,b)$, then
$a=b$.

The elements of the set $R^{a}$, where $a\in A$, are called roots. The root
system $\mathcal{R}$ is called finite if for all $a\in A$ the set $R^{a}$ is
finite. If $\mathcal{R}$ is a root system of type $\mathcal{C}$, then we say
that $\mathcal{W}(\mathcal{R}):=\mathcal{W}(\mathcal{C})$ is the Weyl groupoid
of $\mathcal{R}$.

###  3.2 Weyl groupoid of $sl(m|n)$

Now we are able to construct the Weyl groupoid of the Lie superalgebra
$sl(2|1)$. Note that it follows from [7] that Definition 3.2 of a Weyl
groupoid is true for the simple Lie superalgebra $A(m,n)$ for
$m,n\in\mathbb{Z}_{\geq 0}$, where $A(m,n)$ is defined by $sl(m+1,n+1)$ if
$m\neq n$, and otherwise $A(n,n):=sl(n+1,n+1)/\mathfrak{i}$, where
$\mathfrak{i}$ is its unique one-dimensional ideal. We use notations
$\epsilon_{i}$ ($i\in\overline{1,m}$), $\delta_{j-m}$
($j\in\overline{m+1,m+n}$) from Sections 2 and 3.1.

Let $I=\overline{1,m+n-1}$. The elements of a finite set $D\subset\mathbb{N}$
will be used to label different Dynkin diagrams for $A(m-1,n-1)$. We make a
convention to parameterize elements of the dual space $\mathfrak{d}^{*}$ by
the set $I(m|n)=\\{1,2,...,m+n\\}$, where

$\bar{\epsilon}_{i}:=\begin{cases}\epsilon_{i},&\mbox{ if
}\;i\in\overline{1,m},\\\ \delta_{i-m},&\mbox{ if
}\;i\in\overline{m+1,m+n}.\end{cases}$

Let

$(\tau^{d}=\\{\alpha_{1,d}=\bar{\epsilon}_{i_{1}}-\bar{\epsilon}_{i_{2}},\;\alpha_{2,d}=\bar{\epsilon}_{i_{2}}-\bar{\epsilon}_{i_{3}}\;,...,\;\alpha_{m+n-1,d}=\bar{\epsilon}_{i_{m+n-1}}-\bar{\epsilon}_{i_{m+n}}|\\{i_{1},...,i_{m+n}\\}=I(m|n)\\})_{d\in
D}.$ (3.2)

Notice that $\tau^{d}$ is the basis of $\mathfrak{h}^{*}$ for all ${d\in D}$.

Set

$\tau^{1}=\\{\alpha_{1,1}=\epsilon_{1}-\epsilon_{2},\alpha_{2,1}=\epsilon_{2}-\epsilon_{3},...,\alpha_{m,1}=\epsilon_{m}-\delta_{1},\alpha_{m+1,1}=\delta_{1}-\delta_{2},...,\alpha_{m+n-1,1}=\delta_{n-1}-\delta_{n}\\}.$

We recall that $\tau^{1}$ is called a distinguished simple root system for a
Lie superalgebra of type $sl(m,n)$ ($A(m-1,n-1)$) ([16]).

Consider the family of symmetric matrices
$A_{d}=((\alpha_{i}^{d},\alpha_{j}^{d}))_{i,j\in I}$, where the non-degenerate
symmetric bilinear form
$(\cdot,\cdot):\mathfrak{h}^{*}\times\mathfrak{h}^{*}\to\Bbbk$ is defined as
in Section 2. Define a family of tuples $A=(a_{d}=(A_{d},\tau^{d}))_{d\in D}$.

Let $c_{\alpha,\beta}:=-\text{max}\\{k\in\mathbb{Z}_{\geq
0}\;|\;\beta+k\alpha\in\Delta\\}$ for $\alpha\neq\beta$, and
$c_{\alpha,\alpha}:=2$, where $\alpha,\beta\in\Delta$. Introduce a family of
matrices
$(C^{d}=(c_{\alpha,\beta}^{d}:=c_{\alpha,\beta})_{\alpha,\beta\in\tau^{d}})_{d\in
D}$.

Denote the (usual) left action of the symmetric group $S_{m+n}$ on $I(m|n)$ by
$\triangleright:S_{m+n}\times I(m|n)\to I(m|n)$ and on $\Delta$ by
$\circlearrowleft:S_{m+n}\times\Delta\to\Delta$, where
$s\circlearrowleft(\bar{\epsilon}_{j_{1}}-\bar{\epsilon}_{j_{2}})=\bar{\epsilon}_{s\triangleright
j_{1}}-\bar{\epsilon}_{s\triangleright j_{2}}$, for $s\in
S_{m+n},\;j_{1},j_{2}\in I(m|n)$. Thus define for all $\alpha\in\tau^{d}$
($d\in D$) functions $\rho_{\alpha}^{d}:\\{a_{d}\\}\to\\{a_{b}\\}$, such that
$\rho_{\alpha}^{d}(a_{d})=a_{b}$, where $b\in D$,
$\alpha=\bar{\epsilon}_{j_{1}}-\bar{\epsilon}_{j_{2}}$ and
$\tau^{b}=\\{\alpha_{k}^{{}^{\prime}}:=(j_{1},j_{2})\circlearrowleft\alpha_{k}=\sigma_{\alpha}^{d}(\alpha_{k})\;|\;\alpha_{k}\in\tau^{d},k\in
I\\}$.

Consider the simply connected Cartan scheme

$\mathcal{C}=\mathcal{C}(A,D,\mathfrak{h}^{*},(\tau^{d})_{d\in
D},(\rho^{d}_{\alpha})_{\alpha\in\tau^{d},d\in D},(C^{d})_{d\in D}).$

Notice that $\mathcal{R}=\mathcal{R}(\mathcal{C},(\Delta=\Delta^{a})_{a\in
A})$ is the root system of type $\mathcal{C}$. We call
$\mathcal{W}(\mathcal{R})$ the Weyl groupoid of $sl(m|n)$. It is easy to
construct Weyl groupoid for $sl(2|1)$ (see Fig. 1).

${\underset{\epsilon_{1}-\epsilon_{2}}{\bigcirc}}$${\underset{\epsilon_{2}-\delta_{1}}{\bigotimes}}$${\underset{\epsilon_{1}-\delta_{1}}{\bigotimes}}$${\underset{\delta_{1}-\epsilon_{2}}{\bigotimes}}$${\underset{\delta_{1}-\epsilon_{1}}{\bigotimes}}$${\underset{\epsilon_{1}-\epsilon_{2}}{\bigcirc}}$${\underset{\epsilon_{2}-\epsilon_{1}}{\bigcirc}}$${\underset{\epsilon_{1}-\delta_{1}}{\bigotimes}}$${\underset{\epsilon_{2}-\delta_{1}}{\bigotimes}}$${\underset{\delta_{1}-\epsilon_{1}}{\bigotimes}}$${\underset{\delta_{1}-\epsilon_{2}}{\bigotimes}}$${\underset{\epsilon_{2}-\epsilon_{1}}{\bigcirc}\par\par}$$\scriptstyle{d=1}$$\scriptstyle{\sigma_{\epsilon_{2}-\delta_{1}}^{1}}$$\scriptstyle{\sigma_{\delta_{1}-\epsilon_{2}}^{3}}$$\scriptstyle{d=3}$$\scriptstyle{\sigma_{\epsilon_{1}-\delta_{1}}^{3}}$$\scriptstyle{\sigma_{\delta_{1}-\epsilon_{1}}^{5}}$$\scriptstyle{d=5}$$\scriptstyle{d=2}$$\scriptstyle{\sigma_{\epsilon_{1}-\delta_{1}}^{2}}$$\scriptstyle{\sigma_{\delta_{1}-\epsilon_{1}}^{4}}$$\scriptstyle{d=4}$$\scriptstyle{\sigma_{\epsilon_{2}-\delta_{1}}^{4}}$$\scriptstyle{\sigma_{\delta_{1}-\epsilon_{2}}^{6}}$$\scriptstyle{d=6}$$\scriptstyle{\sigma_{\epsilon_{1}-\epsilon_{2}}^{1}}$$\scriptstyle{\sigma_{\epsilon_{2}-\epsilon_{1}}^{2}}$$\scriptstyle{\sigma_{\epsilon_{1}-\epsilon_{2}}^{5}}$$\scriptstyle{\sigma_{\epsilon_{2}-\epsilon_{1}}^{6}}$

Figure 1: Dynkin Diagrams of $sl(2|1)$

$\mathcal{W}(\mathcal{R})$ is the supercategory generated by morphisms (recall
(3.1))

$\mathcal{B}=\\{\sigma_{\alpha_{i,d}}^{d}\in\text{Hom}(\mathcal{W}(\mathcal{R}))|\;i\in
I,d\in D\\}$ (3.3)

and by relations for $j\in J$

$\sigma_{x}^{d_{2}}\sigma_{\alpha_{i,d_{1}}}^{d_{1}}=id_{a_{d_{1}}},\;\sigma_{\alpha_{i,d_{1}}}^{d_{1}}\sigma_{x}^{d_{2}}=id_{a_{d_{2}}},$
(3.4)

where $x=\sigma_{\alpha_{i,d_{1}}}^{d_{1}}(\alpha_{i,d_{1}})$;

$\sigma_{\alpha_{j,d_{2}}}^{d_{2}}\sigma_{\alpha_{i,d_{1}}}^{d_{1}}=\sigma_{\alpha_{i,d_{2}}}^{d_{2}}\sigma_{\alpha_{j,d_{1}}}^{d_{1}},$
(3.5)

iff $|i-j|\geq 2$ and $|\alpha_{i,d_{1}}|=|\alpha_{j,d_{1}}|=0$;

$\sigma_{\alpha_{j,d_{4}}}^{d_{4}}\sigma_{\alpha_{i,d_{3}}}^{d_{3}}\sigma_{\alpha_{j,d_{2}}}^{d_{2}}\sigma_{\alpha_{i,d_{1}}}^{d_{1}}=\sigma_{\alpha_{i,d_{4}}}^{d_{4}}\sigma_{\alpha_{j,d_{3}}}^{d_{3}}\sigma_{\alpha_{i,d_{2}}}^{d_{2}}\sigma_{\alpha_{j,d_{1}}}^{d_{1}},$
(3.6)

iff $|i-j|\geq 2$;

$\sigma_{\alpha_{i,d_{3}}}^{d_{3}}\sigma_{\alpha_{j,d_{2}}}^{d_{2}}\sigma_{\alpha_{i,d_{1}}}^{d_{1}}=\sigma_{\alpha_{j,d_{3}}}^{d_{3}}\sigma_{\alpha_{i,d_{2}}}^{d_{2}}\sigma_{\alpha_{j,d_{1}}}^{d_{1}},$
(3.7)

iff $|i-j|=1$ and $|\alpha_{i,d_{1}}|=|\alpha_{j,d_{1}}|$;

$\sigma_{\alpha_{j,d_{6}}}^{d_{6}}\sigma_{\alpha_{i,d_{5}}}^{d_{5}}\sigma_{\alpha_{j,d_{4}}}^{d_{4}}\sigma_{\alpha_{i,d_{3}}}^{d_{3}}\sigma_{\alpha_{j,d_{2}}}^{d_{2}}\sigma_{\alpha_{i,d_{1}}}^{d_{1}}=\sigma_{\alpha_{i,d_{6}}}^{d_{6}}\sigma_{\alpha_{j,d_{5}}}^{d_{5}}\sigma_{\alpha_{i,d_{4}}}^{d_{4}}\sigma_{\alpha_{j,d_{3}}}^{d_{3}}\sigma_{\alpha_{i,d_{2}}}^{d_{2}}\sigma_{\alpha_{j,d_{1}}}^{d_{1}},$
(3.8)

iff $|i-j|=1$.

Thus we have theorem (definition)

######  Theorem 3.1.

Weyl groupoid $\mathcal{W}(\mathcal{R})$ has the structure of the braid
groupoid of type $A$, i. e. it is groupoid with generators $\mathcal{B}$ (3.3)
and relations
$\sigma_{\alpha}\sigma_{\beta}...=\sigma_{\beta}\sigma_{\alpha}...$ for all
$\alpha,\beta\in\mathcal{B}$, $\alpha\neq\beta$ with $m(\alpha,\beta)$ factors
on both sides, where $m(\alpha,\beta)$ is the order of
$\sigma_{\alpha}\sigma_{\beta}$ in $\mathcal{W}(\mathcal{R})$.

###### Proof.

The results follows immediately from equations (3.4) - (3.8) and direct
computations. ∎

Let us note that the above described braid groupoid is a natural
generalization of the braid group.

###  3.3 Connection with the supercategory of Lie superalgebras

We use notations introduced in Section 3.2. Fix
$A_{d}=((\alpha_{i}^{d},\alpha_{j}^{d}))_{i,j\in I}$ and the set of simple
roots $\tau^{d}$ for $d\in D$. Define a Lie superalgebra
$\mathfrak{g}(A_{d},\tau^{d})$ to be a Lie superalgebra generated as in
Proposition 2.1. Recall the definition of the supercategory
$\underline{\text{sLieAlg}}$ (see Definition 2.1).

We are able to construct the covariant faithful functor
$F:\mathcal{W}(\mathcal{R})\to\underline{\text{sLieAlg}}$. Thus the action on
objects is given by the formula

$F((A_{d},\tau^{d}))=\mathfrak{g}(A_{d},\tau^{d}),$ (3.9)

where $A_{d}=((\alpha_{i}^{d},\alpha_{j}^{d}))_{i,j\in I}$ and $d\in D$.
Notice that $sl(m,n)=\mathfrak{g}(A_{1},\tau^{1})$, where $\tau^{1}$ be a
distinguished system of simple roots ([16]).

Let $\rho(a_{d_{1}})=a_{d_{2}}$ for $d_{1},d_{2}\in D$. We prove in Theorem
3.2 that $\mathfrak{g}(A_{d_{1}},\tau^{d_{1}})$ and
$\mathfrak{g}(A_{d_{2}},\tau^{d_{2}})$ are isomorphic in
$\underline{\text{sLieAlg}}$. Consider a generator
$\sigma_{\alpha_{i,d_{1}}}^{d_{1}}\in\text{Hom}_{\mathcal{W}(\mathcal{R})}(a_{d_{1}},a_{d_{2}})$
(3.3) for an appropriate $\alpha_{i,d_{1}}\in\tau^{d_{1}}$ ($i\in I$) and fix
an isomorphism

$V_{i,d_{1}}\in\text{Hom}_{\underline{\text{sLieAlg}}}(\mathfrak{g}(A_{d_{1}},\tau^{d_{1}}),\mathfrak{g}(A_{d_{2}},\tau^{d_{2}})).$

Define $F(\sigma_{\alpha_{i,d_{1}}}^{d_{1}})=V_{i,d_{1}}$ and
$F(\sigma_{x}^{d_{2}})=V_{i,d_{2}}$, where
$x=\sigma_{\alpha_{i,d_{1}}}^{d_{1}}(\alpha_{i,d_{1}})$ and
$V_{i,d_{2}}:=V_{i,d_{1}}^{-1}$. Additionally suppose that isomorphisms
$V_{i,d}$ satisfy relations (3.4) - (3.8) for all $i\in I$ and $d\in D$. It is
easy to see that $F$ is indeed the covariant faithful functor that preserves
the structure of the Weyl groupoid. We give an example of the family of
isomorphisms
$\\{F(\sigma)\in\text{Hom}(\underline{\text{sLieAlg}})\\}_{\sigma\in\mathcal{B}}$.
We formulate

######  Theorem 3.2.

There exist the unique covariant faithful functor
$F:\mathcal{W}(\mathcal{R})\to\underline{\text{sLieAlg}}$ which satisfies
equation (3.9) and for all $\sigma_{\alpha_{i,d_{1}}}^{d_{1}}\in\mathcal{B}$
($i\in I$)

$F(\sigma_{\alpha_{i,d_{1}}}^{d_{1}})=L_{i,d_{1}},\;F(\sigma_{x}^{d_{2}})=L_{i,d_{2}}^{-}$
(3.10)

and

$L_{i,d_{1}},L_{i,d_{1}}^{-}:\mathfrak{g}(A_{d_{1}},\tau^{d_{1}})\to\mathfrak{g}(A_{d_{2}},\tau^{d_{2}})$

are unique isomorphisms in $\underline{\text{sLieAlg}}$ satisfying equations
(3.11) - (3.22) below for $|i-j|=1$ and $|i-l|\geq 2$ ($j,l\in I$)

$L_{i,d_{1}}(h_{i,d_{1}})=L_{i,d_{1}}^{-}(h_{i,d_{1}})=-h_{i,d_{2}},$ (3.11)
$L_{i,d_{1}}(h_{j,d_{1}})=L_{i,d_{1}}^{-}(h_{j,d_{1}})=h_{i,d_{2}}+h_{j,d_{2}},\;L_{i,d_{1}}(h_{l,d_{1}})=L_{i,d_{1}}^{-}(h_{l,d_{1}})=h_{l,d_{2}},$
(3.12) $L_{i,d_{1}}(e_{i,d_{1}})=(-1)^{|\alpha_{i,d_{1}}|}f_{i,d_{2}},$ (3.13)
$L_{i,d_{1}}(f_{i,d_{1}})=e_{i,d_{2}},$ (3.14)
$L_{i,d_{1}}^{-}(e_{i,d_{1}})=f_{i,d_{2}},$ (3.15)
$L_{i,d_{1}}^{-}(f_{i,d_{1}})=(-1)^{|\alpha_{i,d_{1}}|}e_{i,d_{2}},$ (3.16)
$L_{i,d_{1}}(e_{j,d_{1}})=-(x,y)[e_{i,d_{2}},e_{j,d_{2}}],$ (3.17)
$L_{i,d_{1}}(f_{j,d_{1}})=[f_{j,d_{2}},f_{i,d_{2}}],$ (3.18)
$L_{i,d_{1}}^{-}(e_{j,d_{1}})=(\alpha_{i,d_{1}},\alpha_{j,d_{1}})(-1)^{|x|(|x|+|y|)+[b=-1]+[b(x,y)=1]}[e_{j,d_{2}},e_{i,d_{2}}],$
(3.19)
$L_{i,d_{1}}^{-}(f_{j,d_{1}})=(-1)^{1+|x||y|+[b=1]+[b(x,y)=-1]}[f_{i,d_{2}},f_{j,d_{2}}],$
(3.20) $L_{i,d_{1}}(e_{l,d_{1}})=L_{i,d_{1}}^{-}(e_{l,d_{1}})=e_{l,d_{2}},$
(3.21) $L_{i,d_{1}}(f_{l,d_{1}})=L_{i,d_{1}}^{-}(f_{l,d_{1}})=f_{l,d_{2}},$
(3.22)

where $x=\sigma_{\alpha_{i,d_{1}}}^{d_{1}}(\alpha_{i,d_{1}})$,
$y=\sigma_{\alpha_{i,d_{1}}}^{d_{1}}(\alpha_{j,d_{1}})$ and
$b=[|\alpha_{i,d_{1}}|=0]-[|\alpha_{i,d_{1}}|=1]$.

One has $L_{i,d_{2}}^{-}=(L_{i,d_{1}})^{-1}$.

###### Proof.

The proof follows from the considerations preceding the statement and from the
direct computations. ∎

######  Remark 3.1.

Note that the inverse to
$L_{i,d_{1}}:\mathfrak{g}(A_{d_{1}},\tau^{d_{1}})\to\mathfrak{g}(A_{d_{2}},\tau^{d_{2}})$
is given by the following equations

$L_{i,d_{1}}^{-1}(h_{i,d_{2}})=-h_{i,d_{1}},\;L_{i,d_{1}}^{-1}(h_{j,d_{2}})=h_{i,d_{1}}+h_{j,d_{1}},\;L_{i,d_{1}}^{-1}(h_{l,d_{2}})=h_{l,d_{1}},$
$L_{i,d_{1}}^{-1}(e_{i,d_{2}})=f_{i,d_{1}},$
$L_{i,d_{1}}^{-1}(f_{i,d_{2}})=(-1)^{|\alpha_{i,d_{1}}|}e_{i,d_{1}},$
$L_{i,d_{1}}^{-1}(e_{j,d_{2}})=(x,y)(-1)^{|\alpha_{i,d_{1}}|(|\alpha_{i,d_{1}}|+|\alpha_{j,d_{1}}|)+[b=-1]+[b(\alpha_{i,d_{1}},\alpha_{j,d_{1}})=1]}[e_{j,d_{1}},e_{i,d_{1}}],$
$L_{i,d_{1}}^{-1}(f_{j,d_{2}})=(-1)^{1+|\alpha_{i,d_{1}}||\alpha_{j,d_{1}}|+[b=1]+[b(\alpha_{i,d_{1}},\alpha_{j,d_{1}})=-1]}[f_{i,d_{1}},f_{j,d_{1}}],$
$L_{i,d_{1}}^{-1}(e_{l,d_{2}})=e_{l,d_{1}},\;L_{i,d_{1}}^{-1}(f_{l,d_{2}})=f_{l,d_{1}}.$

######  Corollary 3.1.

Fix any $d_{1}$ and $d_{2}$ in $D$. Then
$\mathfrak{g}(A_{d_{1}},\tau^{d_{1}})$ and
$\mathfrak{g}(A_{d_{2}},\tau^{d_{2}})$ are isomorphic in
$\underline{\text{sLieAlg}}$.

###### Proof.

The result follows from Theorem 3.2 and the fact that Weyl groupoid
$\mathcal{W}(\mathcal{R})$ is simply connected. ∎

In assumptions made in Theorem 3.2 we prove that isomorphisms $L_{i,d}$ ($i\in
I$, $d\in D$) satisfy equations

######  Lemma 3.1.

$L_{i,d_{2}}^{-}L_{i,d_{1}}=id_{a_{d_{1}}},\;L_{i,d_{1}}L_{i,d_{2}}^{-}=id_{a_{d_{2}}},$
(3.23)

where $x=\sigma_{\alpha_{i,d_{1}}}^{d_{1}}(\alpha_{i,d_{1}})$;

$L_{j,d_{2}}L_{i,d_{1}}=L_{i,d_{2}}L_{j,d_{1}},$ (3.24)

iff $|i-j|\geq 2$ and $|\alpha_{i,d_{1}}|=|\alpha_{j,d_{1}}|=0$;

$L_{j,d_{4}}L_{i,d_{3}}L_{j,d_{2}}L_{i,d_{1}}=L_{i,d_{4}}L_{j,d_{3}}L_{i,d_{2}}L_{j,d_{1}},$
(3.25)

iff $|i-j|\geq 2$;

$L_{i,d_{3}}L_{j,d_{2}}L_{i,d_{1}}=L_{j,d_{3}}L_{i,d_{2}}L_{j,d_{1}},$ (3.26)

iff $|i-j|=1$ and $|\alpha_{i,d_{1}}|=|\alpha_{j,d_{1}}|$;

$L_{j,d_{6}}L_{i,d_{5}}L_{j,d_{4}}L_{i,d_{3}}L_{j,d_{2}}L_{i,d_{1}}=L_{i,d_{6}}L_{j,d_{5}}L_{i,d_{4}}L_{j,d_{3}}L_{i,d_{2}}L_{j,d_{1}},$
(3.27)

iff $|i-j|=1$.

###### Proof.

The result follows from the straightforward computations. ∎

######  Theorem 3.3.

The braid goupoid of type A operates via isomorphisms on Lie superalgebras
$\mathfrak{g}(A_{d},\tau^{d})$ ($d\in D$).

###### Proof.

The result follows from Lemma 3.1 and Theorem 3.1. ∎

We can endow Lie superalgebras $\mathfrak{g}(A_{d},\tau^{d})$ with the
structure of a Lie superbialgebra. Recall that
$A_{d}=((\alpha_{i}^{d},\alpha_{j}^{d}))_{i,j\in I}$. Define the linear
function
$\delta:\mathfrak{g}(A_{d},\tau^{d})\to\mathfrak{g}(A_{d},\tau^{d})\otimes\mathfrak{g}(A_{d},\tau^{d})$
on the generators using formulas (2.4), and extend it to all elements of
$\mathfrak{g}(A_{d},\tau^{d})$ using equation (2.2) and by linearity. Then
$\mathfrak{g}(A_{d},\tau^{d})$ becomes the Lie superbialgebra.

######  Theorem 3.4.

Let $\mathcal{W}(\mathcal{R})$ the Weyl groupoid of $sl(2|1)$. Then the Dynkin
diagrams in $\mathcal{W}(\mathcal{R})$ are isomorphic as graphs if and only if
corresponding Lie superbialgebras are isomorphic.

###### Proof.

We prove that if Dynkin diagrams are isomorphic then corresponding Lie
superbialgebras are isomorphic (see Fig. 1). Consider the unique morphisms

$W_{d_{1},d_{2}}\in\text{Hom}_{\underline{\text{sBiLieAlg}}}(\mathfrak{g}(A_{d_{1}},\tau^{d_{1}}),\mathfrak{g}(A_{d_{2}},\tau^{d_{2}}))$

such that

$W_{d_{1},d_{2}}(h_{i,d_{1}})=h_{i,d_{2}},\;W_{d_{1},d_{2}}(e_{i,d_{1}})=e_{i,d_{2}},$
$W_{d_{1},d_{2}}(f_{i,d_{1}})=f_{i,d_{2}},$

for all $i\in I$ and $(d_{1},d_{2})\in\\{(1,2),(3,4),(5,6)\\}$. It is easy to
see that $W_{d_{1},d_{2}}$ are isomorphisms. Also there exist the unique
isomorphism

$W_{d_{1},d_{2}}(h_{i,d_{1}})=h_{j,d_{2}},\;W_{d_{1},d_{2}}(e_{i,d_{1}})=e_{j,d_{2}},$
$W_{d_{1},d_{2}}(f_{i,d_{1}})=f_{j,d_{2}},$

where $i\neq j$ ($i,j\in I$) for $(d_{1},d_{2})\in\\{(1,5)\\}$. So the
necessity follows.

Now it remains to show that there is no isomorphism

$W_{1,3}\in\text{Hom}_{\underline{\text{sBiLieAlg}}}(\mathfrak{g}(A_{1},\tau^{1}),\mathfrak{g}(A_{3},\tau^{3}))$

as corresponding Dynkin diagrams are not isomorphic. Indeed, let $\phi$ be an
isomorphism. As an $\phi$ is an even morphism then
$\phi(e_{1,1})\in\mathfrak{g}(A_{3},\tau^{3})_{\bar{0}}$ and we have

$\phi(e_{1,1})=\gamma_{1}[e_{1,d_{2}},e_{2,d_{2}}]+\gamma_{2}[f_{2,d_{2}},f_{1,d_{2}}]+\gamma_{3}h,$

where $\gamma_{1},\gamma_{2},\gamma_{3}\in\Bbbk$ and $h$ is an element of
Cartan subalgebra. Then

$\delta_{2}\circ\phi(e_{1,1})=$
$\gamma_{1}\frac{1}{2}(1+(-1)^{|e_{1,d_{2}}||e_{2,d_{2}}|})\cdot$
$(e_{2,d_{2}}\otimes e_{1,d_{2}}-e_{1,d_{2}}\otimes
e_{2,d_{2}}+(h_{1,d_{2}}+h_{2,d_{2}})\otimes[e_{1,d_{2}},e_{2,d_{2}}]-[e_{1,d_{2}},e_{2,d_{2}}]\otimes(h_{1,d_{2}}+h_{2,d_{2}}))+$
$\gamma_{2}\frac{1}{2}(1+(-1)^{|f_{1,d_{2}}||f_{2,d_{2}}|})\cdot$
$(f_{1,d_{2}}\otimes f_{2,d_{2}}-f_{2,d_{2}}\otimes
f_{1,d_{2}}+(h_{1,d_{2}}+h_{2,d_{2}})\otimes[f_{2,d_{2}},f_{1,d_{2}}]-[f_{2,d_{2}},f_{1,d_{2}}]\otimes(h_{1,d_{2}}+h_{2,d_{2}}))=0;$
$\phi\otimes\phi\circ\delta_{1}(e_{1,1})=$
$\frac{1}{2}(\phi(h_{1,1})\otimes\phi(e_{1,1})-\phi(e_{1,1})\otimes\phi(h_{1,1}))\neq
0,$

where $\delta_{1}$ ($\delta_{2}$) is the cocommutator for
$\mathfrak{g}(A_{1},\tau^{1})$ ($\mathfrak{g}(A_{2},\tau^{2})$). Thus we get
the contradiction.

The result follows. ∎

Define the subcategory $\mathcal{SL}$ in the supercategory
$\underline{\text{sLieAlg}}$ as the image of the functor
$F:\mathcal{W}(\mathcal{R})\to\underline{\text{sLieAlg}}$ defined above.
Recall that objects of $\mathcal{SL}$ (3.9) are also Lie superbialgebras. Thus
it follows from Proposition 2.2 that morphisms in $\mathcal{SL}$ (3.10) induce
structures of Lie superbialgebras on objects of $\mathcal{SL}$. Consequently,
$\mathcal{SL}$ can be viewed as the subcategory in the supercategory
$\underline{\text{sBiLieAlg}}$.

##  4 Weyl groupoid of quantum superalgebra $sl(2|1)$ at roots of unity

###  4.1 Quantized universal enveloping superalgebras

Here we recall the notion of quantized universal enveloping superalgebras (for
more detail see [40], [41], [17], [18]).

Let $K=\Bbbk[[h]]$, where $h$ is an indeterminate and view $K$ as a superspace
concentrated in degree $\bar{0}$. Let $M$ be a module over $K$. Consider the
inverse system of $K$-modules

$p_{n}:M_{n}=M/h^{n}M\to M_{n-1}=M/h^{n-1}M.$

Let $\hat{M}=\varprojlim M_{n}$ be the inverse limit. Then $\hat{M}$ has the
natural inverse limit topology (called the $h$-adic topology).

Let $V$ be a $\Bbbk$-superspace. Let $V[[h]]$ to be the set of formal power
series. The superspace $V[[h]]$ is naturally a $K$-module and has a norm given
by

$||v_{n}h^{n}+v_{n+1}h^{n+1}+...||=2^{-n},$

where $v_{n}\neq 0$ and $v_{i}\in V$ for $i\geq n$. The topology defined by
this norm is complete and coincides with the $h$-adic topology. We say that a
$K$-module $M$ is topologically free if it is isomorphic to $V[[h]]$ for some
$\Bbbk$-module $V$.

Let $M$ and $N$ be topologically free $K$-modules. We define the topological
tensor product of $M$ and $N$ to be $\widehat{M\otimes_{K}N}$ which we denote
by $M\otimes N$. It follows that $M\otimes N$ is topologically free and that

$V[[h]]\otimes W[[h]]=(V\otimes W)[[h]]$

for $\Bbbk$-module $V$ and $W$.

We say a (Hopf) superalgebra defined over $K$ is topologically free if it is
topologically free as a $K$-module and the tensor product is the above
topological tensor product.

A quantized universal enveloping (QUE) superalgebra $A$ is a topologically
free Hopf superalgebra over $\Bbbk[[h]]$ such that $A/hA$ is isomorphic as a
Hopf superalgebra to universal enveloping superalgebra $U(\mathfrak{g})$ for
some Lie superalgebra $\mathfrak{g}$. We use the following result proved in
the non-super case in [13] and in the super case in [2].

######  Proposition 4.1.

Let $A$ be a QUE superalgebra: $A/hA\cong U(\mathfrak{g})$. Then the Lie
superalgebra $\mathfrak{g}$ has a natural structure of a Lie superbialgebra
defined by

$\delta(x)=h^{-1}(\Delta(\tilde{x})-\Delta^{op}(\tilde{x}))\mod h,$ (4.1)

where $x\in\mathfrak{g}$, $\tilde{x}\in A$ is a preimage of $x$, $\Delta$ is a
comultiplication in $A$ and
$\Delta^{op}:=\tau_{U(\mathfrak{g}),U(\mathfrak{g})}\circ\Delta$ (for the
definition of $\tau_{U(\mathfrak{g}),U(\mathfrak{g})}$ see (2.3)).

######  Definition 4.1.

Let $A$ be a QUE superalgebra and let $(\mathfrak{g},[\cdot,\cdot],\delta)$ be
the Lie superbialgebra defined in Proposition 4.1. We say that $A$ is a
quantization of the Lie superbialgebra $\mathfrak{g}$.

Let $t$ be an indeterminate. Set

$\begin{bmatrix}m+n\\\
n\end{bmatrix}_{t}=\prod_{i=0}^{n-1}\frac{t^{m+n-i}-t^{-m-n+i}}{t^{i+1}-t^{-i-1}}\in\Bbbk[t],$

where $m,n\in\mathbb{N}$. Denote by

$e^{ht}=\sum_{n\geq 0}\frac{t^{n}h^{n}}{n!}\in\Bbbk[[h]].$ (4.2)

Put $q=e^{h/2}$ and recall notations introduced in Section 2. We need the
following result, see [29] and [41].

######  Theorem 4.1.

Let $(\mathfrak{g},A,\tau)$ be a Lie superalgebra of type $A$, where Cartan
matrix $A$ is symmetrizable, i. e. there are nonzero rational numbers $g_{i}$
for $i\in I$ such that $g_{i}a_{ij}=g_{j}a_{ji}$. There exists an explicit QUE
Hopf superalgebra $U_{h}^{DJ}(\mathfrak{g},A,\tau)$. The Hopf superalgebra
$U_{h}^{DJ}(\mathfrak{g},A,\tau)$ is defined as the $\Bbbk[[h]]$-superalgebra
generated by the elements $h_{i}$, $e_{i}$ and $f_{i}$, where $i\in I$ (all
generators are even except $e_{i}$ and $f_{i}$ for $i\in\tau$ which are odd),
and the relations:

$[h_{i},h_{j}]=0,\;[h_{i},e_{j}]=a_{ij}e_{j},\;[h_{i},f_{j}]=-a_{ij}f_{j},$
$[e_{i},f_{j}]=\delta_{i,j}\frac{q^{g_{i}h_{i}}-q^{-g_{i}h_{i}}}{q^{g_{i}}-q^{-g_{i}}},$

and the quantum Serre-type relations

$e_{i}^{2}=f_{i}^{2}=0\;\text{for}\;i\in I\;\text{such that}\;a_{ii}=0,$
$[e_{i},e_{j}]=[f_{i},f_{j}]=0\;\text{for}\;i,j\in I\;\text{such
that}\;a_{ij}=0\;\text{and}\;i\neq j,$
$\sum_{v=0}^{1+|a_{ij}|}(-1)^{v}\begin{bmatrix}1+|a_{ij}|\\\
v\end{bmatrix}_{q^{g_{i}}}e_{i}^{1+|a_{ij}|-v}e_{j}e_{i}^{v}=\sum_{v=0}^{1+|a_{ij}|}(-1)^{v}\begin{bmatrix}1+|a_{ij}|\\\
v\end{bmatrix}_{q^{g_{i}}}f_{i}^{1+|a_{ij}|-v}f_{j}f_{i}^{v}=0$

for $i\neq j$, $i\notin\tau$ and $i,j\in I$,

$[[[e_{m-1},e_{m}]_{q^{(\epsilon_{m},\epsilon_{m})}},e_{m+1}]_{q^{(\epsilon_{m+1},\epsilon_{m+1})}},e_{m}]=0\;\text{and}$
$[[[f_{m-1},f_{m}]_{q^{(\epsilon_{m},\epsilon_{m})}},f_{m+1}]_{q^{(\epsilon_{m+1},\epsilon_{m+1})}},f_{m}]=0,$

if $m-1,m,m+1\in I$ and $a_{mm}=0$,
$\alpha_{m}=\epsilon_{m}-\epsilon_{m+1}\in\tau$. $[\cdot,\cdot]_{v}$ is the
bilinear form defined by $[x,y]_{v}=xy-(-1)^{|x||y|}vyx$ on homogeneous $x,y$
and $v\in\Bbbk[[h]]$. The comultiplication, counit and antipode are given by

$\Delta(h_{i})=h_{i}\otimes 1+1\otimes h_{i},\;\Delta(e_{i})=e_{i}\otimes
1+q^{g_{i}h_{i}}\otimes e_{i},\;\Delta(f_{i})=f_{i}\otimes
q^{-g_{i}h_{i}}+1\otimes f_{i};$
$\epsilon(h_{i})=\epsilon(e_{i})=\epsilon(f_{i})=0;\;S(h_{i})=-h_{i},\;S(e_{i})=-q^{-g_{i}h_{i}}e_{i},\;S(f_{i})=-f_{i}q^{g_{i}h_{i}},$

where $i\in I$.

###  4.2 Definition of quantum superalgebra at roots of unity

We introduce the quantum superalgebra of $sl(m|n)$ for any Dynkin diagram
using notations from section3.2 and 3.3 (see [8], [42]). Let $q$ be an
algebraically independent and invertible element over $\mathbb{Q}$. Consider
Lie superalgebra $\mathfrak{g}(A_{d},\tau^{d})$ ($d\in D$), where
$\tau^{d}=\\{\alpha_{i,d}\;|\;i\in I\\}$ is the corresponding set of simple
roots and $A_{d}=((\alpha_{i,d},\alpha_{j,d}))_{i,j\in I}$. Let
$\mathfrak{U}_{q}^{d}:=\mathfrak{U}_{q}(\mathfrak{g}(A_{d},\tau^{d}))$ for any
$d\in D$ be the associative superalgebra over $\mathbb{Q}(q)$ with $1$,
generated by $\\{e_{i,d},f_{i,d},k_{i,d},k_{i,d}^{-1}\;|\;i\in I\\}$,
satisfying

$XY=YX\;\text{for}\;X,Y\in\\{k_{i,d},k_{i,d}^{-1}\;|\;i\in I\\},$ (4.3)
$k_{i,d}k_{i,d}^{-1}=k_{i,d}^{-1}k_{i,d}=1,\;e_{i,d}k_{j,d}=q^{-(\alpha_{j}^{d},\alpha_{i}^{d})}k_{j,d}e_{i,d},\;k_{j,d}f_{i,d}=q^{-(\alpha_{j}^{d},\alpha_{i}^{d})}f_{i,d}k_{j,d},$
(4.4)
$[e_{i,d},f_{j,d}]_{1}=e_{i,d}f_{j,d}-(-1)^{|\alpha_{i}^{d}||\alpha_{j}^{d}|}f_{j,d}e_{i,d}=\delta_{i,j}\frac{k_{i,d}-k_{i,d}^{-1}}{q-q^{-1}},$
(4.5) $e_{i,d}^{2}=f_{i,d}^{2}=0,\;\text{if}\;|\alpha_{i,d}|=1,$ (4.6)
$[e_{i,d},[e_{i,d},e_{j,d}]_{q^{(\alpha_{i,d},\alpha_{j,d})}}]_{q^{(\alpha_{i,d},\alpha_{i,d}+\alpha_{j,d})}}=0,\mbox{
if }|\alpha_{i,d}|=0,\;|i-j|=1,$ (4.7)
$[f_{i,d},[f_{i,d},f_{j,d}]_{q^{(\alpha_{i,d},\alpha_{j,d})}}]_{q^{(\alpha_{i,d},\alpha_{i,d}+\alpha_{j,d})}}=0,\mbox{
if }|\alpha_{i,d}|=0,\;|i-j|=1,$ (4.8)
$[[[e_{i-1,d},e_{i,d}]_{q^{(\epsilon_{i},\epsilon_{i})}},e_{i+1,d}]_{q^{(\epsilon_{i+1},\epsilon_{i+1})}},e_{i,d}]=0,\mbox{
if }|\alpha_{i,d}=\epsilon_{i}-\epsilon_{i+1}|=1,$ (4.9)
$[[[f_{i-1,d},f_{i,d}]_{q^{(\epsilon_{i},\epsilon_{i})}},f_{i+1,d}]_{q^{(\epsilon_{i+1},\epsilon_{i+1})}},f_{i,d}]=0,\mbox{
if }|\alpha_{i,d}=\epsilon_{i}-\epsilon_{i+1}|=1,$ (4.10)

where $\delta_{i,j}$ is the Kronecker delta,
$\alpha_{i,d},\alpha_{j,d}\in\tau^{d}$ ($i,j\in I$); $[\cdot,\cdot]_{v}$ is
the bilinear form defined by

$[x,y]_{v}=xy-(-1)^{|x||y|}vyx$ (4.11)

on homogeneous $x,y$ and $v\in\mathbb{Q}(q)$. The parity function is defined
by $|k_{i,d}|=0$ and $|e_{i,d}|=|f_{i,d}|=|\alpha_{i,d}|$, where
$\alpha_{i,d}\in\tau^{d}$ ($i\in I$).

Also $\mathfrak{U}_{q}^{d}(\mathfrak{g})$ is a Hopf superalgebra which
comultiplication $\Delta$, counit $\epsilon$ and antipode $S$ are

$\Delta_{d}(k_{i,d})=k_{i,d}\otimes
k_{i,d},\;\Delta_{d}(e_{i,d})=e_{i,d}\otimes 1+k_{i,d}\otimes
e_{i,d},\;\Delta_{d}(f_{i,d})=f_{i,d}\otimes k_{i,d}^{-1}+1\otimes f_{i,d};$
(4.12)
$\epsilon_{d}(k_{i,d})=1,\;\epsilon_{d}(e_{i,d})=\epsilon_{d}(f_{i,d})=0;\;S_{d}(k_{i,d}^{\pm
1})=k_{i,d}^{\mp
1},\;S_{d}(e_{i,d})=-k_{i,d}^{-1}e_{i,d},\;S_{d}(f_{i,d})=-f_{i,d}k_{i,d},$
(4.13)

where $i\in I$.

######  Proposition 4.2.

There exists the unique injective morphism of Hopf superalgebras for $d\in D$

$f:\mathfrak{U}_{q}(\mathfrak{g}(A_{d},\tau^{d}))\to
U_{h}^{DJ}(\mathfrak{g},A_{d},\tau),$

where $\tau=\\{\alpha\;|\;|\alpha|=1,\;\alpha\in\tau^{d}\\}$, such that for
$i\in I$

$f(q)=e^{\frac{h}{2}},\;f(k_{i})=e^{\frac{hh_{i}}{2}},\;f(k_{i}^{-1})=e^{-\frac{hh_{i}}{2}},\;f(e_{i})=e_{i},\;f(f_{i})=f_{i}.$

###### Proof.

The result follows from the direct computations. ∎

Fix Hopf superalgebra $\mathfrak{U}_{q}(\mathfrak{g}(A_{d},\tau^{d}))$ for
$d\in D$. It follows from Proposition 4.2 that we can consider
$\mathfrak{U}_{q}^{d}$ as a supersubalgebra in
$U_{h}^{DJ}(\mathfrak{g},A_{d},\tau)$. Thus we are able to apply equation
(4.1) to $\mathfrak{U}_{q}^{d}$. Then it easy to see that the Lie superalgebra
$\mathfrak{g}(A_{d},\tau^{d})$ has a natural structure of a Lie superbialgebra
defined by equation (2.4) and extended to all the elements of
$\mathfrak{g}(A_{d},\tau^{d})$ using (2.2).

From now on let $q$ be a root of unity of odd order $p$. Then it is easy to
see that $\mathfrak{U}_{q}^{d}$ can be defined in the same way. Now we
introduce some auxiliary notations, see [37].

######  Notation 4.1.

Let $(\mathfrak{U}_{q}^{d})^{<}$, $(\mathfrak{U}_{q}^{d})^{>}$,
$(\mathfrak{U}_{q}^{d})^{0}$ be the $\mathbb{Q}(q)$-subalgebras of
$\mathfrak{U}_{q}^{d}$ generated respectively by $\\{f_{i,d}\\}_{i\in I}$,
$\\{e_{i,d}\\}_{i\in I}$, $\\{k_{i,d}\\}_{i\in I}$.

Let $\tau^{d}=\\{\alpha_{i,d}\;|\;i\in I\\}$ (3.2) be the set of simple roots
associated with Dynkin diagram labeled by $d\in D$ and
$\Delta^{+}_{d}=\\{\alpha_{j,d}+\alpha_{j+1,d}+...+\alpha_{i,d}\\}_{1\leq
j\leq i<n+m}$. Consider the following total order $"\leq"$ on
$\Delta^{+}_{d}$:

$\alpha_{j,d}+\alpha_{j+1,d}+...+\alpha_{i,d}\leq\alpha_{j^{{}^{\prime}},d}+\alpha_{j^{{}^{\prime}}+1,d}+...+\alpha_{i^{{}^{\prime}},d}\mbox{
iff }j<j^{{}^{\prime}}\mbox{ or }j=j^{{}^{\prime}},i\leq i^{{}^{\prime}}.$
(4.14)

For every $\beta\in\Delta^{+}_{d}$, we choose a decomposition
$\beta=\alpha_{i_{1},d}+...+\alpha_{i_{p},d}$ such that
$[...[e_{\alpha_{i_{1},d}},e_{\alpha_{i_{2},d}},...,e_{\alpha_{i_{p},d}}]$ is
a non-zero root vector $e_{\beta,d}$ of $\mathfrak{g}(A_{d},\tau^{d})$ (here,
$e_{\alpha_{i,d},d}=e_{i,d}$ denotes the standart Chevalley generator of
$\mathfrak{g}(A_{d},\tau^{d})$). Then, we define the $PBW$ basis element
$e_{\beta,d}\in(\mathfrak{U}_{q}^{d})^{>}$ and
$f_{\beta,d}\in(\mathfrak{U}_{q}^{d})^{<}$ via

$e_{\beta,d}:=[...[[e_{i_{1},d},e_{i_{2},d}]_{q^{(\alpha_{i_{1},d},\alpha_{i_{2},d})}},e_{i_{3},d}]_{q^{(\alpha_{i_{1},d}+\alpha_{i_{2},d},\alpha_{i_{3},d})}},...,e_{i_{p},d}]_{q^{(\alpha_{i_{1},d}+\alpha_{i_{2},d}+...+\alpha_{i_{p-1},d},\alpha_{i_{p},d})}},$
$f_{\beta,d}:=[...[[f_{i_{p},d},f_{i_{p-1},d}]_{q^{-(\alpha_{i_{p},d},\alpha_{i_{p-1},d})}},f_{i_{p-2},d}]_{q^{-(\alpha_{i_{p},d}+\alpha_{i_{p-1},d},\alpha_{i_{p-2},d})}},...,f_{i_{1},d}]_{q^{-(\alpha_{i_{p},d}+\alpha_{i_{p-1},d}+...+\alpha_{i_{2},d},\alpha_{i_{1},d})}}.$

In particular, $e_{\alpha_{i,d},d}=e_{i,d}$ and $f_{\alpha_{i,d},d}=f_{i,d}$.
Let $H$ denote the set of all functions $h:\tau^{d}\to\\{0,1,...,p-1\\}$ such
that $h(\alpha)\leq 1$ if $|\alpha|=1$ ($\alpha\in\Delta^{+}_{d}$). The
monomials

$e_{h,d}:=\overrightarrow{\prod_{\beta\in\Delta^{+}_{d}}}e_{\beta,d}^{h(\beta)},\;f_{h,d}:=\overleftarrow{\prod_{\beta\in\Delta^{+}_{d}}}f_{\beta,d}^{h(\beta)},\;\forall
h\in H$ (4.15)

will be called the ordered PBW monomials of $(\mathfrak{U}_{q}^{d})^{>}$ and
$(\mathfrak{U}_{q}^{d})^{<}$. Here, the arrows $\rightarrow$ an $\leftarrow$
over the product signs refer to the total order (4.14) and its opposite,
respectively.

Let $H_{0}$ denote the set of all functions $g:I\to\\{0,1,...,p-1\\}$. The
monomials (note the order of the products is irrelevant, due to (4.3))

$k_{g,d}:=\prod_{i\in I}k_{i,d}^{g(i)},\;\forall g\in H_{0}.$ (4.16)

will be called the PBW monomials of $(\mathfrak{U}_{q}^{d})^{0}$.

Thus define a total order $"\leq"$ on the elements
$\\{e_{\alpha,d},f_{\alpha,d},k_{i,d}^{\pm 1}\;|\;\alpha\in\Delta^{+}_{d},i\in
I\\}\subset\mathfrak{U}_{q}^{d}$: set $k_{i,d}\leq k_{j,d}$, if $i\leq j$
($i,j\in I$); $e_{\alpha,d}\leq e_{\beta,d}$ and $f_{\alpha,d}\leq
f_{\beta,d}$, if $\alpha\leq\beta$ ($\alpha,\beta\in\Delta^{+}_{d}$);
$f_{\alpha,d}<k_{i,d}<e_{\beta,d}$ ($\alpha,\beta\in\Delta^{+}_{d}$, $i\in
I$).

For all $n\in\mathbb{Z}$ set $[n]:=\frac{q^{n}-q^{-n}}{q-q^{-1}}$. Denote by

$exp_{q}(x):=\sum_{n=0}^{\infty}\frac{x^{n}}{(n)_{q}!},$

where $x$ is an indeterminate and for all $k\in\mathbb{N}$ we set
$(k)_{q}:=\frac{q^{k}-1}{q-1}$ and $(0)_{q}!:=1$,
$(n)_{q}!:=(1)_{q}(2)_{q}...(n)_{q},$ if $n\in\mathbb{Z}_{+}$.

Now we define the quantum superalgebras
$U_{q}^{d}:=U_{q}(\mathfrak{g}(A_{d},\tau^{d}))$ of $sl(m|n)$ at roots of
unity for the Dynkin diagrams labeled by $d\in D$, see also [34], Proposition
3.1.

######  Definition 4.2.

For any $d\in D$ let $U_{q}^{d}$ be the quotient of the Hopf superalgebra
$\mathfrak{U}_{q}^{d}$ by the two-sided $\mathbb{Z}_{2}$-graded Hopf ideal
$\mathfrak{I}$ generated by the following elements:

$e_{i,d}^{p},f_{i,d}^{p},$ (4.17)

for all $|\alpha_{i,d}|=0$ ($\alpha_{i,d}\in\tau^{d}$, $i\in I$);

$k_{i,d}^{p}-1$ (4.18)

for all $i\in I$.

For convenience we preserve the same notations for $U_{q}^{d}$ as for
$\mathfrak{U}_{q}^{d}$ ($d\in D$). Notice that Proposition 4.1 is not true for
$U_{q}^{d}$, as when we specialize to a root of unity $q$ then the equation
(4.1) doesn’t hold.

###  4.3 Category of Hopf superalgebras and twists

We consider some categorical definitions and general results about Hopf
superalgebras. Our notations here follow [1].

######  Definition 4.3.

  1. 1.

Let Alg be the strict monoidal category of algebras over the field
$\mathbb{Q}(q)$.

  2. 2.

Let sAlg be the strict monoidal supercategory ([9], Definition 1.4) of unital
associative superalgebras over the field $\mathbb{Q}(q)$. A morphism
$f\in\text{Hom}_{\text{sAlg}}(V,W)$ between superalgebras
$(V,\mu_{V},\eta_{V})$ and $(W,\mu_{W},\eta_{W})$ is a linear map of the
underlying vector spaces such that $f\circ\mu_{V}=\mu_{W}\circ(f\otimes f)$
and $f\circ\eta_{V}=\eta_{W}$.

  3. 3.

Let HAlg be the strict monoidal category of Hopf algebras over the field
$\mathbb{Q}(q)$.

  4. 4.

Let sHAlg be the strict monoidal supercategory of Hopf superalgebras over the
field $\mathbb{Q}(q)$. A morphism $f\in\text{Hom}_{\text{sHAlg}}(V,W)$ between
Hopf superalgebras $(V,\mu_{V},\eta_{V},\Delta_{V},\epsilon_{V},S_{V})$ and
$(W,\mu_{W},\eta_{W},\Delta_{W},\epsilon_{W},S_{W})$ is a linear map of the
underlying vector spaces such that $f\circ\mu_{V}=\mu_{W}\circ(f\otimes f)$,
$f\circ\eta_{V}=\eta_{W}$, $(f\otimes f)\circ\Delta_{V}=\Delta_{W}\circ f$,
$\epsilon_{W}\circ f=\epsilon_{V}$ and $f\circ S_{V}=S_{W}\circ f$.

Let $(H,\mu,\eta,\Delta,\epsilon,S)$ be a Hopf superalgebra in
$\underline{\text{sHAlg}}$. Recall some results about twists, see [14], [30],
[3], [39].

######  Definition 4.4.

A twist for $H$ is an invertible even element $\mathcal{J}\in H\otimes H$
which satisfies

$(\Delta\otimes id_{H})(\mathcal{J})(\mathcal{J}\otimes
1)=(id_{H}\otimes\Delta)(\mathcal{J})(1\otimes\mathcal{J}),$ (4.19)
$(\epsilon\otimes
id_{H})(\mathcal{J})=(id_{H}\otimes\epsilon)(\mathcal{J})=1,$ (4.20)

where $id_{H}$ is the identity map of $H$.

######  Proposition 4.3.

Let $(H,\mu,\eta,\Delta,\epsilon,S)$ be a Hopf (super)algebra in
$\underline{\text{HAlg}}$ ($\underline{\text{sHAlg}}$) and let $\mathcal{J}$
be a twist for $H$. Then there is a new Hopf (super)algebra
$H^{\mathcal{J}}:=(H,\mu,\eta,\Delta^{\mathcal{J}},\epsilon,S^{\mathcal{J}})$
defined by the same (super)algebra and counit, and

$\Delta^{\mathcal{J}}(h):=\mathcal{J}^{-1}(\Delta(h))\mathcal{J},\;S^{\mathcal{J}}(h):=U^{-1}(S(h))U$

for all $h\in H$. Here $U=\mu\circ(S\otimes id_{H})(\mathcal{J})$ and is
invertible. Moreover, $U^{-1}=\mu\circ(id_{H}\otimes S)(\mathcal{J}^{-1})$. If
$H$ is a quasi-cocommutative (braided) Hopf (super)algebra with an universal
$R$-matrix $R$ then $H^{\mathcal{J}}$ is also quasi-cocommutative (braided)
with the universal $R$-matrix $R^{\mathcal{J}}$:

$R^{\mathcal{J}}:=\tau_{H,H}(\mathcal{J}^{-1})R_{H}\mathcal{J},$

where $\tau_{H,H}$ is defined by (2.3).

###### Proof.

The result follows from the definition and properties of a comultiplication,
antipode and universal $R$-matrix. ∎

######  Definition 4.5.

a) The Hopf (super)algebra

$H^{\mathcal{J}}:=(H,\mu,\eta,\Delta^{\mathcal{J}},\epsilon,S^{\mathcal{J}})$
(4.21)

is called the twisted Hopf (super)algebra by the twist $\mathcal{J}$. The same
notation we use for the quasi-cocommutative (braided) Hopf (super)algebra
$H^{\mathcal{J}}:=(H,\mu,\eta,\Delta^{\mathcal{J}},\epsilon,S^{\mathcal{J}},R^{\mathcal{J}})$.
b) We call $\mathcal{J}$ the twist of type $1$.
c) Let $\chi\in\text{Hom}_{\text{Alg}}(V,W)$
($\text{Hom}_{\underline{\text{sAlg}}}(V,W)$ be an isomorphism. Then we call
$\chi$ the twist of type $2$.

######  Proposition 4.4.

Let $\chi\in\text{Hom}_{\text{Alg}}(V,W)$
($\text{Hom}_{\underline{\text{sAlg}}}(V,W)$) be a twist of type $2$. Suppose
that $V$ and $W$ are Hopf (super)algebras, namely

$(V,\mu_{V},\eta_{V},\Delta_{V},\epsilon_{V},S_{V})\text{ and
}(W,\mu_{W},\eta_{W},\Delta_{W},\epsilon_{W},S_{W}).$

Let for any $w\in W$

$\Delta_{W}^{\chi}(w):=(\chi\otimes\chi)\circ\Delta_{V}(\chi^{-1}(w)),\;\epsilon_{W}^{\chi}(w):=\epsilon_{V}\circ\chi^{-1}(w),\;S^{\chi}_{W}(w):=\chi\circ
S_{V}(\chi^{-1}(w)).$

Then
$V^{\chi}:=(W,\mu_{W},\eta_{W},\Delta_{W}^{\chi},\epsilon_{W}^{\chi},S^{\chi}_{W})$
is a Hopf (super)algebra isomorphic to $V$. If $V$ is a quasi-cocommutative
(braided) Hopf (super)algebra with an universal $R$-matrix $R_{V}$ then $W$ is
also quasi-cocommutative (braided) with the universal $R$-matrix
$R_{W}^{\chi}$:

$R_{W}^{\chi}=(\chi\otimes\chi)(R_{V}).$

###### Proof.

The result follows from the definition of a Hopf (super)algebra morphism and
direct computations. ∎

The Hopf (super)algebra

$V^{\chi}:=(W,\mu_{W},\eta_{W},\Delta_{W}^{\chi},\epsilon_{W}^{\chi},S^{\chi}_{W})$
(4.22)

is called the twisted Hopf (super)algebra by the isomorphism $\chi$. The same
notation we use for a quasi-cocommutative (braided) Hopf (super)algebra
$V^{\chi}:=(W,\mu_{W},\eta_{W},\Delta_{W}^{\chi},\epsilon_{W}^{\chi},S^{\chi}_{W},R_{W}^{\chi})$.

Let $C$ be a (super)coalgebra over a field $k$. A nonzero element $g$ in $C$
is said to be a group-like element if $\Delta(g)=g\otimes g$. The set of
group-like elements in $C$ is denoted by $G(C)$. For $g,h\in G(C)$, $c\in C$
is said to be skew primitive, or more precisely, $(g,h)$-skew primitive, if
$\Delta(c)=c\otimes g+h\otimes c$. The set of all $(g,h)$-skew primitive
elements is denoted by $P_{g,h}(C)$.

###  4.4 Lusztig type isomorphisms

In this section we show that morphisms of supercategory
$\mathcal{W}(\mathcal{R})$ can be represented by isomorphisms between the
quantum superalgebras $U_{q}^{d}$, where $d\in D$, in supercategory
$\underline{\text{sAlg}}$. Compare with the Section 3.3, see also [23], [33].

We introduce the covariant faithful functor
$F_{q}:\mathcal{W}(\mathcal{R})\to\underline{\text{sAlg}}$. Fix
$(A_{d},\tau^{d})\in\text{Obj}(\mathcal{W}(\mathcal{R}))$ for $d\in D$. The
action on objects is given for all $d\in D$ by the formula

$F_{q}((A_{d},\tau^{d}))=U_{q}^{d},$ (4.23)

where $A_{d}=((\alpha_{i}^{d},\alpha_{j}^{d}))_{i,j\in I}$ and $d\in D$.

Let $\rho(a_{d_{1}})=a_{d_{2}}$ for $d_{1},d_{2}\in D$. We prove in Theorem
4.2 that $U_{q}^{d_{1}}$ and $U_{q}^{d_{2}}$ are isomorphic in
$\underline{\text{sAlg}}$. Consider a generator
$\sigma_{\alpha_{i,d_{1}}}^{d_{1}}\in\text{Hom}_{\mathcal{W}(\mathcal{R})}(a_{d_{1}},a_{d_{2}})$
(3.3) for an appropriate $\alpha_{i,d_{1}}\in\tau^{d_{1}}$ ($i\in I$) and fix
an isomorphism

$J_{i,d_{1}}\in\text{Hom}_{\underline{\text{sAlg}}}(U_{q}^{d_{1}},U_{q}^{d_{2}}).$

Define $F_{q}(\sigma_{\alpha_{i,d_{1}}}^{d_{1}})=J_{i,d_{1}}$ and
$F_{q}(\sigma_{x}^{d_{2}})=J_{i,d_{2}}$, where
$x=\sigma_{\alpha_{i,d_{1}}}^{d_{1}}(\alpha_{i,d_{1}})$ and
$J_{i,d_{2}}:=J_{i,d_{1}}^{-1}$. Additionally suppose that isomorphisms
$V_{i,d}$ satisfy relations (3.4) - (3.8) for all $i\in I$ and $d\in D$. It is
easy to see that $F_{q}$ is indeed the covariant faithful functor that
preserves the structure of the Weyl groupoid. We give an example of the family
of isomorphisms
$\\{F_{q}(\sigma)\in\text{Hom}(\underline{\text{sAlg}})\\}_{\sigma\in\mathcal{B}}$.
Call them Lusztig type isomorphisms.

Let

$T_{i,d_{1}},T_{i,d_{1}}^{-}:U_{q}^{d_{1}}\to U_{q}^{d_{2}}$

are unique isomorphisms in $\underline{\text{sAlg}}$ satisfying the following
equations for $|i-j|=1$ and $|i-l|\geq 2$ ($j,l\in I$)

$T_{i,d_{1}}(k_{i,d_{1}})=T_{i,d_{1}}^{-}(k_{i,d_{1}})=k_{i,d_{2}}^{-1},$
(4.24)
$T_{i,d_{1}}(k_{j,d_{1}})=T_{i,d_{1}}^{-}(k_{j,d_{1}})=k_{i,d_{2}}k_{j,d_{2}},\;T_{i,d_{1}}(k_{l,d_{1}})=T_{i,d_{1}}^{-}(k_{l,d_{1}})=k_{l,d_{2}},$
(4.25)
$T_{i,d_{1}}(e_{i,d_{1}})=(-1)^{|\alpha_{i,d_{1}}|}f_{i,d_{2}}k_{i,d_{2}}^{b},$
(4.26) $T_{i,d_{1}}(f_{i,d_{1}})=k_{i,d_{2}}^{-b}e_{i,d_{2}},$ (4.27)
$T_{i,d_{1}}^{-}(e_{i,d_{1}})=k_{i,d_{2}}^{-b}f_{i,d_{2}}$ (4.28)
$T_{i,d_{1}}^{-}(f_{i,d_{1}})=(-1)^{|x|}e_{i,d_{2}}k_{i,d_{2}}^{b},$ (4.29)
$T_{i,d_{1}}(e_{j,d_{1}})=-(x,y)[e_{x,d_{2}},e_{y,d_{2}}]_{q^{z}},$ (4.30)
$T_{i,d_{1}}(f_{j,d_{1}})=[f_{j,d_{2}},f_{i,d_{2}}]_{q^{-z}},$ (4.31)
$T_{i,d_{1}}^{-}(e_{j,d_{1}})=(\alpha,\beta)(-1)^{|x|(|x|+|y|)+[b=-1]+[b(x,y)=1]}[e_{j,d_{2}},e_{i,d_{2}}]_{q^{z}},$
(4.32)
$T_{i,d_{1}}^{-}(f_{j,d_{1}})=(-1)^{1+|x||y|+[b=1]+[b(x,y)=-1]}[f_{i,d_{2}},f_{j,d_{2}}]_{q^{-z}},$
(4.33) $T_{i,d_{1}}(e_{l,d_{1}})=T_{i,d_{1}}^{-}(e_{l,d_{1}})=e_{l,d_{2}},$
(4.34) $T_{i,d_{1}}(f_{l,d_{1}})=T_{i,d_{1}}^{-}(f_{l,d_{1}})=f_{l,d_{2}},$
(4.35)

where $x=\sigma_{\alpha_{i,d_{1}}}^{d_{1}}(\alpha_{i,d_{1}})$,
$y=\sigma_{\alpha_{i,d_{1}}}^{d_{1}}(\alpha_{j,d_{1}})$,
$b=[|\alpha_{i,d_{1}}|=0]-[|\alpha_{i,d_{1}}|=1]$ and $z=b(x,y)$.

One has $T_{d_{2},d_{1}}^{-}=(T_{d_{1},d_{2}})^{-1}$.

######  Theorem 4.2.

There exist the unique covariant faithful functor
$F_{q}:\mathcal{W}(\mathcal{R})\to\underline{\text{sAlg}}$ which satisfies
equation (4.23) and for all $\sigma_{\alpha_{i,d_{1}}}^{d_{1}}\in\mathcal{B}$
($i\in I$)

$F_{q}(\sigma_{\alpha_{i,d_{1}}}^{d_{1}})=T_{i,d_{1}},\;F_{q}(\sigma_{x}^{d_{2}})=T_{i,d_{2}}^{-},$
(4.36)

where

$T_{i,d_{1}},T_{i,d_{1}}^{-}:U_{q}^{d_{1}}\to U_{q}^{d_{2}}$

are isomorphisms in $\underline{\text{sAlg}}$ above defined by formulas (4.24)
- (4.35).

###### Proof.

The proof follows from the considerations preceding the statement and from the
direct computations. ∎

######  Remark 4.1.

Note that the inverse to $T_{i,d_{1}}:U_{q}^{d_{1}}\to U_{q}^{d_{2}}$ is given
by the following equations

$T_{i,d_{1}}^{-1}(k_{i,d_{2}})=k_{i,d_{1}}^{-1},\;T_{i,d_{1}}^{-1}(k_{j,d_{2}})=k_{i,d_{1}}k_{j,d_{1}},\;T_{i,d_{1}}^{-1}(k_{l,d_{2}})=k_{l,d_{1}},$
$T_{i,d_{1}}^{-1}(e_{i,d_{2}})=k_{i,d_{1}}^{-b}f_{i,d_{1}},$
$T_{i,d_{1}}^{-1}(f_{i,d_{2}})=(-1)^{|\alpha_{i,d_{1}}|}e_{i,d_{1}}k_{i,d_{1}}^{b},$
$T_{i,d_{1}}^{-1}(e_{j,d_{2}})=(x,y)(-1)^{|\alpha_{i,d_{1}}|(|\alpha_{i,d_{1}}|+|\alpha_{j,d_{1}}|)+[b=-1]+[b(\alpha_{i,d_{1}},\alpha_{j,d_{1}})=1]}[e_{j,d_{1}},e_{i,d_{1}}]_{q^{b(\alpha_{i,d_{1}},\alpha_{j,d_{1}})}},$
$T_{i,d_{1}}^{-1}(f_{j,d_{2}})=(-1)^{1+|\alpha_{i,d_{1}}||\alpha_{j,d_{1}}|+[b=1]+[b(\alpha_{i,d_{1}},\alpha_{j,d_{1}})=-1]}[f_{i,d_{1}},f_{j,d_{1}}]_{q^{-b(\alpha_{i,d_{1}},\alpha_{j,d_{1}})}},$
$T_{i,d_{1}}^{-1}(e_{l,d_{2}})=e_{l,d_{1}},\;T_{i,d_{1}}^{-1}(f_{l,d_{2}})=f_{l,d_{1}}.$

######  Corollary 4.1.

Fix any $d_{1}$ and $d_{2}$ in $D$. Then $U_{q}^{d_{1}}$ and $U_{q}^{d_{2}}$
are isomorphic in $\underline{\text{sAlg}}$.

###### Proof.

The result follows from Theorem 4.2 and the fact that Weyl groupoid
$\mathcal{W}(\mathcal{R})$ is simply connected. ∎

In assumptions made in Theorem 4.2 we prove that isomorphisms $T_{i,d}$ ($i\in
I$, $d\in D$) satisfy equations

######  Lemma 4.1.

$T_{i,d_{2}}^{-}T_{i,d_{1}}=id_{a_{d_{1}}},\;T_{i,d_{1}}T_{i,d_{2}}^{-}=id_{a_{d_{2}}},$
(4.37)

where $x=\sigma_{\alpha_{i,d_{1}}}^{d_{1}}(\alpha_{i,d_{1}})$;

$T_{j,d_{2}}T_{i,d_{1}}=T_{i,d_{2}}T_{j,d_{1}},$ (4.38)

iff $|i-j|\geq 2$ and $|\alpha_{i,d_{1}}|=|\alpha_{j,d_{1}}|=0$;

$T_{j,d_{4}}T_{i,d_{3}}T_{j,d_{2}}L_{i,d_{1}}=T_{i,d_{4}}T_{j,d_{3}}T_{i,d_{2}}T_{j,d_{1}},$
(4.39)

iff $|i-j|\geq 2$;

$T_{i,d_{3}}T_{j,d_{2}}T_{i,d_{1}}=T_{j,d_{3}}T_{i,d_{2}}T_{j,d_{1}},$ (4.40)

iff $|i-j|=1$ and $|\alpha_{i,d_{1}}|=|\alpha_{j,d_{1}}|$;

$T_{j,d_{6}}T_{i,d_{5}}T_{j,d_{4}}T_{i,d_{3}}T_{j,d_{2}}T_{i,d_{1}}=T_{i,d_{6}}T_{j,d_{5}}T_{i,d_{4}}T_{j,d_{3}}T_{i,d_{2}}T_{j,d_{1}},$
(4.41)

iff $|i-j|=1$.

###### Proof.

The result follows from the straightforward computations. ∎

We will call the algebraic structure defined in the lemma 4.1 by the formulas
(4.37 – 4.41) a braided groupoid of type A.

######  Theorem 4.3.

The braid groupoid of type A operates via isomorphisms on quantum
superalgebras $U_{q}^{d}$ ($d\in D$).

###### Proof.

The result follows from Lemma 4.1 and Theorem 4.2. ∎

Define the subcategory $\mathcal{QS}$ in the supercategory
$\underline{\text{sAlg}}$ as the image of the functor
$F_{q}:\mathcal{W}(\mathcal{R})\to\underline{\text{sAlg}}$ defined above.
Recall that objects of $\mathcal{QS}$ (4.23) are also Hopf superalgebras
defined by (4.12) - (4.13). Thus it follows from Proposition 4.4 that
morphisms in $\mathcal{QS}$ (4.36) are also morphisms in supercategory
$\underline{\text{sHAlg}}$. Consequently, $\mathcal{QS}$ is also the
subcategory in the supercategory $\underline{\text{sHAlg}}$. Recall that we
defined in the analogous way the subcategory $\mathcal{SL}$ in the
supercategory $\underline{\text{sBiLieAlg}}$, see Section 3.3.

######  Proposition 4.5.

Categories $\mathcal{QS}$ and $\mathcal{SL}$ are equivalent, where the
equivalence $\mathcal{H}:\mathcal{QS}\to\mathcal{SL}$ is defined on objects by
$\mathcal{H}(U_{q}^{d})=\mathfrak{g}(A_{d},\tau^{d})$ and on morphisms by
$\mathcal{H}(id_{U_{q}^{d}})=id_{\mathfrak{g}(A_{d},\tau^{d})}$ and
$\mathcal{H}(T_{i,d_{1}})=L_{i,d_{1}}$ ($d,d_{1},d_{2}\in D$).

###### Proof.

It is easy to see that the functor $\mathcal{H}$ is full, faithful and dense.
The result follows. ∎

###  4.5 PBW basis of $U_{q}^{d}$

We build for any $d\in D$ the PBW basis of $U_{q}^{d}$ in the case of
$sl(2|1)$. Remind the notations and conventions introduced in 4.1. See also
[37], [38].

######  Theorem 4.4.

The elements

$\mathcal{Y}=\\{f_{h_{-},d}\cdot k_{h_{0},d}\cdot
e_{h_{+},d}\;|\;h_{-},h_{+}\in H,h_{0}\in H_{0}\\}$

form a $\mathbb{Q}(q)$-basis of the quantum superalgebra $U_{q}^{d}$, where
$d\in D$.

###### Proof.

The statement immediately follows from the proof ([34], Theorem $3.1$). We
need only to add extra relations 4.43 and check that the result remains true.
Therefore, we give only a sketch of the proof.

Consider a $\mathbb{Q}(q)$ super vector space $L$ generated by
$X=\\{e_{\alpha,d},f_{\alpha,d},k_{i,d}^{\pm
1}\;|\;\alpha\in\Delta^{+}_{d},i\in I\\}$. Introduce a pair $(T(L),i)$ where
$T(L)$ is the tensor superalgebra of the vector superspace $L$ and $i$ is the
canonical inclusion of $L$ in $T(L)$. We identify for convenience $X$ and
$i(X)$. Rewrite equations (4.3) - (4.8) and (4.17) - (4.18) in $T(L)$ in the
following way

$a\otimes b-(-1)^{|a||b|}q^{\delta(a,b)}b\otimes a-[a,b]_{q^{\delta(a,b)}}=0,$
(4.42)

where $a,b\in X,\;[a,b]_{q^{\delta(a,b)}}\in T(L),\;\delta:X\times
X\to\\{-2,-1,0,1,2\\}$,

$a^{\otimes p}-c_{a}=0,$ (4.43)

where $a\in X$, $|a|=0$ and $c_{a}\in\mathbb{Q}(q)\subset T(L)$. Denote by $J$
a $\mathbb{Z}_{2}$-graded two-sided ideal in $T(L)$ generated by relations
(4.42) and (4.43). Notice that $U_{q}^{d}\cong T(L)/J$.

The index of $x_{i_{1}}\otimes x_{i_{2}}\otimes...\otimes x_{i_{n}}\in T(L)$
is defined to be the number of pairs $(l,m)$ with $l<m$ but
$x_{i_{l}}>x_{i_{m}}$, where $x_{i_{j}}\in X$, $i_{j}\in\Delta^{+}_{d}\cup I$
and $j\in\mathbb{N}$. We adopt in a natural way the definition of the index on
elements of $U_{q}^{d}$. Denote by $G$ the monomials having index $0$. Notice
that $G=\mathcal{Y}$ in $U_{q}^{d}$. Thus, we want to prove that $G$ forms the
basis of $U_{q}^{d}$ considered as the $\mathbb{Q}(q)$-superspace.

Notice that each element in $U_{q}^{d}$ is a $\mathbb{Q}(q)$-linear
combination of unit and standard monomials. Indeed, it is easy to prove by
induction on degree and index of elements in $U_{q}^{d}$ that this is the
case.

Further show that elements of $G$ are linear independent in $U_{q}^{d}$. Let
$R=\mathbb{Q}(q)[z_{1},...,z_{|X|}]$ be the polynomial ring. Endow $R$ with
the structure of the superalgebra by defining the parity function
$|z_{i}|=|f_{\alpha,d}|$, $|z_{j+|\Delta^{+}_{d}|}|=|k_{j,d}|$ and
$|z_{i+|I|+|\Delta^{+}_{d}|}|=|e_{\alpha^{{}^{\prime}},d}|$, where
$i\in\\{1,...,|\Delta^{+}_{d}|\\}$, $\alpha\in\Delta^{+}_{d}$ follow in
descending order, $j\in I$ and $\alpha^{{}^{\prime}}\in\Delta^{+}_{d}$ follow
in ascending order. Now we want to construct a morphism of superspaces
$U_{q}^{d}\to R$ which restriction on $G$ is a monomorphism that takes all the
elements of $G$ to linear independent polynomials in $R$. Then the result
follows. Thus, we proof that there is a superspace morphism $\theta:T(L)\to R$
which satisfies the following relations

$\theta(1)=1,\;\theta(f_{\alpha,d})=z_{i},\;\theta(k_{j,d})=z_{j+|\Delta^{+}_{d}|},\;\theta(e_{\alpha^{{}^{\prime}},d})=z_{i+|I|+|\Delta^{+}_{d}|},$

where $i\in\\{1,...,|\Delta^{+}_{d}|\\}$, $\alpha\in\Delta^{+}_{d}$ follow in
descending order, $j\in I$ and $\alpha^{{}^{\prime}}\in\Delta^{+}_{d}$ follow
in ascending order,

$\theta(x_{i_{1}}\otimes x_{i_{2}}\otimes...\otimes
x_{i_{n}})=z_{i_{1}}z_{i_{2}}...z_{i_{n}},\;\text{if }x_{i_{1}}\leq
x_{i_{2}}\leq...\leq x_{i_{n}},$ $\theta(x_{i_{1}}\otimes
x_{i_{2}}\otimes...\otimes x_{i_{k}}\otimes x_{i_{k+1}}\otimes...\otimes
x_{i_{n}})-(-1)^{|x_{i_{k}}||x_{i_{k+1}}|}q^{\delta(x_{i_{k}},x_{i_{k+1}})}\theta(x_{i_{1}}\otimes
x_{i_{2}}\otimes...\otimes x_{i_{k+1}}\otimes x_{i_{k}}\otimes...\otimes
x_{i_{n}})=$ $=\theta(x_{i_{1}}\otimes
x_{i_{2}}\otimes...\otimes[x_{i_{k}},x_{i_{k+1}}]\otimes...\otimes x_{i_{n}})$

for all $x_{i_{1}},x_{i_{2}},...,x_{i_{n}}\in X$ and $1\leq k<n$, where
$x_{i_{j}}\in X$, $i_{j}\in\Delta^{+}_{d}\cup I$ and $j\in\mathbb{N}$,

$\theta(x^{\otimes p})=c_{x},$

where $x\in X,\;|x|=0$ and $c_{x}\in\mathbb{Q}(q)$.

Recall that $T^{0}(L)=\mathbb{Q}(q)1$ and $T^{n}(L)=\bigotimes_{i=1}^{n}L$,
where $n\in\mathbb{N}$. Denote by $T^{n,j}(L)$ a linear subspace $T^{n}(L)$
spanned by all monomials $x_{i_{1}}\otimes x_{i_{2}}\otimes...\otimes
x_{i_{n}}$, which have index less or equal to $j$. Thus,

$T^{n,0}(L)\subset T^{n,1}(L)\subset...\subset T^{n}(L).$

We define $\theta:T^{0}(L)\to R$ by $\theta(1)=1$. Suppose inductively that
$\theta:T^{0}(L)\oplus T^{1}(L)...\oplus T^{n-1}(L)\to R$ has already been
defined satisfying the required conditions. We will show that $\theta$ can be
extended to $\theta:T^{0}(L)\oplus T^{1}(L)...\oplus T^{n}(L)\to R$. We define
$\theta:T^{n,0}(L)\to R$ by

$\theta(x_{i_{1}}\otimes x_{i_{2}}\otimes...\otimes
x_{i_{n}})=z_{i_{1}}z_{i_{2}}...z_{i_{n}}$

for standard monomials of degree $n$. We suppose $\theta:T^{n,i-1}\to R$ has
already been defined, thus giving a superspace morphism from
$\theta:T^{0}(L)\oplus T^{1}(L)...\oplus T^{n-1}(L)\oplus T^{n,i-1}(L)\to R$
satisfying the required conditions. We wish to define $\theta:T^{n,i}(L)\to
R$.

Assume that the monomial $x_{i_{1}}\otimes x_{i_{2}}\otimes...\otimes
x_{i_{n}}$ has the index $i\geq 1$ and let $x_{i_{k}}\geq x_{i_{k+1}}$. Then
define

$\theta(x_{i_{1}}\otimes...\otimes x_{i_{k}}\otimes
x_{i_{k+1}}\otimes...\otimes
x_{i_{n}})=\theta(x_{i_{1}}\otimes...\otimes[x_{i_{k}},x_{i_{k+1}}]\otimes...\otimes
x_{i_{n}})+$ (4.44)
$+(-1)^{|x_{i_{k}}||x_{i_{k+1}}|}q^{\delta(x_{i_{k}},x_{i_{k+1}})}\theta(x_{i_{1}}\otimes...\otimes
x_{i_{k+1}}\otimes x_{i_{k}}\otimes...\otimes x_{i_{n}}).$

This definition is correct as both terms on the right side of the equation
belong to a super vector space
$T^{0}(L)+T^{1}(L)+...+T^{n-1}(L)+T^{n,i-1}(L)$. We state that the definition
4.44 doesn’t depend on the choise of the pair $(x_{i_{k}},x_{i_{k+1}})$, where
$x_{i_{k}}>x_{i_{k+1}}$. Let $(x_{i_{j}},x_{i_{j+1}})$ be another pair, where
$x_{i_{j}}>x_{i_{j+1}}$. There are two different possible situations: 1.
$x_{i_{j}}>x_{i_{k+1}}$, 2. $x_{i_{j}}=x_{i_{k+1}}$. It is easy to see that
the statement is true in both cases.

Further define

$\theta(x_{i_{1}}\otimes...\otimes x_{i_{k}}\otimes x^{\otimes p}\otimes
x_{i_{k+p+1}}\otimes...\otimes
x_{i_{n}})=c_{x}\theta(x_{i_{1}}\otimes...\otimes x_{i_{k}}\otimes
x_{i_{k+p+1}}\otimes...\otimes x_{i_{n}}),$ (4.45)

where $p\leq n$, $x\in X$, $|x|=0$ and $c_{x}\in\mathbb{Q}(q)$. Let the
monomial $x_{i_{1}}\otimes...\otimes x_{i_{k}}\otimes x^{\otimes p}\otimes
x_{i_{k+p+1}}\otimes...\otimes x_{i_{n}}$ have the index $i\geq 1$. Then it is
easy to see that the order of application of equations (4.44) and 4.45 doesn’t
affect on result. Notice, in this connection, that

$\theta(x^{p}\otimes y)=\theta(y\otimes x^{p})=c_{x}\theta(y),$

if $x>y$, where $x,y\in X$, $|x|=0$ and $c_{x}\in\mathbb{Q}(q)$,

$\theta(y\otimes x^{p})=\theta(x^{p}\otimes y)=c_{x}\theta(y),$

if $y>x$, where $x,y\in X$, $|x|=0$ and $c_{x}\in\mathbb{Q}(q)$.

Thus we have defined a map $\theta:T^{n,i}(L)\to R$. A linear extension of
this map gives us $\theta:\sum_{j=0}^{n-1}T^{j}(L)\oplus T^{n,i}(L)\to R$,
which satisfies the required conditions. Since $T^{n}=T^{n,r}$ for
sufficiently large $r$, we can consider a map
$\theta:\sum_{j=0}^{n}T^{j}(L)\to R$. Since
$T(L)=T^{0}\oplus\sum_{\begin{subarray}{c}i\in\mathbb{N}\end{subarray}}T^{i}(L)$,
we get a map $\theta:T(L)\to R$, which satisfies the required conditions. It
is easy to see that $\theta:T(L)\to R$ annihilates $J$. Thus, $\theta$ induces
the required superspace morphism $\bar{\theta}:T(L)/J\to R$, that is
$\bar{\theta}:U_{q}^{d}\to R$.

∎

###  4.6 Hopf superalgebra structure and universal $R$-matrix

We describe how the standard Hopf superalgebra structures associated with each
Dynkin diagram are related.

Isomorphisms described in Theorem 4.2 induce Hopf superalgebra structures
being twists of type $2$, see Section 4.3. We want to understand how the new
Hopf superalgebra structure is related to the standard one defined by
equations (4.12) - (4.13).

Let $\rho(a_{d_{1}})=a_{d_{2}}$ for $d_{1},d_{2}\in D$. Consider a generator
$\sigma_{\alpha_{i,d_{1}}}^{d_{1}}\in\text{Hom}_{\mathcal{W}(\mathcal{R})}(a_{d_{1}},a_{d_{2}})$
(3.3) for an appropriate $\alpha_{i,d_{1}}\in\tau^{d_{1}}$ ($i\in I$) and fix
the isomorphism $F_{q}(\sigma_{\alpha_{i,d_{1}}}^{d_{1}})=T_{i,d_{1}}$ (4.2).
We put

$J_{\alpha_{i,d_{1}}}=\sum_{i=0}^{p-1}(-1)^{i}q^{-(x,x)i(i-1)/2}\frac{(q-q^{-1})^{i}}{(i)_{q^{-(x,x)}}!}T_{i,d_{1}}^{i}(f_{i,d_{1}})\otimes
T_{i,d_{1}}^{i}(e_{i,d_{1}})=$
$=exp_{q^{-(x,x)}}((q-q^{-1})T_{i,d_{1}}(f_{i,d_{1}})\otimes
T_{i,d_{1}}(e_{i,d_{1}}))^{-1},$

if $|\alpha_{i,d_{1}}|=0$;

$J_{\alpha_{i,d_{1}}}=1\otimes 1+(q-q^{-1})f_{i,d_{2}}\otimes
e_{i,d_{2}}=exp_{q^{-(x,x)}}((q-q^{-1})f_{i,d_{2}}\otimes e_{i,d_{2}}),$

if $|\alpha_{i,d_{1}}|=1$, where
$x=\sigma_{\alpha_{i,d_{1}}}^{d_{1}}(\alpha_{i,d_{1}})$.

######  Lemma 4.2.

$J_{\alpha_{i,d_{1}}}$ is a twist of type 1 for
$(U_{q}^{d_{2}})^{T_{i,d_{1}}}$. Moreover, the twist of type 1 for
$(U_{q}^{d_{1}})^{T_{i,d_{2}}^{-}}$ is given by
$J_{\alpha_{i,d_{2}}}^{-}=(T_{i,d_{2}}^{-}\otimes
T_{i,d_{2}}^{-})(J_{\alpha_{i,d_{1}}}^{-1})$.

###### Proof.

First note that $J_{\alpha_{i,d_{1}}}$ is an even element.

Let $|\alpha_{i,d_{1}}|=0$. Then, the inverse is given by

$J_{\alpha_{i,d_{1}}}^{-1}=exp_{q^{-(x,x)}}((q-q^{-1})T_{i,d_{1}}(f_{i,d_{1}})\otimes
T_{i,d_{1}}(e_{i,d_{1}})).$

Also in this case

$J_{\alpha_{i,d_{2}}}^{-}=(T_{i,d_{2}}^{-}\otimes
T_{i,d_{2}}^{-})(J_{\alpha_{i,d_{1}}}^{-1})=exp_{q^{-(x,x)}}((q-q^{-1})f_{i,d_{1}}\otimes
e_{i,d_{1}})$

and

$(J_{\alpha_{i,d_{2}}}^{-})^{-1}=(T_{i,d_{2}}^{-}\otimes
T_{i,d_{2}}^{-})(J_{\alpha_{i,d_{1}}})=exp_{q^{-(x,x)}}((q-q^{-1})f_{i,d_{1}}\otimes
e_{i,d_{1}})^{-1}.$

Let $|\alpha_{i,d_{1}}|=1$. Then, the inverse is given by

$J_{\alpha_{i,d_{1}}}^{-1}=exp_{q^{-(x,x)}}((-1)(q-q^{-1})f_{i,d_{2}}\otimes
e_{i,d_{2}}).$

Thus we have

$J_{\alpha_{i,d_{2}}}^{-}=(T_{i,d_{2}}^{-}\otimes
T_{i,d_{2}}^{-})(J_{\alpha_{i,d_{1}}}^{-1})=exp_{q^{-(x,x)}}((q-q^{-1})T_{i,d_{2}}^{-}(f_{i,d_{2}})\otimes
T_{i,d_{2}}^{-}(e_{i,d_{2}}))^{-1}$

and

$(J_{\alpha_{i,d_{2}}}^{-})^{-1}=exp_{q^{-(x,x)}}((q-q^{-1})T_{i,d_{2}}^{-}(f_{i,d_{2}})\otimes
T_{i,d_{2}}^{-}(e_{i,d_{2}})).$

We have to check that conditions in Definiton 4.4 are satisfied. The result
follows from direct computations.

∎

Now we want to show that for the case of a quantized superalgebra
$(U_{q}^{d_{2}})^{T_{i,d_{1}}}$ the twisting by the two-tensor
$J_{\alpha_{i,d_{1}}}$ coincides with the $U_{q}^{d_{2}}$. We formulate

######  Lemma 4.3.

The twisted Hopf superalgebra
$((U_{q}^{d_{2}})^{T_{i,d_{1}}})^{J_{\alpha_{i,d_{1}}}}$ coincides with Hopf
superalgebra $U_{q}^{d_{2}}$.

###### Proof.

$J_{\alpha_{i,d_{1}}}$ is the twist of type 1 by Lemma 4.2.

Now we consider how changes the coproduct and antipode of
$(U_{q}^{d_{2}})^{T_{i,d_{1}}}$ after twisting by $J_{\alpha_{i,d_{1}}}$. It
is easy to check that

$\Delta_{d_{2}}(h)=J_{\alpha_{i,d_{1}}}^{-1}\Delta_{d_{2}}^{T_{i,d_{1}}}(h)J_{\alpha_{i,d_{1}}}$

for all $h\in U_{q}^{d_{2}}$. Since an antipode is determined uniquely by a
coproduct, we have

$S_{d_{2}}(h)=U^{-1}S_{d_{2}}^{T_{i,d_{1}}}(h)U,$

for all $h\in U_{q}^{d_{2}}$, where
$U=\mu_{U_{q}^{d_{2}}}\circ(S_{d_{2}}^{T_{i,d_{1}}}\otimes
id_{U_{q}^{d_{2}}})(J_{\alpha_{i,d_{1}}})$.

The result follows by Proposition 4.3. ∎

Let $a_{d_{1}}=(A_{d_{1}},\tau^{d_{1}})$ and
$a_{d_{n}}=(A_{d_{n}},\tau^{d_{n}})$ be arbitrary objects in
$\mathcal{W}(\mathcal{R})$ for $d_{1},d_{n}\in D$ ($n\in\mathbb{N}$). It
follows from the definition of $\mathcal{W}(\mathcal{R})$ and equations (3.3)
- (3.8) that there is a morphism
$\theta\in\text{Hom}_{\mathcal{W}(\mathcal{R})}(a_{d_{1}},a_{d_{n}})$. Let
$\theta=\sigma_{\alpha_{i_{n-1},d_{n-1}}}^{d_{n-1}}...\sigma_{\alpha_{i_{2},d_{2}}}^{d_{2}}\sigma_{\alpha_{i_{1},d_{1}}}^{d_{1}}$,
where
$\sigma_{\alpha_{i_{k}},d_{k}}^{d_{k}}\in\text{Hom}_{\mathcal{W}(\mathcal{R})}(a_{d_{k}},a_{d_{k+1}})$
($i_{k}\in I$, $d_{k}\in D$, $\alpha_{i_{k}}\in\tau^{d_{k}}$ and
$k,n\in\mathbb{N}$). It follows from Corollary 4.1 that the functor
$F_{q}:\mathcal{W}(\mathcal{R})\to\underline{\text{sAlg}}$ induces a Lusztig
type isomorphism $T_{d_{1},d_{n}}:U_{q}^{d_{1}}\to U_{q}^{d_{n}}$ in
$\underline{\text{sAlg}}$ such that $F_{q}(\theta)=T_{d_{1},d_{n}}$ and
$T_{d_{1},d_{n}}=T_{i_{n-1},d_{n-1}}...T_{i_{2},d_{2}}T_{i_{1},d_{1}}$.

Thus we can consider Hopf superalgebra $(U_{q}^{d_{1}})^{P_{\theta,d_{1}}}$,
which is defined by formula

$(U_{q}^{d_{1}})^{P_{\theta,d_{1}}}=(((((((U_{q}^{d_{1}})^{T_{i_{1},d_{1}}})^{J_{\alpha_{i_{1},d_{1}}}})^{T_{i_{2},d_{2}}})^{J_{\alpha_{i_{2},d_{2}}}})^{...})^{T_{i_{n-1},d_{n-1}}})^{J_{\alpha_{i_{n-1},d_{n-1}}}}.$
(4.46)

We can summarize our considerations in the

######  Theorem 4.5.

Hopf superalgebra $(U_{q}^{d_{1}})^{P_{\theta,d_{1}}}$ coincides with the Hopf
superalgebra $U_{q}^{d_{n}}$.

###### Proof.

The result follows from Lemma 4.3 by induction on $n$. ∎

We use notations introduced in Section 4.3. It follows from the same arguments
as in the [11] that $G(U_{q}^{d})$ is generated by the group-like elements
$k_{i}$ ($i\in I$) in the Hopf superalgebra $U_{q}^{d}$. To get all Hopf
superalgebra isomorphisms of $U_{q}^{d}$ for $d\in D$, we need to describe all
skew primitive elements $P_{1,g}(U_{q}^{d})$ and $P_{g,1}(U_{q}^{d})$ ($g\in
G(U_{q}^{d})$).

######  Lemma 4.4.

We have

$P_{1,g}(U_{q}^{d})=\begin{cases}\mathbb{Q}(q)(1-g)\oplus\mathbb{Q}(q)e_{i}\oplus\mathbb{Q}(q)gf_{i},&g=k_{i}\;(1\leq
i<m+n),\\\ \mathbb{Q}(q)(1-g),&\mbox{otherwise};\end{cases}$ (4.47)
$P_{g,1}(U_{q}^{d})=\begin{cases}\mathbb{Q}(q)(1-g)\oplus\mathbb{Q}(q)f_{i}\oplus\mathbb{Q}(q)e_{i}g,&g=k_{i}^{-1}\;(1\leq
i<m+n),\\\ \mathbb{Q}(q)(1-g),&\mbox{otherwise}.\end{cases}$ (4.48)

###### Proof.

The proof is similar to that in [11], [25]. ∎

Fix $d_{1}\in D$ and consider the quantum superalgebra $U_{q}^{d_{1}}$. Now we
want to describe quantum superalgebras $U_{q}^{d_{2}}$ ($d_{2}\in D$) that are
isomorphic to $U_{q}^{d_{1}}$ as Hopf superalgebras. We can describe Dynkin
diagram associated with $U_{q}^{d_{1}}$ ($U_{q}^{d_{2}}$) by the set of simple
roots

$\tau^{d_{1}}=\\{\alpha_{i,d_{1}}\;|\;i\in
I\\}\;(\tau^{d_{2}}=\\{\alpha_{i,d_{2}}\;|\;i\in I\\}).$

Let $a=(a_{1},...,a_{m+n-1})\in(\mathbb{Q}(q)^{*})^{m+n-1}$. Suppose that
$(\alpha_{i,d_{1}},\alpha_{j,d_{1}})=(\alpha_{i,d_{2}},\alpha_{j,d_{2}})$ for
all $i,j\in I$. Define $\phi_{a}:U_{q}^{d_{1}}\to U_{q}^{d_{2}}$ by

$\phi_{a}(k_{i,d_{1}})=k_{i,d_{2}},\;\phi_{a}(e_{i,d_{1}})=a_{i}e_{i,d_{2}},\;\phi_{a}(f_{i,d_{1}})=a_{i}^{-1}f_{i,d_{2}}.$
(4.49)

Suppose that
$(\alpha_{i,d_{1}},\alpha_{j,d_{1}})=(\alpha_{m+n-i,d_{2}},\alpha_{m+n-j,d_{2}})$
for all $i,j\in I$. We define $\phi^{{}^{\prime}}:U_{q}^{d_{1}}\to
U_{q}^{d_{2}}$ by

$\phi^{{}^{\prime}}(k_{i,d_{1}})=k_{m+n-i,d_{2}},\;\phi^{{}^{\prime}}(e_{i,d_{1}})=e_{m+n-i,d_{2}},\;\phi^{{}^{\prime}}(f_{i,d_{1}})=f_{m+n-i,d_{2}}.$
(4.50)

Suppose that
$(\alpha_{i,d_{1}},\alpha_{j,d_{1}})=-(\alpha_{i,d_{2}},\alpha_{j,d_{2}})$ for
all $i,j\in I$. We define $\phi^{{}^{\prime\prime}}:U_{q}^{d_{1}}\to
U_{q}^{d_{2}}$ by

$\phi^{{}^{\prime\prime}}(k_{i,d_{1}})=k_{i,d_{2}},\;\phi^{{}^{\prime\prime}}(e_{i,d_{1}})=k_{i,d_{2}}f_{i,d_{2}},\;\phi^{{}^{\prime\prime}}(f_{i,d_{1}})=e_{i,d_{2}}k_{i,d_{2}}^{-1}.$
(4.51)

Suppose that
$(\alpha_{i,d_{1}},\alpha_{j,d_{1}})=-(\alpha_{m+n-i,d_{2}},\alpha_{m+n-j,d_{2}})$
for all $i,j\in I$. We define $\phi^{{}^{\prime\prime\prime}}:U_{q}^{d_{1}}\to
U_{q}^{d_{2}}$ by

$\phi^{{}^{\prime\prime\prime}}(k_{i,d_{1}})=k_{m+n-i,d_{2}},\;\phi^{{}^{\prime\prime\prime}}(e_{i,d_{1}})=k_{m+n-i,d_{2}}f_{m+n-i,d_{2}},\;\phi^{{}^{\prime\prime\prime}}(f_{i,d_{1}})=e_{m+n-i,d_{2}}k_{m+n-i,d_{2}}^{-1}.$
(4.52)

######  Theorem 4.6.

Let $\phi_{a}$, $\phi^{{}^{\prime}}$, $\phi^{{}^{\prime\prime}}$ and
$\phi^{{}^{\prime\prime\prime}}$ be as above. Then these maps define Hopf
superalgebra isomorphisms.

###### Proof.

It follows immediately from the definition that $\phi_{a}$,
$\phi^{{}^{\prime}}$, $\phi^{{}^{\prime\prime}}$ and
$\phi^{{}^{\prime\prime\prime}}$ are $\mathbb{Z}_{2}$-grading and even maps.
The result follows from the direct computations. ∎

Compare the following result with Theorem 4 in [11].

######  Theorem 4.7.

Let $U_{q}^{d_{1}}$ and $U_{q}^{d_{2}}$ be Hopf superalgebras associated with
Dynkin diagrams $d_{1}$ and $d_{2}\in D$ resprectively. Then there exists an
isomorphism
$\phi\in\text{Hom}_{\underline{\text{sHAlg}}}(U_{q}^{d_{1}},U_{q}^{d_{2}})$ if
and only if $d_{1}$ and $d_{2}$ are isomorphic as graphs and

  1. 1.

$\phi=\phi_{a}$, if
$(\alpha_{i,d_{1}},\alpha_{j,d_{1}})=(\alpha_{i,d_{2}},\alpha_{j,d_{2}})$ for
all $i,j\in I$;

  2. 2.

$\phi=\phi^{{}^{\prime}}\circ\phi_{a}$, if
$(\alpha_{i,d_{1}},\alpha_{j,d_{1}})=(\alpha_{m+n-i,d_{2}},\alpha_{m+n-j,d_{2}})$
for all $i,j\in I$;

  3. 3.

$\phi=\phi^{{}^{\prime\prime}}\circ\phi_{a}$, if
$(\alpha_{i,d_{1}},\alpha_{j,d_{1}})=-(\alpha_{i,d_{2}},\alpha_{j,d_{2}})$ for
all $i,j\in I$;

  4. 4.

$\phi=\phi^{{}^{\prime\prime\prime}}\circ\phi_{a}$, if
$(\alpha_{i,d_{1}},\alpha_{j,d_{1}})=-(\alpha_{m+n-i,d_{2}},\alpha_{m+n-j,d_{2}})$
for all $i,j\in I$.

###### Proof.

For any $i\in I$, we have

$\Delta_{d_{2}}(\phi(e_{i,d_{1}}))=(\phi\otimes\phi)\Delta_{d_{1}}(e_{i,d_{1}})=\phi(e_{i,d_{1}})\otimes
1+\phi(k_{i,d_{1}})\otimes\phi(e_{i,d_{1}}).$

It follows from the definition of the quantum superalgebra in Section 4.2 that
there are no other invertible elements in $U_{q}^{d}$ except for that in
$G(U_{q}^{d})$ for $d\in D$. Thus $\phi(k_{i,d_{1}})\in G(U_{q}^{d_{2}})$.
Moreover, as $\phi$ is an isomorphism, we have from (4.47) that there exists
some $\bar{i}=\rho(i)\in I$ such that $\phi(k_{i,d_{1}})=k_{\bar{i},d_{2}}$
for $\rho:I\to I$. It is clear that $\rho$ is bijective. By Lemma 4.4 we have

$\phi(e_{i,d_{1}})=a_{i}e_{\bar{i},d_{2}}+b_{i}k_{\bar{i},d_{2}}f_{\bar{i},d_{2}}+c_{i}(1-k_{\bar{i},d_{2}}),$

for suitable $a_{i},b_{i},c_{i}\in\mathbb{Q}(q)$. Note that
$e_{\bar{i},d_{2}}$, $k_{\bar{i},d_{2}}f_{\bar{i},d_{2}}$ and
$1-k_{\bar{i},d_{2}}$ are linear independent.

Recall that $q^{2}\neq 1$. Consider the equation

$\phi(k_{i,d_{1}}e_{j,d_{1}}k_{i,d_{1}}^{-1})=q^{(\alpha_{i,d_{1}},\alpha_{j,d_{1}})}\phi(e_{j,d_{1}}).$
(4.53)

We have

$0=a_{j}(q^{(\alpha_{\bar{i},d_{2}},\alpha_{\bar{j},d_{2}})}-q^{(\alpha_{i,d_{1}},\alpha_{j,d_{1}})})=b_{j}(q^{-(\alpha_{\bar{i},d_{2}},\alpha_{\bar{j},d_{2}})}-q^{(\alpha_{i,d_{1}},\alpha_{j,d_{1}})})=c_{j}(1-q^{(\alpha_{i,d_{1}},\alpha_{j,d_{1}})}).$
(4.54)

Let $i\neq j$. Note that
$(\alpha_{\bar{i},d_{2}},\alpha_{\bar{j},d_{2}})=0,1\mbox{ or }-1$. There is
at least one odd vertex in Dynkin diagram. Let $|e_{j,d_{1}}|=1$ and
$|i-j|=1$. Then
$|(\alpha_{i,d_{1}},\alpha_{j,d_{1}})|=|(\alpha_{\bar{i},d_{2}},\alpha_{\bar{j},d_{2}})|=1$.
Thus $c_{j}=0$ and $a_{j}\neq 0$, $b_{j}=0$ or $a_{j}=0$, $b_{j}\neq 0$. Thus
if $|e_{j,d_{1}}|=1$

$\phi(e_{j,d_{1}})=a_{j}e_{\bar{j},d_{2}}\mbox{ or
}\phi(e_{j,d_{1}})=b_{j}k_{\bar{j},d_{2}}f_{\bar{j},d_{2}}.$

Moreover,

$\phi(e_{j,d_{1}}^{2})=a_{j}^{2}e_{\bar{j},d_{2}}^{2}=0\mbox{ or
}\phi(e_{j,d_{1}})=q^{(\alpha_{\bar{j},d_{2}},\alpha_{\bar{j},d_{2}})}b_{j}^{2}k_{\bar{j},d_{2}}^{2}f_{\bar{j},d_{2}}^{2}=0.$
(4.55)

It follows that $\phi(e_{j,d_{1}})$ is an odd generator. Denote by
$I=I_{\bar{0},d_{1}}\cup I_{\bar{1},d_{1}}=I_{\bar{0},d_{2}}\cup
I_{\bar{1},d_{2}}$ the sets of even and odd indices of generators in
$U_{q}^{d_{1}}$ and $U_{q}^{d_{2}}$ respectively. Then we get from (4.55) that
$|I_{\bar{1},d_{2}}|\geq|I_{\bar{1},d_{1}}|$. As $\phi$ is the isomorphism
using the same argument, we have $|I_{\bar{1},d_{1}}|\geq|I_{\bar{1},d_{2}}|$.
Thus
$|I_{\bar{1},d_{1}}|=|I_{\bar{1},d_{2}}|\Rightarrow|I_{\bar{0},d_{1}}|=|I_{\bar{0},d_{2}}|$.

Consider the equation (4.53) and let $i=j$ and $|e_{j,d_{1}}|=0$. It follows
that $\phi(e_{j,d_{1}})$ is an even generator and
$|(\alpha_{i,d_{1}},\alpha_{j,d_{1}})|=|(\alpha_{\bar{i},d_{2}},\alpha_{\bar{j},d_{2}})|=2$.
Then we get from the equation (4.54) that if $|e_{j,d_{1}}|=0$

$\phi(e_{j,d_{1}})=a_{j}e_{\bar{j},d_{2}}\mbox{ or
}\phi(e_{j,d_{1}})=b_{j}k_{\bar{j},d_{2}}f_{\bar{j},d_{2}}.$

Case 1. Fix $j\in I$ such that $|e_{j,d_{1}}|=1$ and let
$\phi(e_{j,d_{1}})=a_{j}e_{\bar{j},d_{2}}$. We claim that $\forall i\in I$, we
have $\phi(e_{i,d_{1}})=a_{i}e_{\bar{i},d_{2}}$ and $\rho(i)=i$ or
$\rho(i)=m+n-i$.

Consider the equations

$\phi(k_{i,d_{1}}e_{j,d_{1}}k_{i,d_{1}}^{-1})=q^{(\alpha_{i,d_{1}},\alpha_{j,d_{1}})}\phi(e_{j,d_{1}}),\;k_{\bar{i},d_{2}}a_{j}e_{\bar{j},d_{2}}k_{\bar{i},d_{2}}^{-1}=q^{(\alpha_{\bar{i},d_{2}},\alpha_{\bar{j},d_{2}})}a_{j}e_{\bar{j},d_{2}}.$

Thus $\forall i\in I$

$q^{(\alpha_{\bar{i},d_{2}},\alpha_{\bar{j},d_{2}})}=q^{(\alpha_{i,d_{1}},\alpha_{j,d_{1}})},\;(\alpha_{\bar{i},d_{2}},\alpha_{\bar{j},d_{2}})=(\alpha_{i,d_{1}},\alpha_{j,d_{1}}).$

Moreover, if $|i-j|=1$, we have $(\alpha_{i,d_{1}},\alpha_{j,d_{1}})\neq 0$
and $\bar{i}=\bar{j}-1$ or $\bar{i}=\bar{j}+1$.

It follows from

$\phi(k_{j,d_{1}}e_{i,d_{1}}k_{j,d_{1}}^{-1})=q^{(\alpha_{i,d_{1}},\alpha_{j,d_{1}})}\phi(e_{i,d_{1}})$

that for $|i-j|=1$

$0=a_{i}(q^{(\alpha_{\bar{i},d_{2}},\alpha_{\bar{j},d_{2}})}-q^{(\alpha_{i,d_{1}},\alpha_{j,d_{1}})})=b_{i}(q^{-(\alpha_{\bar{i},d_{2}},\alpha_{\bar{j},d_{2}})}-q^{(\alpha_{i,d_{1}},\alpha_{j,d_{1}})}).$

Hence, $\phi(e_{i,d_{1}})=a_{i}e_{\bar{i},d_{2}}$.

Now we suppose that $\bar{i}=\bar{j}-1$. By the same argument
$\overline{i-1}=\bar{i}-1$ or $\overline{i-1}=\bar{i}+1=\bar{j}$. But $\rho$
is the bijection and this implies that $\overline{i-1}=\bar{i}-1$. Thus we use
induction on $r$ to prove that $\phi(e_{r,d_{1}})=a_{i}e_{\bar{r},d_{2}}$ in
the case $1\leq r\leq j$; the case $j+1\leq r\leq m+n-1$ is similar. Using the
same arguments as before it is easy to see that the result follows. Moreover,
$\forall i\in I$ as

$\rho(1)<\rho(2)<...<\rho(j)<...<\rho(m+n-1)$

we have $\rho(i)=i$.

Now we suppose that $\bar{i}=\bar{j}+1$. By the same argument, we have
$\overline{i-1}=\bar{i}+1$. Thus we use induction on $r$ to prove that
$\phi(e_{r,d_{1}})=a_{i}e_{\bar{r},d_{2}}$ in the case $1\leq r\leq j$; the
case $j+1\leq r\leq m+n-1$ is similar. Using the same arguments as before it
is easy to see that the result follows. Moreover, $\forall i\in I$ as

$\rho(1)>\rho(2)>...>\rho(j)>...>\rho(m+n-1)$

we have $\rho(i)=m+n-i$.

Case 2. Fix $j\in I$ such that $|e_{j,d_{1}}|=1$ and let
$\phi(e_{j,d_{1}})=b_{j}k_{\bar{j},d_{2}}f_{\bar{j},d_{2}}$. We claim that
$\forall i\in I$, we have
$\phi(e_{i,d_{1}})=b_{i}k_{\bar{i},d_{2}}f_{\bar{i},d_{2}}$ and $\rho(i)=i$ or
$\rho(i)=m+n-i$.

Consider the equation

$\phi(k_{i,d_{1}}e_{j,d_{1}}k_{i,d_{1}}^{-1})=q^{(\alpha_{i,d_{1}},\alpha_{j,d_{1}})}\phi(e_{j,d_{1}}),\;k_{\bar{i},d_{2}}b_{j}k_{\bar{j},d_{2}}f_{\bar{j},d_{2}}k_{\bar{i},d_{2}}^{-1}=q^{-(\alpha_{\bar{i},d_{2}},\alpha_{\bar{j},d_{2}})}b_{j}k_{\bar{j},d_{2}}f_{\bar{j},d_{2}}.$

Thus $\forall i\in I$

$q^{(\alpha_{i,d_{1}},\alpha_{j,d_{1}})}=q^{-(\alpha_{\bar{i},d_{2}},\alpha_{\bar{j},d_{2}})},\;(\alpha_{i,d_{1}},\alpha_{j,d_{1}})=-(\alpha_{\bar{i},d_{2}},\alpha_{\bar{j},d_{2}}).$

Moreover, if $|i-j|=1$, we have $(\alpha_{i,d_{1}},\alpha_{j,d_{1}})\neq 0$
and $\bar{i}=\bar{j}-1$ or $\bar{i}=\bar{j}+1$.

It follows from

$\phi(k_{j,d_{1}}e_{i,d_{1}}k_{j,d_{1}}^{-1})=q^{(\alpha_{i,d_{1}},\alpha_{j,d_{1}})}\phi(e_{i,d_{1}})$

that for $|i-j|=1$

$0=a_{i}(q^{(\alpha_{\bar{i},d_{2}},\alpha_{\bar{j},d_{2}})}-q^{(\alpha_{i,d_{1}},\alpha_{j,d_{1}})})=b_{i}(q^{-(\alpha_{\bar{i},d_{2}},\alpha_{\bar{j},d_{2}})}-q^{(\alpha_{i,d_{1}},\alpha_{j,d_{1}})}).$

Hence, $\phi(e_{i,d_{1}})=b_{i}k_{\bar{i},d_{2}}f_{\bar{i},d_{2}}$.

Now we suppose that $\bar{i}=\bar{j}-1$. By the same argument
$\overline{i-1}=\bar{i}-1$ or $\overline{i-1}=\bar{i}+1=\bar{j}$. But $\rho$
is the bijection and this implies that $\overline{i-1}=\bar{i}-1$. Thus we use
induction on $r$ to prove that
$\phi(e_{r,d_{1}})=b_{r}k_{\bar{r},d_{2}}f_{\bar{r},d_{2}}$ in the case $1\leq
r\leq j$; the case $j+1\leq r\leq m+n-1$ is similar. Using the same arguments
as before it is easy to see that the result follows. Moreover, $\forall i\in
I$ as

$\rho(1)<\rho(2)<...<\rho(j)<...<\rho(m+n-1)$

we have $\rho(i)=i$.

Now we suppose that $\bar{i}=\bar{j}+1$. By the same argument, we have
$\overline{i-1}=\bar{i}+1$. Thus we use induction on $r$ to prove that
$\phi(e_{r,d_{1}})=b_{r}k_{\bar{r},d_{2}}f_{\bar{r},d_{2}}$ in the case $1\leq
r\leq j$; the case $j+1\leq r\leq m+n-1$ is similar. Using the same arguments
as before it is easy to see that the result follows. Moreover, $\forall i\in
I$ as

$\rho(1)>\rho(2)>...>\rho(j)>...>\rho(m+n-1)$

we have $\rho(i)=m+n-i$.

Consider the equation

$\phi(k_{i,d_{1}}f_{j,d_{1}}k_{i,d_{1}}^{-1})=q^{-(\alpha_{i,d_{1}},\alpha_{j,d_{1}})}\phi(f_{j,d_{1}}).$

We use (4.48) to obtain that
$\phi(f_{j,d_{1}})=a_{j}^{{}^{\prime}}f_{j,d_{2}}$ or
$\phi(f_{j,d_{1}})=a_{j}^{{}^{\prime}}f_{m+n-j,d_{2}}$ in Case 1 and
$\phi(f_{j,d_{1}})=b_{j}^{{}^{\prime}}e_{j,d_{2}}k_{j,d_{2}}^{-1}$ or
$\phi(f_{j,d_{1}})=b_{j}^{{}^{\prime}}e_{m+n-j,d_{2}}k_{m+n-j,d_{2}}^{-1}$ in
Case 2.

Finally, consider the equation

$\phi(e_{i,d_{1}}f_{j,d_{1}}-(-1)^{|\alpha_{i,d_{1}}||\alpha_{j,d_{1}}|}f_{j,d_{1}}e_{i,d_{1}})=\delta_{i,j}\phi(\frac{k_{i,d_{1}}-k_{i,d_{1}}^{-1}}{q-q^{-1}}).$

We obtain that $a_{i}^{{}^{\prime}}=a_{i}^{-1}$ and
$b_{i}^{{}^{\prime}}=b_{i}^{-1}$ for all $i\in I$.

This impiles that if there exists an isomorphism
$\phi\in\text{Hom}_{\underline{\text{sHAlg}}}(U_{q}^{d_{1}},U_{q}^{d_{2}})$
then Dynkin diagrams associated with $U_{q}^{d_{1}}$ and $U_{q}^{d_{2}}$ are
isomorphic as graphs and

  1. 1.

$\phi=\phi_{a}$, if
$(\alpha_{i,d_{1}},\alpha_{j,d_{1}})=(\alpha_{i,d_{2}},\alpha_{j,d_{2}})$ for
all $i,j\in I$;

  2. 2.

$\phi=\phi^{{}^{\prime}}\circ\phi_{a}$, if
$(\alpha_{i,d_{1}},\alpha_{j,d_{1}})=(\alpha_{m+n-i,d_{2}},\alpha_{m+n-j,d_{2}})$
for all $i,j\in I$;

  3. 3.

$\phi=\phi^{{}^{\prime\prime}}\circ\phi_{a}$, if
$(\alpha_{i,d_{1}},\alpha_{j,d_{1}})=-(\alpha_{i,d_{2}},\alpha_{j,d_{2}})$ for
all $i,j\in I$;

  4. 4.

$\phi=\phi^{{}^{\prime\prime\prime}}\circ\phi_{a}$, if
$(\alpha_{i,d_{1}},\alpha_{j,d_{1}})=-(\alpha_{m+n-i,d_{2}},\alpha_{m+n-j,d_{2}})$
for all $i,j\in I$.

The sufficience follows from Theorem 4.6.

∎

Let $\text{Aut}_{\underline{\text{sHAlg}}}(U_{q}^{d})$ ($d\in D$) be the group
of Hopf superalgebra automorphisms of $U_{q}^{d}$ with the associated Dynkin
diagram described by the set of simple roots
$\tau^{d}=\\{\alpha_{i,d}\;|\;i\in I\\}$. Consider the conditions imposed on
roots described in the formulation of Theorem 4.7. Also recall that $q$ is a
root of unity of odd order $p$.

######  Corollary 4.2.

  1. 1.

$\text{Aut}_{\underline{\text{sHAlg}}}(U_{q}^{d})\cong\mathbb{Z}/2\mathbb{Z}\times(\mathbb{Q}(q)^{*})^{m+n-1}$,
if the condition 2 is satisfied;

  2. 2.

$\text{Aut}_{\underline{\text{sHAlg}}}(U_{q}^{d})\cong\mathbb{Z}/2p\mathbb{Z}\times(\mathbb{Q}(q)^{*})^{m+n-1}$,
if the condition 4 is satisfied;

  3. 3.

$\text{Aut}_{\underline{\text{sHAlg}}}(U_{q}^{d})\cong(\mathbb{Q}(q)^{*})^{m+n-1}$,
otherwise.

###### Proof.

We use the same notations as in Theorem 4.7.

Consider the case 1 (2). Let $\mathbb{Z}/2\mathbb{Z}=<x>$
($\mathbb{Z}/2p\mathbb{Z}=<x>$). Recall that in the condition 2 (4) of Theorem
4.7 are explicitly descibed all possible automorphisms for this case. Thus we
are able to construct the function
$\gamma_{1}:\text{Aut}_{\underline{\text{sHAlg}}}(U_{q}^{d})\to\mathbb{Z}/2\mathbb{Z}\times(\mathbb{Q}(q)^{*})^{m+n-1}$
($\gamma_{2}:\text{Aut}_{\underline{\text{sHAlg}}}(U_{q}^{d})\to\mathbb{Z}/2p\mathbb{Z}\times(\mathbb{Q}(q)^{*})^{m+n-1}$)
defined by

$\gamma_{1}(\phi_{a})=1\times a,\;\gamma_{1}(\phi^{{}^{\prime}})=x\times
1\;(\gamma_{2}(\phi_{a})=1\times
a,\;\gamma_{2}(\phi^{{}^{\prime\prime\prime}})=x\times 1).$

Also note that $\phi_{a}$ and $\phi^{{}^{\prime}}$
($\phi^{{}^{\prime\prime\prime}}$) commute. Applying $\phi^{{}^{\prime}}$
($\phi^{{}^{\prime\prime\prime}}$) to the generators of $U_{q}^{d}$ (see
(4.50) and (4.52) respectively) we see that $\phi^{{}^{\prime}}$ has order $2$
($\phi^{{}^{\prime\prime\prime}}$ has order $2p$). It is easy to see that
$\gamma_{1}$ ($\gamma_{2}$) induces the group isomorphism.

Finally consider the case 3. Note that functions described by the condition 3
of Theorem 4.7 cannot exist when we consider the automorphisms of Hopf
superalgebra structure on $U_{q}^{d}$. Thus there is only one possible case
left, namely the condition 1 of Theorem 4.7. Construct the function
$\gamma_{3}:\text{Aut}_{\underline{\text{sHAlg}}}(U_{q}^{d})\to(\mathbb{Q}(q)^{*})^{m+n-1}$
defined by

$\gamma_{3}(\phi_{a})=a.$

It is clear that $\gamma_{3}$ induces the group isomorphism. ∎

We use notations introduced in Section 3.2. Consider the set of simple roots
$\tau^{d}=\\{\alpha_{i,d}\;|\;i\in I\\}$ that corresponds to the Dynkin
diagram of $U_{q}^{d}$ ($d\in D$). Recall that $\epsilon_{i}$
($i\in\overline{1,m}$) and $\delta_{i}$ ($i\in\overline{1,n}$) are elements of
the dual space $\mathfrak{d}^{*}$. Denote by $\bar{\epsilon}_{i}$ ($i\in
I(m|n)$) an element of $L\cup L^{{}^{\prime}}$, where
$L=\\{\epsilon_{i}\;|\;i\in\overline{1,m}\\}$ and
$L^{{}^{\prime}}=\\{\delta_{i}\;|\;i\in\overline{1,n}\\}$. Let $S_{m}$
($S_{n}$) be the symmetric group. Define the action of $S_{m}\times S_{n}$
over $L\cup L^{{}^{\prime}}$: if $(s_{1},s_{2})\in S_{m}\times S_{n}$ and
$\bar{\epsilon}_{i}\in L\cup L^{{}^{\prime}}$ ($i\in I(m|n)$) then

$(s_{1},s_{2})\diamond(\bar{\epsilon}_{i})=\begin{cases}\epsilon_{s_{1}(i)},&\mbox{if
}\;\bar{\epsilon}_{i}\in L,\\\ \delta_{s_{2}(i)},&\mbox{if
}\;\bar{\epsilon}_{i}\in L^{{}^{\prime}}.\end{cases}$

We can use this definition to induce the corresponding action of $S_{m}\times
S_{n}$ over $\tau^{d}$ in the obvious way, i. e. to induce the function
$\diamond:(S_{m}\times S_{n})\times\tau^{d}\to\tau^{d}$.

Additionally consider the case $m=n$. We define the map $\boxplus:L\cup
L^{{}^{\prime}}\to L\cup L^{{}^{\prime}}$ in the following way: if
$\bar{\epsilon}_{i}\in L\cup L^{{}^{\prime}}$ ($i\in I(m|n)$) then

$\boxplus(\bar{\epsilon}_{i})=\begin{cases}\delta_{i},&\mbox{if
}\;\bar{\epsilon}_{i}\in L,\\\ \epsilon_{i},&\mbox{if }\;\bar{\epsilon}_{i}\in
L^{{}^{\prime}}.\end{cases}$

We can use this definition to induce the function
$\boxplus:\tau^{d}\to\tau^{d}$ in the obvious way.

It is easy to prove the following result

######  Theorem 4.8.

A) Fix $d\in D$. Consider the Dynkin diagram of the Hopf superalgebra
$U_{q}^{d}$

$(\bar{\epsilon}_{1}-\bar{\epsilon}_{2},\bar{\epsilon}_{2}-\bar{\epsilon}_{3},...,\bar{\epsilon}_{m+n-1}-\bar{\epsilon}_{m+n}).$

Then Dynkin diagrams of all isomorphic Hopf superlagebras have the form

  1. 1.

$(s\diamond(\bar{\epsilon}_{1}-\bar{\epsilon}_{2}),s\diamond(\bar{\epsilon}_{2}-\bar{\epsilon}_{3}),...,s\diamond(\bar{\epsilon}_{m+n-1}-\bar{\epsilon}_{m+n}));$

  2. 2.

$(s\diamond(\bar{\epsilon}_{m+n}-\bar{\epsilon}_{m+n-1}),...,s\diamond(\bar{\epsilon}_{3}-\bar{\epsilon}_{2}),s\diamond(\bar{\epsilon}_{2}-\bar{\epsilon}_{1}));$

  3. 3.

$(s\diamond(\boxplus(\bar{\epsilon}_{1})-\boxplus(\bar{\epsilon}_{2})),s\diamond(\boxplus(\bar{\epsilon}_{2})-\boxplus(\bar{\epsilon}_{3})),...,s\diamond(\boxplus(\bar{\epsilon}_{m+n-1})-\boxplus(\bar{\epsilon}_{m+n}))),$

if $m=n$ and all simple roots in $\tau^{d}$ are odd;

  4. 4.

$(s\diamond(\boxplus(\bar{\epsilon}_{m+n})-\boxplus(\bar{\epsilon}_{m+n-1})),...,s\diamond(\boxplus(\bar{\epsilon}_{3})-\boxplus(\bar{\epsilon}_{2})),s\diamond(\boxplus(\bar{\epsilon}_{2})-\boxplus(\bar{\epsilon}_{1}))),$

if $m=n$;

for all $s\in S_{m}\times S_{n}$.

B) Hopf superalgebra structures of $U_{q}(sl(m|n))$ are enumerated by the
Dynkin diagrams of $U_{q}^{d}$ ($d\in D$) up to isomorphism in the sense of
Theorem 4.7.

###### Proof.

First we prove part A. We want to construct Dynkin diagrams associated with
all Hopf superalgebras $U_{q}^{d^{{}^{\prime}}}$ ($d^{{}^{\prime}}\in D$) that
are isomorphic to $U_{q}^{d}$. Therefore we consider each isomorphism
described in Theorem 4.7 and construct Dynkin diagrams associated with
isomorphic Hopf superalgebras. It is easy to see that isomorhisms described in
conditions 1 \- 4 in Theorem 4.7 give rise to Dynkin diagrams described by
conditions 1 \- 4 stated in part A respectively. The result follows.

Now we prove part B. Fix $d\in D$. Apply Theorem 4.7 to get all Hopf
superalgebras $U_{q}^{d^{{}^{\prime}}}$ ($d^{{}^{\prime}}\in D$) isomorphic to
$U_{q}^{d}$. Recall Corollary 4.1 which states that there exists an
isomorphism of superalgebras
$\gamma_{d^{{}^{\prime}}}:U_{q}^{d^{{}^{\prime}}}\to U_{q}(sl(m|n))$ for all
$d^{{}^{\prime}}\in D$. By Proposition 4.4 we are able to endow
$U_{q}(sl(m|n))$ with the structure of Hopf superalgebra which is isomorphic
to that on $U_{q}^{d}$ using $\gamma_{d}$ as a twist of type $2$ (see
Definition 4.5). Thereom 4.7 guarantees us that in this way we get all non-
isomorphic structures of Hopf superalgebra on $U_{q}(sl(m|n))$ which are are
induced by $U_{q}^{d^{{}^{\prime}}}$ for all $d^{{}^{\prime}}\in D$. The
result follows. ∎

###  4.7 Unique Hopf superalgebra structures of $U_{q}(sl(2|1))$

We describe all unique Hopf superalgebra structures of $U_{q}(sl(2|1))$ and
their universal $R$-matices using Theorem 4.8 and Fig. 1.

Condsider Lie superalgebra $sl(2|1)$. Recall that
$\tau^{1}=\\{\alpha_{1,1}=\epsilon_{1}-\epsilon_{2},\alpha_{2,1}=\epsilon_{2}-\delta_{1}\\}$,
$\alpha_{3,1}=\alpha_{1,1}+\alpha_{2,1}$ and
$\tau^{3}=\\{\alpha_{1,3}=\epsilon_{1}-\delta_{1},\alpha_{2,3}=\delta_{1}-\epsilon_{2}\\}$,
$\alpha_{3,3}=\alpha_{1,3}+\alpha_{2,3}$ (see Fig. (1)).

Notice that there are just two non-isomorphic quantum Hopf superalgebras of
$U_{q}(sl(2|1))$ associated with Dynkin diagrams. Indeed, all quantum Hopf
superalgebras associated with Dynkin diagrams labeled by $d\in\\{1,2,5,6\\}$
are isomorphic by Theorem 4.8. We get the same result for quantum Hopf
superalgebras associated with Dynkin diagrams labeled by $d\in\\{3,4\\}$.
Moreover, we know from Theorem 4.8 that quantum Hopf superalgebras
$U_{q}^{d_{1}}$ and $U_{q}^{d_{2}}$ with $d_{1}\in\\{1,2,5,6\\}$ and
$d_{2}\in\\{3,4\\}$ are non-isomorphic (compare with Theorem 3.4).

We know that the universal $R$-matrix $\bar{R}_{1}$ of $U_{q}(sl(2|1))$ (see
[34], Theorem 3.4) is the even invertible element

$\bar{R}_{1}=\tilde{R}K,$

where

$\tilde{R}=exp_{q^{2}}((q-q^{-1})[e_{1,1},e_{2,1}]_{q^{-1}}\otimes[f_{2,1},f_{1,1}]_{q})exp_{q^{2}}((q-q^{-1})e_{2,1}\otimes
f_{2,1})\times$ $\times exp_{q^{2}}((-1)(q-q^{-1})e_{1,1}\otimes
f_{1,1})exp_{q^{2}}((-1)q(q-q^{-1})^{3}[e_{1,1},e_{2,1}]_{q^{-1}}e_{2,1}\otimes[f_{2,1},f_{1,1}]_{q}f_{2,1}),$
$K=p^{-2}\sum_{0\leq i_{1},j_{1},i_{2},j_{2}\leq
p-1}q^{i_{1}(2i_{2}-j_{2})-j_{1}i_{2}}k_{1,1}^{i_{2}}k_{2,1}^{j_{2}}\otimes
k_{1,1}^{i_{1}}k_{2,1}^{j_{1}}.$

It follows from Theorem 4.5 that $R$-matrix $\bar{R_{3}}$ for $U_{q}^{3}$ with
the standard Hopf superalgebra structure defined by equations (4.12) - (4.13)
is

$\bar{R_{3}}=(\tau_{U_{q}^{3},U_{q}^{3}}\circ
J_{\alpha_{2,1}}^{-1})\bar{R}^{T_{2,1}}J_{\alpha_{2,1}},$

where

$\bar{R}^{T_{2,1}}=(T_{2,1}\otimes
T_{2,1})(\bar{R}_{1})=\tilde{R}^{T_{2,1}}K^{T_{2,1}},$
$\tilde{R}^{T_{2,1}}=exp_{q^{2}}((q-q^{-1})[[e_{2,3},e_{1,3}]_{q^{-1}},f_{2,3}k_{2,3}^{-1}]_{q^{-1}}\otimes[k_{2,3}e_{2,3},[f_{1,3},f_{2,3}]_{q}]_{q})\times$
$exp_{q^{2}}((q-q^{-1})(-1)f_{2,3}k_{2,3}^{-1}\otimes k_{2,3}e_{2,3})\times$
$exp_{q^{2}}((q-q^{-1})[e_{2,3},e_{1,3}]_{q^{-1}}\otimes[f_{1,3},f_{2,3}]_{q})\times$
$exp_{q^{2}}(q(q-q^{-1})^{3}[[e_{2,3},e_{1,3}]_{q^{-1}},f_{2,3}k_{2,3}^{-1}]_{q^{-1}}f_{2,3}k_{2,3}^{-1}\otimes[k_{2,3}e_{2,3},[f_{1,3},f_{2,3}]_{q}]_{q}k_{2,3}e_{2,3}),$
$K^{T_{2,1}}=p^{-2}\sum_{0\leq i_{1},r_{1},i_{2},r_{2}\leq
p-1}q^{i_{1}r_{2}+i_{2}r_{1}}k_{1,3}^{i_{2}}k_{2,3}^{r_{2}}\otimes
k_{1,3}^{i_{1}}k_{2,3}^{r_{1}},$ $J_{\alpha_{2,1}}=1\otimes
1+(q-q^{-1})f_{2,3}\otimes e_{2,3},$ $J_{\alpha_{2,1}}^{-1}=1\otimes
1-(q-q^{-1})f_{2,3}\otimes e_{2,3}.$

It follows from the direct computations that

$\bar{R_{3}}=(\tau_{U_{q}^{3},U_{q}^{3}}\circ
J_{\alpha_{2,1}}^{-1})\tilde{R}^{T_{2,1}}J_{\alpha_{2,1}}K^{T_{2,1}}.$

## Acknowledgements

The authors are grateful to Egor Dotsenko for a discussion of the results of
this article, as well as for clarifying the relation between the quantum Weyl
groupoid and the Casimir connection.

## Funding

This work is funded by Russian Science Foundation, scientific grant
21-11-00283.

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