# Quasi-bialgebras from set-theoretic type solutions of the Yang-Baxter
equation

Anastasia Doikou, Alexandros Ghionis and Bart Vlaar  Department of
Mathematics, Heriot-Watt University, Edinburgh EH14 4AS $\&$ Maxwell Institute
for Mathematical Sciences, Edinburgh, UK<EMAIL_ADDRESS><EMAIL_ADDRESS>Max Planck Institute for Mathematics Vivatsgasse 7, 53111 Bonn, Germany
<EMAIL_ADDRESS>
###### Abstract.

We examine classes of quantum algebras emerging from involutive, non-
degenerate set-theoretic solutions of the Yang-Baxter equation and their
$q$-analogues. After providing some universal results on quasi-bialgebras and
admissible Drinfeld twists we show that the quantum algebras produced from
set-theoretic solutions and their $q$-analogues are in fact quasi-triangular
quasi-bialgebras. Specific illustrative examples compatible with our generic
findings are worked out. In the $q$-deformed case of set-theoretic solutions
we also construct admissible Drinfeld twists similar to the set-theoretic
ones, subject to certain extra constraints dictated by the $q$-deformation.
These findings greatly generalize recent relevant results on set-theoretic
solutions and their $q$-deformed analogues.

2010 MCS: 16T20, 16T25, 17B37, 20G42
Keywords: Quasi-bialgebras, set-theoretic Yang-Baxter equation, Drinfeld
twists

## Introduction

111Data availability statement: Data sharing not applicable to this article as
no datasets were generated or analysed during the current study.

The primary focus in this article is the investigation of classes of quantum
algebras arising from set-theoretic solutions of the Yang-Baxter equation
(YBE) [5, 41] and their $q$-deformed analogues, and their connections to
quasi-triangular quasi-bialgebras. The problem of identifying and classifying
set-theoretic solutions of the Yang-Baxter equation was first posed by
Drinfeld in early 90s [12] and ever since a considerable research activity has
been developed on this topic (see for instance [15, 16, 24], [1, 34, 40]). A
lot of attention has been focused recently on certain algebraic structures
that generalize nilpotent rings, called braces and generate all involutive,
non-degenerate solutions of the YBE [35, 36]. Skew-braces on the other hand
are used [23] to describe non-involutive set-theoretic solutions of the YBE,
and they may be instrumental in identifying universal $R$-matrices. This
rising research field has been particularly fertile and numerous relevant
studies have been produced over the past several years (see for instance [2,
3, 6, 7], [9]-[11], [18]-[20], [21, 22], [23, 25, 26, 30], [37]-[39]).

Novel connections between set-theoretic solutions, quantum integrable systems
and the associated quantum algebras were uncovered in [9, 10]. More precisely,
quantum groups associated to Baxterized solutions of the Yang-Baxter equation
coming from braces were derived via the FRT construction [17], new classes of
quantum discrete integrable systems with periodic and open boundary conditions
were produced and the symmetries of these integrable systems were also
identified. Furthermore, the explicit forms of admissible twists for
involutive set-theoretic solutions of the YBE were derived and their
admissibility was proven in [11]. Admissible twists for non-involutive set-
theoretic solutions coming from skew braces were also subsequently introduced
in [22].

The notion of an admissible twist ${\cal F}$, which links distinct Hopf or
quasi-Hopf algebras was originally introduced by Drinfeld in [14]. If the Hopf
algebra is in addition equipped with a quasi-triangular structure, i.e. a
universal $R$-matrix exists, then the twisted Hopf algebra also has a quasi-
triangular structure. In Drinfeld’s works such twists as well as the
associated universal $R$-matrices are always considered to have semi-classical
limits [13, 14], i.e. they can be expressed as formal series expansions in
powers of some deformation parameter, with the leading term being the unit
element. In the analysis of [9, 10, 11] on the other hand Baxterized
$R$-matrices coming from set-theoretic solutions of the Yang-Baxter equation
were identified, being of the form $R(\lambda)=r+{1\over\lambda}{\cal P}$,
where $r$ is the set-theoretic solution of the Yang-Baxter equation and ${\cal
P}$ is the permutation operator. Interestingly, the $r$-matrix in this case
does not contain any free parameter (deformation parameter), and consequently
the $R$-matrix has no semi-classical analogue. A similar observation can be
made about the associated admissible twist [11]. This is a crucial difference
between the studies in [9, 10, 11] and Drinfeld’s analysis [14]. Moreover,
whenever the notion of the twist is discussed in Drinfeld’s original work and
in the literature in general a quite restrictive action of the co-unit on the
twist is almost always assumed, i.e. $(\mbox{id}\otimes\epsilon){\cal
F}=(\epsilon\otimes\mbox{id}){\cal F}=1_{\cal A}$ (${\cal A}$ is the
associated quantum algebra). It is however worth noting that in [14] Drinfeld
describes how to use certain twists without this restricted counit action to
twist quasi-bialgebras with nontrivial unit constraints to quasi-bialgebras to
trivial constraints. For our purposes here we allow quasi-bialgebras with
nontrivial unit constraints. We will further discuss this point in subsection
1.2.

It was shown in [11] that all involutive, set-theoretic solutions of the YBE
can be obtained from the permutation operator via a suitable admissible twist
that was explicitly derived. The action of the co-unit on the twist in this
case was also identified and is given as $(\mbox{id}\otimes\epsilon){\cal
F}=1_{\cal A}$ and $(\epsilon\otimes\mbox{id}){\cal F}\neq 1_{\cal A}.$
Bearing these findings in mind, we relax in this study the restrictive
condition $(\mbox{id}\otimes\epsilon){\cal F}=(\epsilon\otimes\mbox{id}){\cal
F}=1_{\cal A}$ and consider a general scenario, which leads to intriguing new
results. One would expect that the quantum algebra emerging from any
$R$-matrix that satisfies the Yang-Baxter equation would be a bialgebra. It
was noted however in [11], using a special class of set-theoretic solutions,
that the corresponding quantum algebra was not co-associative and the related
associator was derived. This was indeed the first hint that set-theoretic
solutions of the YBE give rise to quasi-bialgebras. In this article we greatly
extend these findings after providing some universal results on quasi-
bialgebras and admissible Drinfeld twists.

More specifically, we describe below what is achieved in each of the
subsequent sections.

  1. (1)

In section 1 we show some new results on quasi-bialgebras and general
admissible Drinfelds’s twists that lay out the main frame for studying quantum
algebras arising from the set-theoretic and $q$-deformed set-theoretic
solutions of the YBE. More precisely, in subsection 1.1 before we present the
main results we first recall fundamental definitions on quasi-bialgebras and
quasi-triangular quasi-bialgebras. We then show that given certain imposed
conditions the non-associative version of the YBE reduces to the familiar YBE;
however the underlying quantum algebra is still a quasi-bialgebra.

In subsection 1.2 we move on to study generic admissible twists by relaxing
the conditions $(\mbox{id}\otimes\epsilon){\cal
F}=(\epsilon\otimes\mbox{id}){\cal F}=1_{\cal A}$ and we also examine the
consequences of such a general choice. We then give particular emphasis on
cases where the twists send a quasi-bialgebra to a bialgebra. These cases are
relevant to the findings of sections 2 and 3. An explicit illustrative example
is also worked out at the end of the subsection.

  2. (2)

In section 2 we focus on the quantum algebras emerging from involutive, non-
degenerate set-theoretic solutions of the YBE. More specifically, in
subsection 2.1 we present some background information regarding set-theoretic
solutions of the Yang-Baxter equation as well as a brief review on the recent
findings of [9] on the links between set-theoretic solutions of the Yang-
Baxter equation and quantum algebras.

In subsection 2.2 we review basic definitions and facts about the
$\mathfrak{gl}_{n}$ Yangian, useful for the purposes of our analysis. We first
recall the definition of the $\mathfrak{gl}_{n}$ Yangian, which is most
relevant in our present investigation [13, 8]. We then recall that the Yangian
is a Hopf algebra and we comment on the action of the antipode after suitably
twisting the algebra.

In subsection 2.3 we study set-theoretic solutions of the Yang-Baxter equation
associated to quasi-bialgebras. We first review some fundamental results on
the admissible Drinfeld twist for involutive, set-theoretic solution of the
YBE derived in [11]. Specifically, it was shown in [11] that Baxterized set-
theoretic solutions are always coming from the $\mathfrak{gl}_{n}$ Yangian
$R$-matrix via a suitable twist. We show here a key proposition by introducing
a family of group-like elements commuting with the Yangian $R$-matrix, which
leads to the main statement that the respective twisting of the
$\mathfrak{gl}_{n}$ Yangian yields a quasi-triangular quasi-bialgebra in
accordance to the findings of section 1. A special case of set-theoretic
solution called the Lyubashenko solution [12] is also presented as an
illustrative example.

  3. (3)

In section 3 we discuss the $q$-generalizations of set-theoretic solutions.
Although these generalizations strictly speaking are not set-theoretic
solutions of the Yang-Baxter equation they are certainly inspired by the
results of [10] and [11]. First, in subsection 3.1 we provide basic
definitions and information regarding the algebra
${\mathfrak{U}}_{q}(\widehat{\mathfrak{gl}_{n}})$ [13, 27, 28] and we also
comment on the action of the antipode after suitably twisting the algebra.

In subsection 3.2, motivated by the set-theoretic solutions and the associated
twists [11], we generalize results regrading the twist of the
${\mathfrak{U}}(\widehat{\mathfrak{gl}_{n}})$ $R$-matrix. We exploit the
admissible twists found for the set-theoretic solutions to derive generalized
$q$-deformed solutions. The findings of this section greatly generalize the
preliminary results of [10] and produce a generic class of solutions, i.e. the
$q$-deformed alanogues of the set-theoretic solutions. The $q$-analogue of
Lyuabshenko’s solution, first introduced in [10], is also discussed as an
example of this construction.

##  1\. Quasi-triangular quasi-bialgebras

In this section we recall fundamental definitions on quasi-bialgebras and
quasi-triangular quasi-bialgebras (see also for instance [4, 8, 14, 29, 32]),
and we show various novel propositions on special classes of quasi-bialgebras
as well as on generic admissible Drinfeld twists. Particular emphasis is given
on Drinfeld twists that send a quasi-bialgebra to a bialgebra as this case is
pertinent to the findings of sections 2 and 3.

###  1.1. Quasi-bialgebras and the usual YBE

Before we present the main results we first recall some fundamental
definitions on quasi-bialgebras and the associated notion of quasi-
triangularity. Throughout Section 1, $k$ is any field. Later on we will
restrict to the case $k=\mathbb{C}$. It is worth noting that if ${\cal A}$ is
a $k$-algebra then ${\cal A}\otimes{\cal A}$ (in fact ${\cal A}^{\otimes n},\
n\in\\{1,2,3,\ldots\\}$) is a $k$-algebra in a natural way. If $V$ and $W$ are
${\cal A}$-modules, then $V\otimes W$ is an ${\cal A}\otimes{\cal A}$-module
and not necessarily an ${\cal A}$-module. The following definition due to
Drinfeld [14] provides a general setup where the category $\mbox{Rep}({\cal
A})$ of all ${\cal A}$-modules naturally has a tensor structure.

######  Definition 1.1.

A quasi-bialgebra $\big{(}{\cal A},\Delta,\epsilon,\Phi,c_{r},c_{l}\big{)}$ is
a unital associative algebra ${\cal A}$ over some field $k$ with the following
algebra homomorphisms:

  * •

the co-product $\Delta:{\cal A}\to{\cal A}\otimes{\cal A}$

  * •

the co-unit $\epsilon:{\cal A}\to k$

together with the invertible element $\Phi\in{\cal A}\otimes{\cal
A}\otimes{\cal A}$ (the associator) and the invertible elements
$c_{l},c_{r}\in{\cal A}$ (unit constraints), such that:

  1. (1)

$(\mbox{id}\otimes\Delta)\Delta(a)=\Phi\Big{(}(\Delta\otimes\mbox{id})\Delta(a)\Big{)}\Phi^{-1},$
$\forall a\in{\cal A}.$

  2. (2)

$\Big{(}(\mbox{id}\otimes\mbox{id}\otimes\Delta)\Phi\Big{)}\Big{(}(\Delta\otimes\mbox{id}\otimes\mbox{id})\Phi\Big{)}=\Big{(}1\otimes\Phi\Big{)}\Big{(}(\mbox{id}\otimes\Delta\otimes\mbox{id})\Phi\Big{)}\Big{(}\Phi\otimes
1\Big{)}.$

  3. (3)

$(\epsilon\otimes\mbox{id})\Delta(a)=c_{l}^{-1}ac_{l}$ and
$(\mbox{id}\otimes\epsilon)\Delta(a)=c_{r}^{-1}ac_{r},$ $\forall a\in{\cal
A}.$

  4. (4)

$(\mbox{id}\otimes\epsilon\otimes\mbox{id})\Phi=c_{r}\otimes c_{l}^{-1}.$

In the special case where $\Phi=1\otimes 1\otimes 1$ one recovers a bialgebra,
i.e. co-associativity is restored.

Before we move on to the definition of a quasi-triangular quasi-bialgebra we
first introduce some useful notation. Let $\sigma:{\cal A}\otimes{\cal
A}\to{\cal A}\otimes{\cal A}$ be the “flip” map, such that $a\otimes b\mapsto
b\otimes a$ $\forall a,b\in{\cal A},$ then we set
$\Delta^{(op)}:=\sigma\circ\Delta.$ A quasi-bialgebra is called cocommutative
if $\Delta^{(op)}=\Delta.$ We also consider the general element
$A=\sum_{j}a_{j}\otimes b_{j}\in{\cal A}\otimes{\cal A}$, then in the “index”
notation we denote: $A_{12}:=\sum_{j}a_{j}\otimes b_{j}\otimes 1,$
$A_{23}:=\sum_{j}1\otimes a_{j}\otimes b_{j}$ and
$A_{13}:=\sum_{j}a_{j}\otimes 1\otimes b_{j}.$ In fact, given an algebra
${\cal A}$ with homomorphisms $\Delta:{\cal A}\to{\cal A}\otimes{\cal A}$ and
$\epsilon:{\cal A}\to k$, the conditions (1)-(4) listed in the Definition 1.1
are equivalent to the statement that the category of ${\cal A}$-modules,
equipped with the tensor product of the category of $k$-linear spaces, is
itself a tensor category (see e.g. [29, Proposition XV.1.2]).

By means of the axioms of Definition 1.1 we can derive further counit formulas
for the associator and unit constraints.

######  Lemma 1.2.

Let $\big{(}{\cal A},\Delta,\epsilon,\Phi,c_{r},c_{l}\big{)}$ be a quasi-
bialgebra, then:

$\displaystyle(\epsilon\otimes\mbox{id}\otimes\mbox{id})\Phi=\Delta(c_{l}^{-1})(c_{l}\otimes
1),\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
(\mbox{id}\otimes\mbox{id}\otimes\epsilon)\Phi=(1\otimes
c_{r}^{-1})\Delta(c_{r})$ (1.1) $\displaystyle\epsilon(c_{l})=\epsilon(c_{r})$
(1.2)

###### Proof.

By applying the counit on the second and third tensor factors in axioms (2)
and using axiom (3), we obtain (1.1). Hence, applying the counit on the
remaining (or third) tensor factor in axiom (4) and using axiom (3) we obtain
(1.2). ∎

######  Remark 1.3.

We may define the normalized quantities
$\hat{c}_{l}=\epsilon(c_{l}^{-1})c_{l}$ and
$\hat{c}_{r}=\epsilon(c_{r}^{-1})c_{r}.$ Then $\big{(}{\cal
A},\Delta,\epsilon,\Phi,\hat{c}_{r},\hat{c}_{l}\big{)}$ is a quasi-bialgebra
over $k$ with $\epsilon(\hat{c}_{l})=\epsilon(\hat{c}_{r})=1;$ indeed the
normalization of $c_{l},\ c_{r}$ leaves the quasi-bialgebra axioms invariant.
Considering the latter statement we may assume, without loss of generality,
that $\epsilon(c_{r})=\epsilon(c_{l})=1$. In this case axiom (4) of Definition
(1.1) implies that we may recover $c_{l}$ and $c_{r}$ from $\Phi$ via

$c_{l}^{-1}=\big{(}\epsilon\otimes\epsilon\otimes\mbox{id}\big{)}\Phi,\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
c_{r}=\big{(}\mbox{id}\otimes\epsilon\otimes\epsilon\big{)}\Phi.$ (1.3)

Henceforth, we will always assume that $\epsilon(c_{r})=\epsilon(c_{l})=1$ and
define a quasi-bialgebra structure just by specifying $({\cal
A},\Delta,\epsilon,\Phi);$ in particular, we impose the axioms of Definition
1.1, where $c_{l}$ and $c_{r}$ are shorthand notations defined in (1.3). It is
worth emphasizing that the common assumption for quasi-bialgebras is
$c_{l}=c_{r}=1$ [14], so even though we have a counit-normalization here we
still discuss a more general situation (see also later in the text the
relevant Corollary 1.8).

Following [14], see also [29], the notion of quasitriangularity for bialgebras
extends to quasi-bialgebras and can be defined category-theoretically. The
following definition captures the equivalent algebraic characterization.

######  Definition 1.4.

A quasi-bialgebra $\big{(}{\cal A},\Delta,\epsilon,\Phi\big{)}$ is called
quasi-triangular (or braided) if an invertible element ${\cal R}\in{\cal
A}\otimes{\cal A}$ (universal $R$-matrix) exists, such that

  1. (1)

$\Delta^{(op)}(a){\cal R}={\cal R}\Delta(a),$ $\forall a\in{\cal A}.$

  2. (2)

$(\mbox{id}\otimes\Delta){\cal R}=\Phi^{-1}_{231}{\cal R}_{13}\Phi_{213}{\cal
R}_{12}\Phi^{-1}_{123}.$

  3. (3)

$(\Delta\otimes\mbox{id}){\cal R}=\Phi_{312}{\cal R}_{13}\Phi_{132}^{-1}{\cal
R}_{23}\Phi_{123}.$

By employing conditions (1)-(3) of Definition 1.4 and condition (3) of
Definition 1.1 one deduces that $(\epsilon\otimes\mbox{id}){\cal
R}=c_{r}^{-1}c_{l}$ and $(\mbox{id}\otimes\epsilon){\cal R}=c_{l}^{-1}c_{r}$.
Moreover, by means of conditions (1)-(3) of Definition 1.4, it follows that
${\cal R}$ satisfies a non-associative version of the Yang-Baxter equation

${\cal R}_{12}\Phi_{312}{\cal R}_{13}\Phi_{132}^{-1}{\cal
R}_{23}\Phi_{123}=\Phi_{321}{\cal R}_{23}\Phi^{-1}_{231}{\cal
R}_{13}\Phi_{213}{\cal R}_{12}.$ (1.4)

If ${\cal A}$ is a quasi-bialgebra in Definition 1.4 then $\big{(}{\cal
A},\Delta,\epsilon,\Phi,{\cal R}\big{)}$ is a quasi-triangular quasi-
bialgebra. The quasi-bialgebra is called triangular if in addition to the
conditions (1)-(3) of Definition 1.4, ${\cal R}$ also satisfies ${\cal
R}_{21}{\cal R}_{12}=1$.

In the special case where $\Phi=1\otimes 1\otimes 1$ one recovers a quasi-
triangular bialgebra. Indeed, the co-associativity is restored
$(\mbox{id}\otimes\Delta)\Delta(a)=(\Delta\otimes\mbox{id})\Delta(a),$ and
$c_{r}=c_{l}=1.$ Also, $(\mbox{id}\otimes\Delta){\cal R}={\cal R}_{13}{\cal
R}_{12},$ $(\Delta\otimes\mbox{id}){\cal R}={\cal R}_{13}{\cal R}_{23},$ and
the universal ${\cal R}$-matrix satisfies the usual Yang-Baxter equation

${\cal R}_{12}{\cal R}_{13}{\cal R}_{23}={\cal R}_{23}{\cal R}_{13}{\cal
R}_{12}.$ (1.5)

In the remainder of this paper we consider two special cases of the general
algebraic setting for quasi-triangular quasi-bialgebras as described above.
This specific setup, introduced in the following proposition, will be central
in proving general key properties in this section and will be particularly
relevant to the findings of sections 2 and 3.

######  Proposition 1.5.

Let $\big{(}{\cal A},\Delta,\epsilon,\Phi,{\cal R}\big{)}$ be a quasi-
triangular quasi-bialgebra, then the following two statements hold:

  1. (1)

Suppose $\Phi$ satisfies the condition (in the index notation)
$\Phi_{213}{\cal R}_{12}={\cal R}_{12}\Phi_{123},$ then

$\displaystyle(\mbox{id}\otimes\Delta){\cal R}=\Phi^{-1}_{231}{\cal
R}_{13}{\cal R}_{12},\quad(\Delta\otimes\mbox{id}){\cal R}={\cal R}_{13}{\cal
R}_{23}\Phi_{123},$ (1.6)

and the universal ${\cal R}$ matrix satisfies the usual Yang-Baxter equation.
Also, $(\epsilon\otimes\mbox{id}){\mathcal{R}}=c_{l},$ $\leavevmode\nobreak\
(\mbox{id}\otimes\epsilon){\cal R}=c_{l}^{-1}$ and $c_{r}=1_{\cal A}.$

  2. (2)

Suppose $\Phi$ satisfies the condition $\Phi_{132}{\cal R}_{23}={\cal
R}_{23}\Phi_{123},$ then

$\displaystyle(\mbox{id}\otimes\Delta){\cal R}={\cal R}_{13}{\cal
R}_{12}\Phi^{-1}_{123},\quad(\Delta\otimes\mbox{id}){\cal R}=\Phi_{312}{\cal
R}_{13}{\cal R}_{23},$ (1.7)

and the universal ${\cal R}$ matrix satisfies the usual Yang-Baxter equation.
Also, $(\epsilon\otimes\mbox{id}){\mathcal{R}}=c_{r}^{-1},$
$\leavevmode\nobreak\ (\mbox{id}\otimes\epsilon){\cal R}=c_{r}$ and
$c_{l}=1_{\cal A}.$

###### Proof.

The proof of (1.6) and (1.7) is straightforward by means of relations (2) and
(3) of Definition 1.4 and the conditions of cases (1), (2) of the Proposition.

By considering case (1) of Proposition 1.5 we conclude that the modified Yang-
Baxter equation (1.4) becomes

${\cal R}_{12}{\cal R}_{13}{\cal R}_{23}\Phi_{123}={\cal R}_{23}{\cal
R}_{13}{\cal R}_{12}\Phi_{123},$ (1.8)

and because $\Phi$ is an invertible element, equation (1.8) reduces to the
usual Yang-Baxter equation (1.5). We end up to the same conclusion by
considering case (2).
The remaining statements are shown as follows:

  1. (1)

We act on the second of the equations (1.6) with
$\mbox{id}\otimes\epsilon\otimes\mbox{id}$:

$\displaystyle(\mbox{id}\otimes\epsilon\otimes\mbox{id})\big{(}(\Delta\otimes\mbox{id}){\cal
R}\big{)}=(\mbox{id}\otimes\epsilon\otimes\mbox{id})\big{(}{\cal R}_{13}{\cal
R}_{23}\Phi_{123}\big{)}\ \Rightarrow$ $\displaystyle(c_{r}^{-1}\otimes
1){\cal R}_{13}(c_{r}\otimes 1)={\cal
R}_{13}\big{(}(\mbox{id}\otimes\epsilon\otimes\mbox{id}){\cal
R}_{23}\big{)}(c_{r}\otimes c_{l}^{-1})\Rightarrow$ $\displaystyle{\cal
R}_{13}^{-1}(c_{r}^{-1}\otimes 1){\cal R}_{13}(1\otimes
c_{l})=(\mbox{id}\otimes\epsilon\otimes\mbox{id}){\cal R}_{23}.$ (1.9)

We now act with $\epsilon$ on the first position of the tensor product (1.9)
and we obtain (recall $\epsilon(c_{l})=\epsilon(c_{r})=1$)
$(\epsilon\otimes\mbox{id}){\cal R}=c_{l}.$ By substituting the latter on
(1.9) we conclude that $c_{r}=1_{\cal A}.$

Similarly, we can act on the first of equs. $(\ref{2})$ with
$\mbox{id}\otimes\mbox{id}\otimes\epsilon,$ and arrive at

$(\mbox{id}\otimes\mbox{id}\otimes\epsilon){\cal R}_{13}=(c_{l}^{-1}\otimes
1){\cal R}_{12}(1\otimes c_{r}){\cal R}_{12}^{-1},$

which after acting with $\epsilon$ on the second position of the tensor
product leads to $(\mbox{id}\otimes\epsilon){\cal R}=c_{l}^{-1}$ and
$c_{r}=1_{\cal A}$.

  2. (2)

Likewise, assuming that $\Phi_{132}{\cal R}_{23}={\cal R}_{23}\Phi_{123}$
holds we show, following the logic of the proof above, that
$(\epsilon\otimes\mbox{id}){\cal R}=c_{r}^{-1},$ $\leavevmode\nobreak\
(\mbox{id}\otimes\epsilon){\cal R}=c_{r}$ and $c_{l}=1_{\cal A}.$ ∎

######  Remark 1.6.

Let $u\in{\cal A}$ be a group-like element, i.e. $u$ is invertible and
$\Delta(u)=u\otimes u,$ and consequently $\epsilon(u)=1.$ Let also
$\Phi=1\otimes 1\otimes u^{-1}$ and ${\cal R}=u^{-1}\otimes u.$ With this
choice of $\Phi$ and ${\cal R}$ a quasi-triangular quasi-bialgebra structure
is defined. Indeed, the case (1) of Proposition 1.5 holds, i.e. conditions
(1.6) are satisfied and ${\cal R}$ trivially satisfies the Yang-Baxter
equation. Also, $c_{r}=1_{{\cal A}},$ $c_{l}=u$ and
$(\epsilon\otimes\mbox{id}){\cal R}=u,$ $(\mbox{id}\otimes\epsilon){\cal
R}=u^{-1}.$

###  1.2. Drinfeld twists

One of the main results on quasi-triangular (quasi-)bialgebras as shown by
Drinfeld [13, 14] is the fact that the property of being quasi-triangular
(quasi-)bialgebra is preserved by twisting (see also [31]). As far as we can
tell whenever the notion of Drinfeld twist is discussed in the literature a
trivial action of the co-unit on the twist is almost always assumed. In the
following proposition we are going to relax this condition and consider the
most general scenario. In [14] (Drinfeld 1989, Section 1) it is explained how
to use certain Drinfeld twists without this restricted counit action to twist
quasi-bialgebras with nontrivial unit constraints to quasi-bialgebras to
trivial constraints. For our purposes it is convenient to allow quasi-
bialgebras with nontrivial unit constraints. We come back to this point later.
We are then going to discuss a specific case that is associated to the study
of quantum algebras emerging from set-theoretic solutions of the Yang-Baxter
equation corresponding also to the conditions of Proposition 1.5.

The following result is the natural extension of Drinfeld’s result [14] on
twists for quasi-bialgebras to the case of nontrivial unit constraints. Recall
that we assume that, without loss of generality, the unit constraints are
expressed in terms of the associator by means of equation (1.3).

######  Proposition 1.7.

Let $\big{(}{\cal A},\Delta,\epsilon,\Phi,{\cal R}\big{)}$ be a quasi-
triangular quasi-bialgebra and let ${\cal F}\in{\cal A}\otimes{\cal A}$ be an
invertible element, such that

$\displaystyle\Delta_{\cal F}(a)={\cal F}\Delta(a){\cal
F}^{-1},\leavevmode\nobreak\ \leavevmode\nobreak\ \forall a\in{\cal A}$ (1.10)
$\displaystyle\Phi_{\cal F}({\cal F}\otimes
1)\big{(}(\Delta\otimes\mbox{id}){\cal F}\big{)}=(1\otimes{\cal
F})\big{(}(\mbox{id}\otimes\Delta){\cal F}\big{)}\Phi$ (1.11)
$\displaystyle{\cal R}_{\cal F}={\cal F}^{(op)}{\cal R}{\cal F}^{-1},$ (1.12)

where ${\cal F}^{(op)}:=\sigma({\cal F}).$ Then $\big{(}{\cal A},\Delta_{\cal
F},\epsilon,\Phi_{\cal F},{\cal R}_{\cal F}\big{)}$ is also a quasi-triangular
quasi-bialgebra.

###### Proof.

Let as consider the general twist ${\cal F}=\sum_{j}f_{j}\otimes g_{j}.$ In
terms of the invertible elements $v:=\sum_{j}\epsilon(f_{j})g_{j},$
$\leavevmode\nobreak\ w:=\sum_{j}\epsilon(g_{j})f_{j},$ we have

$(\epsilon\otimes\mbox{id}){\cal F}=v,\quad(\mbox{id}\otimes\epsilon){\cal
F}=w.$ (1.13)

Axioms (1)-(2) of Definition 1.1 as well as axioms of Definition 1.4 are
checked as in the original proof by Drinfeld (see also [14, 29]). For these
the action of the counit on the twist is never used. The check of these axioms
is somehow tedious, but nevertheless straightforward. Let us for instance
check the axioms (1)-(3) of Definition 1.4:

  * •

Axiom (1) of Definition 1.4: we compute

$\displaystyle\Delta^{(op)}_{\mathcal{F}}(a){\mathcal{R}}_{\mathcal{F}}$
$\displaystyle=$ $\displaystyle{\cal F}^{(op)}\Delta^{(op)}(a)({\cal
F}^{(op)})^{-1}{\cal F}^{(op)}{\cal R}{\cal F}^{-1}$ (1.14) $\displaystyle=$
$\displaystyle{\cal F}^{(op)}{\cal R}\Delta(a){\cal
F}^{-1}={\mathcal{R}}_{\mathcal{F}}\Delta_{\mathcal{F}}(a).$

  * •

Axioms (2) and (3) of Definition 1.4: we compute

$(\mbox{id}\otimes\Delta_{\cal F}){\cal R}_{\cal
F}=\big{(}(\mbox{id}\otimes{\cal F})(\mbox{id}\otimes\Delta){\cal
F}^{(op)}\big{)}\big{(}(\mbox{id}\otimes\Delta){\cal
R}\big{)}\big{(}(\mbox{id}\otimes\Delta){\cal F}^{-1}(\mbox{id}\otimes{\cal
F}^{-1})\big{)}.$

It is convenient to introduce some useful notation:

${\cal F}_{1,23}:=(\mbox{id}\otimes\Delta){\cal F},\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ {\cal F}_{12,3}:=(\Delta\otimes\mbox{id}){\cal F},$
(1.15)

and by the quasi-bialgebra axioms ${\cal F}_{21,3}{\cal R}_{12}={\cal
R}_{12}{\cal F}_{12,3},$ $\ {\cal F}_{1,32}{\cal R}_{23}={\cal R}_{23}{\cal
F}_{1,23};$ also recall $({\cal R}_{\cal F})_{12}={\cal F}_{21}{\cal
R}_{12}{\cal F}_{12}^{-1}.$ We then denote according to the index notation
$(\mbox{id}\otimes\Delta){\cal F}^{(op)}={\cal F}_{23,1},$ and we re-express
condition (1.11) as ${\cal F}_{12}{\cal F}_{12,3}={\cal F}_{23}{\cal
F}_{1,23}\Phi_{123},$ then

$\displaystyle(\mbox{id}\otimes\Delta_{\cal F}){\cal R}_{\cal F}$
$\displaystyle=$ $\displaystyle{\cal F}_{23}{\cal
F}_{23,1}\big{(}(\mbox{id}\otimes\Delta){\cal R}\big{)}{\cal
F}_{1,23}^{-1}{\cal F}_{23}^{-1}$ (1.16) $\displaystyle=$
$\displaystyle(\Phi_{\cal F})^{-1}_{231}{\cal F}_{31}{\cal
F}_{2,31}\Phi_{231}\big{(}\Phi_{231}^{-1}{\cal R}_{13}\Phi_{213}{\cal
R}_{12}\Phi^{-1}_{123}\big{)}{\cal F}_{1,23}^{-1}{\cal F}_{23}^{-1}$
$\displaystyle=$ $\displaystyle(\Phi_{\cal F})^{-1}_{231}({\cal R}_{\cal
F})_{13}{\cal F}_{13}{\cal F}_{2,13}\Phi_{213}{\cal R}_{12}{\cal
F}_{123}^{-1}$ $\displaystyle=$ $\displaystyle(\Phi_{\cal F})^{-1}_{231}({\cal
R}_{\cal F})_{13}(\Phi_{\cal F})^{-1}_{213}{\cal F}_{21}{\cal F}_{21,3}{\cal
R}_{12}{\cal F}_{123}^{-1}$ $\displaystyle=$ $\displaystyle(\Phi_{\cal
F})^{-1}_{231}({\cal R}_{\cal F})_{13}(\Phi_{\cal F})^{-1}_{213}({\cal
R}_{\cal F})_{12}{\cal F}_{12}{\cal F}_{12,3}\Phi^{-1}_{123}{\cal
F}_{1,23}^{-1}{\cal F}_{23}^{-1}$ $\displaystyle=$ $\displaystyle(\Phi_{\cal
F})^{-1}_{231}({\cal R}_{\cal F})_{13}(\Phi_{\cal F})^{-1}_{213}({\cal
R}_{\cal F})_{12}(\Phi_{\cal F})^{-1}_{123}.$

Similarly, for $(\Delta_{\cal F}\otimes\mbox{id}){\cal R}_{\cal F}.$

We shall now examine axioms (3)-(4) of Definition 1.1, and working out the
action of the counit on the twisted ${\cal R}$-matrix.

  * •

Axiom (3) of Definition 1.1: we recall (1.13) and axiom (3) of Definition 1.1,
then $(\epsilon\otimes\mbox{id})\Delta_{\cal
F}(a)=(\epsilon\otimes\mbox{id})\big{(}{\cal F}\Delta(a){\cal
F}^{-1}\big{)}=(c^{\cal F}_{l})^{-1}ac^{\cal F}_{l},$ where $c^{\cal
F}_{l}=c_{l}v^{-1}$. Similarly, $(\mbox{id}\otimes\epsilon)\Delta_{\cal
F}(a)=(c^{\cal F}_{r})^{-1}ac^{\cal F}_{r},$ where $c^{\cal
F}_{r}=c_{r}w^{-1}$, $\forall a\in{\cal A}.$

  * •

Axiom (4) of Definition 1.1: we recall that
$(\mbox{id}\otimes\epsilon\otimes\mbox{id})(1\otimes{\cal F})=1\otimes v,$
$(\mbox{id}\otimes\epsilon\otimes\mbox{id})({\cal F}\otimes 1)=w\otimes 1,$
and $(\mbox{id}\otimes\epsilon\otimes\mbox{id})(\mbox{id}\otimes\Delta){\cal
F}=(1\otimes c_{l}^{-1}){\cal F}(1\otimes c_{l}),$
$(\mbox{id}\otimes\epsilon\otimes\mbox{id})(\Delta\otimes\mbox{id}){\cal
F}=(c_{r}^{-1}\otimes 1){\cal F}(c_{r}\otimes 1),$ bearing also in mind (1.11)
and the fact that $(\mbox{id}\otimes\epsilon\otimes\mbox{id})\Phi=c_{r}\otimes
c_{l}^{-1},$ we deduce that
$(\mbox{id}\otimes\epsilon\otimes\mbox{id})\Phi_{\cal F}=c^{\cal
F}_{r}\otimes(c^{\cal F}_{l})^{-1},$ and this concludes our proof as all the
axioms of the quasi-triangular quasi-bialgebra are satisfied.

We also examine $(\mbox{id}\otimes\epsilon){\cal R}_{\cal F}$ and
$(\epsilon\otimes\mbox{id}){\cal R}_{\cal F},$ recalling (1.12) and (1.13) we
conclude that $(\mbox{id}\otimes\epsilon){\cal R}_{\cal
F}=v\big{(}(\mbox{id}\otimes\epsilon){\cal R}\big{)}w^{-1}$ and
$(\epsilon\otimes\mbox{id}){\cal R}_{\cal
F}=w\big{(}(\epsilon\otimes\mbox{id}){\cal R}\big{)}v^{-1}.$ When one of the
quasi-bialgebras is a bialgebra, e.g. set $\Phi_{\cal F}=1\otimes 1\otimes 1$
($c^{\cal F}_{l}=c^{\cal F}_{r}=1$), then $(\epsilon\otimes\mbox{id}){\cal
R}_{\cal F}=(\mbox{id}\otimes\epsilon){\cal R}_{\cal F}=1,$ and consequently
$(\mbox{id}\otimes\epsilon){\cal R}=v^{-1}w$ and
$(\epsilon\otimes\mbox{id}){\cal R}=w^{-1}v.$ Recalling also that $c^{\cal
F}_{l}=c_{l}v^{-1}$ and $c^{\cal F}_{r}=c_{r}w^{-1}$ we deduce that $c_{l}=v$
and $c_{r}=w.$

When also $\Phi=1\otimes 1\otimes 1,$ i.e. we are dealing with two bialgebras
then we also have $(\epsilon\otimes\mbox{id}){\cal
R}=(\mbox{id}\otimes\epsilon){\cal R}=1$ and hence $v=w,$ and due to Axiom (3)
of (quasi) bialgebras we immediately deduce that $u=1.$ ∎

We note that in the case of a triangular quasi-bialgebra the extra condition
${\cal R}^{(op)}{\cal R}=1_{{\cal A}\otimes{\cal A}}$ holds, and due to (1.12)
we deduce that ${\cal R}_{\cal F}^{(op)}{\cal R}_{\cal F}=1_{{\cal
A}\otimes{\cal A}},$ so triangularity is also preserved.

We examine in the next Corollary a special case of the Proposition 1.7, namely
the case where the Drinfeld twist is a special pure tensor (see also [14],
Dinfeld 1989).

######  Corollary 1.8.

Let $\big{(}{\cal A},\Delta,\epsilon,\Phi,{\cal R}\big{)}$ be a quasi-
triangular quasi-bialgebra and consider the special twist ${\cal
F}=c_{r}\otimes c_{l},$ where $c_{r}$ and $c_{l}$ are derived from $\Phi$
according to Remark 1.3. Then the twisted quasi-triangular quasi-bialgebra
$\big{(}{\cal A},\Delta_{\cal F},\epsilon,\Phi_{\cal F},{\cal R}_{\cal
F}\big{)}$ [14], has trivial unit constraints i.e. $c^{\cal F}_{l}=c^{\cal
F}_{r}=1_{\cal A}.$

###### Proof.

For any quasi-triangular quasi-bialgebra $(\epsilon\otimes id){\cal
R}=c_{r}^{-1}c_{l}$ and $(id\otimes\epsilon){\cal R}=c_{l}^{-1}c_{r}.$
According to Remark 1.3 any quasi-triangular quasi-bialgebra can be suitably
counit- normalized such that $\epsilon(c_{r})=\epsilon(c_{l})=1.$ We now
consider the special twist ${\cal F}=c_{r}\otimes c_{l}$, which in the context
of quasi-triangular quasi-bialgebras is admissible. The ${\cal F}$-twisted
quasi-triangular quasi-bialgebra has an $R$-matrix: ${\cal R}_{\cal
F}=(c_{l}\otimes c_{r}){\cal R}(c_{r}^{-1}\otimes c_{l}^{-1}),$ which
satisfies: $(\epsilon\otimes\mbox{id}){\cal R}_{\cal
F}=(\mbox{id}\otimes\epsilon){\cal R}_{\cal F}=1$. The latter statement
together with axiom (3) of Definition 1.1 lead to the “trivialization” of the
quasi-bialgebra, i.e. $c^{\cal F}_{l}=c^{\cal F}_{r}=1_{\cal A}.$ ∎

Note that twists can be in general composed, i.e. if the twist ${\cal F}$
sends quasi-bialgebra ${\cal A}$ to quasi-bialgebra ${\cal B}$, with trivial
unit constraints, and the twist ${\cal G}$ sends ${\cal B}$ to a third quasi-
bialgebra ${\cal C}$, then the product ${\cal G}{\cal F}$ is a twist as well,
sending ${\cal A}$ to ${\cal C}$. This type of factorization might make the
technicalities of describing certain admissible twists a bit easier to follow,
breaking it up into possibly more manageable building blocks. In particular,
if ${\cal A}$ is an arbitrary quasi-triangular quasi-bialgebra and ${\cal C}$
is a quasi-triangular bialgebra, then there always exist a quasi-triangular
quasi-bialgebra ${\cal B}$ with trivial unit constraints, and twists ${\cal
F}$ and ${\cal G}$ such that:

  1. (1)

${\cal F}$ is as described in Corollary 1.8,

  2. (2)

${\cal G}$ satisfies $(\epsilon\otimes id){\cal G}=(id\otimes\epsilon){\cal
G}=1$,

  3. (3)

${\cal B}$ is obtained from ${\cal A}$ by twisting with ${\cal F}$,

  4. (4)

${\cal C}$ is obtained from ${\cal B}$ by twisting with ${\cal G}$.

We shall be focusing henceforth on situations of twisting between a
(quasi-)bialgebra and a bialgebra in accordance to Propositions 1.7 for the
special case of (quasi-)bialgebras of 1.5, specifically $\Phi_{\cal
F}=1\otimes 1\otimes 1$ and $\Phi\neq 1\otimes 1\otimes 1.$ We are considering
thus two specific cases that consist the appropriate framework to describe the
quantum algebras [9, 10] emerging from set-theoretic solutions of the Yang-
Baxter equation expressed as suitable Drinfeld twists [11] as will be
discussed in the subsequent section. We introduce now some useful notation and
a worked out example appropriate for our frame here when examining quantum
algebras emerging form set-theoretic solutions of the YBE and their
$q$-analogues (Sections 2 and 3), compatible also with the analysis in [11].

######  Remark 1.9.

According to Proposition 1.5 we distinguish two cases: Let $\big{(}{\cal
A},\Delta,\epsilon,\Phi,{\cal R}\big{)}$ be a quasi-triangular quasi-bialgebra
and $\big{(}{\cal A},\Delta_{\cal F},\epsilon,{\cal R}_{\cal F}\big{)}$ a
quasi-triangular bialgebra and let the conditions of Proposition 1.5 hold. We
first recall the useful notation introduced in the proof of Proposition 1.7:
${\cal F}_{1,23}:=(\mbox{id}\otimes\Delta){\cal F}$ and ${\cal
F}_{12,3}:=(\Delta\otimes\mbox{id}){\cal F},$ and by the quasi-bialgebra
axioms ${\cal F}_{21,3}{\cal R}_{12}={\cal R}_{12}{\cal F}_{12,3},$ $\ {\cal
F}_{1,32}{\cal R}_{23}={\cal R}_{23}{\cal F}_{1,23}.$

  1. (1)

If the associator satisfies $\Phi_{213}{\cal R}_{12}={\cal R}_{12}\Phi_{123},$
we denote (index notation):

${\cal F}^{*}_{12,3}:=\big{(}(\Delta\otimes\mbox{id}){\cal
F}\big{)}\Phi^{-1},$ (1.17)

then condition (1.11) can be re-expressed as ${\cal F}_{123}:={\cal
F}_{23}{\cal F}_{1,23}={\cal F}_{12}{\cal F}_{12,3}^{*}.$ Due to constraint
(1.6) we also deduce ${\cal F}^{*}_{21,3}{\cal R}_{12}={\cal R}_{12}{\cal
F}^{*}_{12,3}.$ This is compatible also with the first part of Proposition
1.5.

  2. (2)

If the associator satisfies $\Phi_{132}{\cal R}_{23}={\cal R}_{23}\Phi_{123},$
we denote

${\cal F}^{*}_{1,23}=\big{(}(\mbox{id}\otimes\Delta){\cal F}\big{)}\Phi,$
(1.18)

then condition (1.11) is re-expressed as ${\cal F}_{123}:={\cal F}_{23}{\cal
F}^{*}_{1,23}={\cal F}_{12}{\cal F}_{12,3}.$ In this case, due to constraint
(1.6), we also deduce ${\cal F}^{*}_{1,32}{\cal R}_{23}={\cal R}_{23}{\cal
F}^{*}_{1,23}.$ This is compatible with the second part of Proposition 1.5.

######  Remark 1.10.

Under the conditions of Proposition 1.5 the universal ${\cal R}$-matrix
satisfies the usual YBE, and hence the object ${\mathrm{T}}_{1,23}:={\cal
R}_{13}{\cal R}_{12}$ satisfies the RTT relation [17]:
${\cal R}({\mathrm{T}}\otimes
1)(1\otimes{\mathrm{T}})=(1\otimes{\mathrm{T}})({\mathrm{T}}\otimes 1){\cal
R}$. Also, due to the properties described in Remark 1.9:

$({\mathrm{T}}_{\cal F})_{1,23}:=({\cal R}_{\cal F})_{13}({\cal R}_{\cal
F})_{12}={\cal F}_{231}{\mathrm{T}}_{1,23}{\cal F}_{123}^{-1}.$ (1.19)

The above is checked as in [11] (the $N$-generalization also holds as
described in [11]).

Before we focus on set-theoretic solutions of the YBE we will first work out a
particular example compatible with the special case of Remark 1.6.

######  Example 1.11.

Let $\big{(}{\cal A},\Delta_{{\cal A}},\epsilon,{\cal R}_{\cal A}\big{)}$ be a
quasi-triangular bialgebra. Let also $u\in{\cal A}\otimes{\cal A}$ be a group-
like element, i.e. $u$ is invertible and $\Delta_{\cal A}(u)=u\otimes u,$ and
consequently $\epsilon(u)=1,$ $\leavevmode\nobreak\ \big{[}\Delta_{\cal
A}(u),\ {\cal R}_{\cal A}\big{]}=0.$ Consider also the Drinfeld twist ${\cal
F}=1\otimes u:$ ${\cal R}={\cal F^{(op)}}^{-1}{\cal R}_{\cal A}{\cal F}$ and
$\Delta(a)={\cal F}^{-1}\Delta_{\cal A}(a){\cal F},$ $a\in{\cal A}.$

Then $\big{(}{\cal A},\Delta,\epsilon,\Phi,{\cal R}\big{)}$ is a quasi-
triangular quasi-bialgebra:

  * •

$\Phi=1\otimes 1\otimes u^{-1}.$

  * •

$c_{r}=1,\ c_{l}=u.$

  * •

$(\Delta\otimes\mbox{id}){\cal R}={\cal R}_{13}{\cal R}_{23}\Phi_{123}$ and
$(\mbox{id}\otimes\Delta)=\Phi^{-1}_{231}{\cal R}_{13}{\cal R}_{12}.$

We recall the twist: ${\cal F}=1\otimes u,$ we can then readily check that
$(\epsilon\otimes\mbox{id}){\cal F}=u,$ and via $\epsilon(u)=1,$ we conclude
$(\mbox{id}\otimes\epsilon){\cal F}=1.$ Even though the condition
$(\epsilon\otimes\mbox{id}){\cal F}=1,$ is now relaxed, the twist is still
admissible, indeed this is a rather trivial statement: ${\cal F}$ satisfies
the following

${\cal F}_{123}:={\cal F}_{12}{\cal F}_{12,3}^{*}={\cal F}_{23}{\cal
F}_{1,23}$ (1.20)

where ${\cal F}_{1,23}:=(\mbox{id}\otimes\Delta){\cal F}=1\otimes u\otimes u$
and ${\cal F}^{*}_{12,3}:=\big{(}(\Delta\otimes\mbox{id}){\cal
F}\big{)}\Phi^{-1}=1\otimes 1\otimes u^{2}.$ The $N$-fold twist is then
defined as

${\cal F}_{12...N}:={\cal F}_{23..N}{\cal F}_{1,23..N}={\cal
F}_{12...N-1}{\cal F}_{12...N-1,N},$ (1.21)

more specifically ${\cal F}_{12..N}=1\otimes u\otimes
u^{2}\otimes\ldots\otimes u^{N-1}.$ The twist is admissible and hence the YBE
is also satisfied by ${\cal R}$ (see also [12] and Proposition 1.7).

Notice also that $\Phi_{213}{\cal R}_{12}={\cal R}_{12}\Phi_{123},$ so
Proposition 1.5 holds as well as Proposition 1.7 and Remark 1.9, then all the
axioms of Definitions 1.1 and 1.4 are satisfied, indeed:

  * •

$(\mbox{id}\otimes\Delta)\Delta(a)=(1\otimes 1\otimes
u^{-1})\big{(}(\Delta\otimes\mbox{id})\Delta(a)\big{)}(1\otimes 1\otimes u),$
$\forall a\in{\cal A}.$

  * •

Axiom 2 of Definition 1.1 is trivially satisfied.

  * •

$(\epsilon\otimes\mbox{id})\Delta(a)=u^{-1}au$ and
$(\mbox{id}\otimes\epsilon)\Delta(a)=a$

  * •

$(\mbox{id}\otimes\epsilon\otimes\mbox{id})\Phi=1\otimes u^{-1}.$

  * •

$\Delta^{(op)}(a){\cal R}={\cal R}\Delta(a),$ $\forall a\in{\cal A}.$

  * •

$(\mbox{id}\otimes\Delta){\cal R}=\Phi_{231}^{-1}{\cal R}_{13}{\cal R}_{12}\ $
and $\ (\Delta\otimes\mbox{id}){\cal R}={\cal R}_{13}{\cal R}_{12}\Phi_{123}.$

And due to Proposition 1.5 the ${\cal R}$-matrix satisfies the Yang-Baxter
equation. It is also straightforward to show, using that $\epsilon$ is an
algebra homomorphism, that $(\epsilon\otimes\mbox{id}){\cal R}=u,\
(\mbox{id}\otimes\epsilon){\cal R}=u^{-1},$ as expected due to Proposition
1.5. In this example it is clear that in the quasi-bialgebra setting Drinfeld
twists do not require a counit constraint.

##  2\. Quasi-bialgebras from Yangians

In this section we focus on the quantum algebras emerging from involutive,
non-degenerate, set-theoretic solutions of the YBE. From now on we work over
the field $k=\mathbb{C}.$ It was shown in [11] after identifying a suitable
admissible twist that Baxterized set-theoretic solutions are always coming
from the $\mathfrak{gl}_{n}$ Yangian $R$-matrix via the mentioned twist. We
will show here that the respective twisting of the $\mathfrak{gl}_{n}$ Yangian
leads to a quasi-triangular quasi-bialgebra in accordance to the findings of
section 1.

###  2.1. Set-theoretic solutions of the YBE

We present in this section basic background information regarding set-
theoretic solutions of the Yang-Baxter equation as well as a brief review on
the recent findings of [9] on the links between set-theoretic solutions of the
Yang-Baxter equation and quantum algebras.

Let $X=\\{{\mathrm{x}}_{1},{\mathrm{x}}_{2},\ldots,{\mathrm{x}}_{n}\\}$ be a
finite set and ${\check{r}}:X\times X\rightarrow X\times X,$ such that

${\check{r}}(x,y)=\big{(}\sigma_{x}(y),\tau_{y}(x)\big{)}.$

We say that $\check{r}$ is non-degenerate if $\sigma_{x}$ and $\tau_{y}$ are
bijective functions. Also, the solution $(X,\check{r})$ is involutive:
$\check{r}(\sigma_{x}(y),\tau_{y}(x))=(x,y)$, ($\check{r}^{2}(x,y))=(x,y)$).
We focus on non-degenerate, involutive solutions of the set-theoretic braid
equation:

$({\check{r}}\times id_{X})(id_{X}\times{\check{r}})({\check{r}}\times
id_{X})=(id_{X}\times{\check{r}})({\check{r}}\times
id_{X})(id_{X}\times{\check{r}}).$

Let $V$ be the space of dimension equal to the cardinality of $X$, and with a
slight abuse of notation, let $\check{r}$ also denote the matrix associated to
the linearisation of ${\check{r}}$ on $V={\mathbb{C}}X$ (see [39] for more
details), i.e. $\check{r}$ is the $n^{2}\times n^{2}$ matrix:

${\check{r}}=\sum_{x,y\in X}e_{x,\sigma_{x}(y)}\otimes e_{y,\tau_{y}(x)}.$
(2.1)

where $e_{x,y}$ is the $n\times n$ matrix:
$(e_{x,y})_{z,w}=\delta_{x,z}\delta_{y,w}$. The matrix $\check{r}:V\otimes
V\rightarrow V\otimes V$ satisfies as expected the (constant) Braid equation:

$({\check{r}}\otimes I_{V})(I_{V}\otimes{\check{r}})({\check{r}}\otimes
I_{V})=(I_{V}\otimes{\check{r}})({\check{r}}\otimes
I_{V})(I_{V}\otimes{\check{r}}),$

where $I_{V}$ the identity matrix. Note also that $\check{r}$ is involutive,
i.e. ${\check{r}}^{2}=I_{V\otimes V}.$

We define also, $r={\mathcal{P}}{\check{r}}$, where
${\mathcal{P}}=\sum_{x,y\in X}e_{x,y}\otimes e_{y,x}$ is the permutation
operator; consequently ${r}=\sum_{x,y\in X}e_{y,\sigma_{x}(y)}\otimes
e_{x,\tau_{y}(x)}.$ The Yangian [41] is a special case:
$\check{r}=\sum_{x,y\in X}e_{x,y}\otimes e_{y,x}.$

We recall now the Yang-Baxter equation in the braid form in the presence of
spectral parameters $\lambda_{1},\ \lambda_{2}$
($\delta=\lambda_{1}-\lambda_{2}$) :

$\check{R}_{12}(\delta)\ \check{R}_{23}(\lambda_{1})\
\check{R}_{12}(\lambda_{2})=\check{R}_{23}(\lambda_{2})\
\check{R}_{12}(\lambda_{1})\ \check{R}_{23}(\delta).$ (2.2)

where $\check{R}:V\otimes\to V\otimes V$ and let in general
$\check{R}=\sum_{j}a_{j}\otimes b_{j},$ then in the index notation
$\check{R}_{12}=\sum_{j}a_{j}\otimes b_{j}\otimes I_{V},$
$\check{R}_{23}=\sum_{j}I_{V}\otimes a_{j}\otimes b_{j}$ and
$\check{R}_{13}=\sum_{j}a_{j}\otimes I_{V}\otimes b_{j}.$

We focus henceforth on involutive, non-degenerate set-theoretic solutions of
the YBE, given by (2.1). The set-theoretic solution $\check{r}$ (2.1) is a
representation of the $A$-type Hecke algebra for $q=1$ (see also [9]), as
$\check{r}$ satisfies the braid relations and also $\check{r}^{2}=I_{X\otimes
X}.$ The set-theoretic $\check{r}$ provides a representation of the $A$-type
Hecke algebra, hence Baxterized solutions of the Yang-Baxter equation can be
derived [9]:

$\check{R}(\lambda)=\lambda\check{r}+I_{X}\otimes I_{X},$ (2.3)

where $I_{X}$ is the identity matrix of dimension equal to the cardinality of
the set $X$. Let also $R(\lambda)={\cal P}\check{R}(\lambda)$, (recall the
permutation operator ${\cal P}=\sum_{x,y}e_{x,y}\otimes e_{y,x}$), then the
following properties for $R$-matrices coming from set-theoretic solutions were
shown in [9]:

  1. (1)

Unitarity: $R_{12}(\lambda)\ R_{21}(-\lambda)=(-\lambda^{2}+1)I_{X\otimes X}.$

  2. (2)

Crossing Unitarity: $R_{12}^{t_{1}}(\lambda)\
R_{12}^{t_{2}}(-\lambda-n)=\lambda(-\lambda-n)I_{X\otimes X}.$

  3. (3)

$R_{12}^{t_{1}t_{2}}(\lambda)=R_{21}(\lambda),$ where ${}^{t_{1,2}}$ denotes
transposition on the first, second space respectively.

We give a brief account on the quantum algebra emerging from Baxterized set-
theoretic solutions of the Yang-Baxter equation. Our approach on deriving the
quantum group associated to set-theoretic solutions [9, 10] is based on the
FRT construction [17], which is in a sense dual to the Hopf algebraic
description [13]. Given a solution of the Yang-Baxter equation, the quantum
algebra is defined via the fundamental relation [17](we have multiplied the
familiar RTT relation with the permutation operator):

$\check{R}_{12}(\lambda_{1}-\lambda_{2})\ L_{1}(\lambda_{1})\
L_{2}(\lambda_{2})=L_{1}(\lambda_{2})\ L_{2}(\lambda_{1})\
\check{R}_{12}(\lambda_{1}-\lambda_{2}),$ (2.4)

where
$\check{R}(\lambda)\in\mbox{End}({\mathbb{C}}^{n})\otimes\mbox{End}({\mathbb{C}}^{n})$,
$\ L(\lambda)\in\mbox{End}({\mathbb{C}}^{n})\otimes{\cal A}$ and ${\cal A}$ is
the quantum algebra defined by (2.4). We focus on solutions given by (2.3),
(2.1). The defining relations of the corresponding quantum algebra were
derived in [9]:

The quantum algebra associated to the set-theoretic $R$-matrix (2.3), (2.1) is
defined by generators $L^{(m)}_{zw},\ z,w\in X$, and defining relations

$\displaystyle
L_{z,w}^{(n)}L_{\hat{z},\hat{w}}^{(m)}-L_{z,w}^{(m)}L_{\hat{z},\hat{w}}^{(n)}$
$\displaystyle=$ $\displaystyle
L^{(m)}_{z,\sigma_{w}(\hat{w})}L^{(n+1)}_{\hat{z},\tau_{\hat{w}}(w)}-L^{(m+1)}_{z,\sigma_{w}(\hat{w})}L^{(n)}_{\hat{z},\tau_{\hat{w}}(w)}$
(2.5) $\displaystyle-$ $\displaystyle
L^{(n+1)}_{\sigma_{z}(\hat{z}),w}L^{(m)}_{\tau_{\hat{z}}(z),\hat{w}}+L^{(n)}_{\sigma_{z}(\hat{z},)w}L^{(m+1)}_{\tau_{\hat{z}}(z),\hat{w}}.$

The proof is based on the fundamental relation (2.4) and the form of the
Baxterized set-theoretic $R$-matrix (for the detailed proof see [9]). Recall
also that in the index notation we define $\check{R}_{12}=\check{R}\otimes
1_{\cal A}$:

$\displaystyle L_{1}(\lambda)=\sum_{z,w\in X}e_{z,w}\otimes I_{X}\otimes
L_{z,w}(\lambda),\ \quad L_{2}(\lambda)=\sum_{z,w\in X}I_{X}\otimes
e_{z,w}\otimes L_{z,w}(\lambda)$ (2.6)

where $L_{z,w}(\lambda)=\sum_{m=0}^{\infty}\lambda^{-m}L_{z,w}^{(m)}$ and
$L_{z,w}^{(m)}$ are the generators of the affine algebra ${\cal A}$ and
$\check{R}$ is given in (2.3), (2.1). Note that the element
${\mathrm{T}}_{1,23}(\lambda)=L_{13}(\lambda)L_{12}(\lambda),$ also satisfies
(2.4) [17, 13], i.e it is a tensor representation of the quantum algebra.

###  2.2. The Yangian ${\cal Y}(\mathfrak{gl}_{n})$

We now present a brief review on the $\mathfrak{gl}_{n}$ Yangian useful for
our purposes here. We first briefly recall the definition of the
$\mathfrak{gl}_{n}$ Yangian, which is most relevant in our present
investigation (for a review on Yangians see e.g. [13, 8, 33]). We also review
the Yangian as a Hopf algebra and we then comment on the action of the
antipode after a suitable twist of the algebra.

######  Definition 2.1.

The $\mathfrak{gl}_{n}$ Yangian ${\cal Y}(\mathfrak{gl}_{n})$, is a non-
abelian algebra with generators ${\mathrm{Q}}_{ab}^{(p)}$,
$p\in\big{\\{}1,2,\ldots\big{\\}}$, $a,b\in\big{\\{}1,\ 2,\dots,n\big{\\}}$
and defining relations given below

$\displaystyle\Big{[}{\mathrm{Q}}_{ab}^{(1)},\
{\mathrm{Q}}_{cd}^{(1)}\Big{]}=\delta_{cb}{\mathrm{Q}}^{(1)}_{ad}-\delta_{ad}{\mathrm{Q}}^{(1)}_{cb}$
(2.7) $\displaystyle\Big{[}{\mathrm{Q}}_{ab}^{(1)},\
{\mathrm{Q}}_{cd}^{(2)}\Big{]}=\delta_{cb}{\mathrm{Q}}^{(2)}_{ad}-\delta_{ad}{\mathrm{Q}}^{(2)}_{cb}$
(2.8) $\displaystyle\Big{[}{\mathrm{Q}}_{ab}^{(2)},\
{\mathrm{Q}}_{cd}^{(2)}\Big{]}=\delta_{cb}{\mathrm{Q}}^{(3)}_{ad}-\delta_{ad}{\mathrm{Q}}^{(3)}_{cb}+{1\over
4}{\mathrm{Q}}_{ad}^{(1)}(\sum_{e}{\mathrm{Q}}_{ce}^{(1)}{\mathrm{Q}}_{eb}^{(1)})-{1\over
4}(\sum_{e}{\mathrm{Q}}_{ae}^{(1)}{\mathrm{Q}}_{ed}^{(1)}){\mathrm{Q}}_{cb}^{(1)}$
(2.9)

also relations

$\displaystyle\Big{[}{\mathrm{Q}}_{ab}^{(1)},\
\Big{[}{\mathrm{Q}}_{cd}^{(2)},\
{\mathrm{Q}}_{ef}^{(2)}\Big{]}\Big{]}-\Big{[}{\mathrm{Q}}_{ab}^{(2)},\
\Big{[}{\mathrm{Q}}_{cd}^{(1)},\ {\mathrm{Q}}_{ef}^{(2)}\Big{]}\Big{]}=$
$\displaystyle{1\over 4}\sum_{p,q}\Big{(}\Big{[}{\mathrm{Q}}_{ab}^{(1)},\
\Big{[}{\mathrm{Q}}_{cp}^{(1)}{\mathrm{Q}}_{pd}^{(1)},\
{\mathrm{Q}}_{eq}^{(1)}{\mathrm{Q}}_{qf}^{(1)}\Big{]}\Big{]}-\Big{[}{\mathrm{Q}}_{ap}^{(1)}{\mathrm{Q}}_{pb}^{(1)},\
\Big{[}{\mathrm{Q}}_{cd}^{(1)},\
{\mathrm{Q}}_{eq}^{(1)}{\mathrm{Q}}_{qf}^{(1)}\Big{]}\Big{]}\Big{)},$ (2.10)
$\displaystyle\ldots\mbox{(higher orders)}$

It will be useful for what follows to introduce at this point the definition
of a quasi-Hopf algebra [13].

######  Definition 2.2.

A quasi-Hopf algebra ${\cal A}$ is a quasi-bialgebra $\big{(}{\cal
A},\Delta,\epsilon,\Phi\big{)}$ for which there exist
$\alpha,\beta\in{\mathcal{A}}$ and a bijective algebra anti-homomorphism
$S:{\cal A}\to{\cal A}$ (the antipode) such that

  1. (1)

$\sum_{j}S(f_{j})\alpha h_{j}=\epsilon(w)\alpha\ $ and $\ \sum_{j}f_{j}\beta
S(h_{j})=\epsilon(w)\beta$
where $\Delta(w)=\sum_{j}f_{j}\otimes h_{j},$ $\forall w\in{\cal A}.$

  2. (2)

$\sum_{j}x_{j}\beta S(y_{j})\alpha z_{j}=1_{\cal A}\ $ and $\
\sum_{j}S(\hat{x}_{j})\alpha\hat{y}_{j}\beta S(\hat{z}_{j})=1_{\cal A}$
where $\Phi=\sum_{j}x_{j}\otimes y_{j}\otimes z_{j}$ and
$\Phi^{-1}=\sum_{j}\hat{x}_{j}\otimes\hat{y}_{j}\otimes\hat{z}_{j}.$

Quasi-Hopf algebras generalize Hopf algebras in the same way that quasi-
bialgebras generalize bialgebras.

The Yangian $\big{(}{\cal
Y}(\mathfrak{gl}_{n}),\Delta_{Y},\epsilon,S_{Y},{\cal R}_{Y}\big{)}$ is a
quasi-triangular Hopf algebra over ${\mathbb{C}}$ equipped with:

  * •

A coproduct $\Delta_{Y}:{\cal Y}(\mathfrak{gl}_{n})\rightarrow{\cal
Y}(\mathfrak{gl}_{n})\otimes{\cal Y}(\mathfrak{gl}_{n})$ such that

$\displaystyle\Delta_{Y}({\mathrm{Q}}_{ab}^{(1)})$ $\displaystyle=$
$\displaystyle{\mathrm{Q}}_{ab}^{(1)}\otimes
1+1\otimes{\mathrm{Q}}_{ab}^{(1)}$ (2.11)
$\displaystyle\Delta_{Y}({\mathrm{Q}}_{ab}^{(2)})$ $\displaystyle=$
$\displaystyle{\mathrm{Q}}_{ab}^{(2)}\otimes
1+1\otimes{\mathrm{Q}}_{ab}^{(2)}+{1\over
2}\sum_{d=1}^{n}({\mathrm{Q}}_{ad}^{(1)}\otimes{\mathrm{Q}}_{db}^{(1)}-{\mathrm{Q}}_{db}^{(1)}\otimes{\mathrm{Q}}_{ad}^{(1)}).$
(2.12)

The $l$-fold co-product ($l$ is an integer greater than 2)
$\Delta_{Y}^{(l)}:{\cal Y}(\mathfrak{gl}_{n})\ \to\ {\cal
Y}^{\otimes(l)}(\mathfrak{gl}_{n})$ is defined as

$\Delta_{Y}^{(l)}=(\mbox{id}\otimes\Delta_{Y}^{(l-1)})\Delta_{Y}=(\Delta_{Y}^{(l-1)}\otimes\mbox{id})\Delta_{Y}.$
(2.13)

  * •

A co-unit $\epsilon:{\cal Y}(\mathfrak{gl}_{n})\to{\mathbb{C}}$, such that

$\epsilon({\mathrm{Q}}^{(1)}_{ab})=\epsilon({\mathrm{Q}}^{(2)}_{ab})=0,$
(2.14)

  * •

An antipode $S_{Y}:{\cal Y}(\mathfrak{gl}_{n})\to{\cal Y}(\mathfrak{gl}_{n})$
such that

$S_{Y}({\mathrm{Q}}^{(1)}_{ab})=-{\mathrm{Q}}_{ab}^{(1)},\qquad
S_{Y}({\mathrm{Q}}^{(2)}_{ab})=-{\mathrm{Q}}^{(2)}_{ab}+{1\over
2}{\mathrm{Q}}^{(1)}_{ab}$ (2.15)

  * •

Also, there exists an invertible element ${\cal R}_{Y}\in{\cal
Y}(\mathfrak{gl}_{n})\otimes{\cal Y}(\mathfrak{gl}_{n})$ (the universal ${\cal
R}$-matrix), such that

    1. (1)

${\cal
R}_{Y}\Delta_{Y}({\mathrm{Q}}_{ab}^{(p)})=\Delta_{Y}^{(op)}({\mathrm{Q}}_{ab}^{(p)}){\cal
R}_{Y}.$

    2. (2)

$(\Delta_{Y}\otimes\mbox{id}){\cal R}_{Y}=({\cal R}_{Y})_{13}({\cal
R}_{Y})_{23}$ and $(\mbox{id}\otimes\Delta_{Y}){\cal R}_{Y}=({\cal
R}_{Y})_{13}({\cal R}_{Y})_{12}$.

We recall now Example 1.11 and we briefly discuss the notion of the antipode
in the following:

######  Remark 2.3.

We specialize Example 1.11 to the case of the Yangian ${\cal
Y}(\mathfrak{gl}_{n})$, i.e. $\big{(}{\cal
Y}(\mathfrak{gl}_{n}),\Delta,\epsilon,\Phi,{\cal R}\big{)},$ where
$\Phi=1\otimes 1\otimes u^{-1}$ for some group-like element $u,$ is a quasi-
triangular quasi-bialgebra. We now want to test whether $\big{(}{\cal
Y}(\mathfrak{gl}_{n}),\Delta,\epsilon,\Phi,{\cal R}\big{)}$ is a quasi-Hopf
algebra, i.e. if there exist $S$, $\alpha$, $\beta$ such that axioms (1) and
(2) in Definitio 2.2 hold.

Suppose that the group-like element $u$ exists, then $\Delta(u)=u\otimes u$
and:

  1. (1)

$S(u)\alpha u=\alpha$ and $u\beta S(u)=\beta,$ which lead to $S(u)=\alpha
u^{-1}\alpha^{-1}=\beta^{-1}u^{-1}\beta.$

  2. (2)

$\beta\alpha u^{-1}=1_{Y}$ and $\alpha\beta S(u)=1_{Y},$ which lead to
$u=\beta\alpha$ and $S(u)=\beta^{-1}\alpha^{-1}.$

All the above equations are self-consistent. Similarly, we can show that
$S(u^{-1})=\alpha\beta$

We now check the axioms of Definition 2.2 for the primitive elements
${\mathrm{Q}}_{ab}^{(1)},$ with a coproduct after twisting given as
$\Delta({\mathrm{Q}}_{ab}^{(1)})={\mathrm{Q}}_{ab}^{(1)}\otimes 1+1\otimes
u^{-1}{\mathrm{Q}}_{ab}^{(1)}u$:

  1. (1)

$S({\mathrm{Q}}_{ab}^{(1)})\alpha+\alpha u^{-1}{\mathrm{Q}}_{ab}^{(1)}u=0$ and
${\mathrm{Q}}_{ab}^{(1)}\beta+\beta S(u^{-1}{\mathrm{Q}}_{ab}^{(1)}u)=0,$
which lead to (by requiring also that $S$ is an algebra anti-homomorphism):
$S({\mathrm{Q}}_{ab}^{(1)})=-\alpha u^{-1}{\mathrm{Q}}_{ab}^{(1)}u\alpha^{-1}$
and $S({\mathrm{Q}}_{ab}^{(1)})=-\alpha{\mathrm{Q}}_{ab}^{(1)}\alpha^{-1}.$
The two latter expressions lead to
$\big{[}{\mathrm{Q}}_{ab}^{(1)},u\big{]}=0,$ which is not true in general.

The axioms for the antipode restrict $u$ to be in the center of the algebra.
This is very limiting, and in general is not true. Indeed, a simple class of
such non-central group-like elements can be defined as $u=e^{Q_{ab}^{(1)}},$
where recall $Q_{ab}^{(1)}$ are primitive elements of the algebra. Also, a
specific example of a represented non-central $u$ is used in the next
subsection for a special class of set-theoretic solutions.

To conclude, we are not able to define $S({\mathrm{Q}}_{ab}^{(1)})$ in a
consistent way for a generic $u$ by strictly following the axioms of
Definition 2.2. A generalization on the axiomatic formulation for the antipode
in quasi-Hopf algebras with nontrivial unit constraints may be in order,
however this issue will be thoroughly studied elsewhere.

It is useful for the purposes of the present investigation to introduce the
evaluation representation $\pi_{\lambda}:{\cal
Y}(\mathfrak{gl}_{n})\to\mbox{End}({\mathbb{C}}^{n})$,
$\lambda\in{\mathbb{C}},$ such that

$\pi_{\lambda}({\mathrm{Q}}_{ab}^{(1)})=e_{a,b},\quad\pi_{\lambda}({\mathrm{Q}}_{ab}^{(2)})=f_{a,b}$
(2.16)

where we define $f_{a,b}:=\lambda e_{a,b}$. Let $R_{Y}:V\otimes V\ toV\otimes
V$ be the $R$-matrix associated to Yangian with explicit form
$R_{Y}(\lambda)=I^{\otimes 2}+{1\over\lambda}{\cal P},$ where ${\cal P}$ is
the permutation operator and $I$ is the $n\times n$ identity matrix.

We also introduce the following convenient notation:

$\big{(}\pi_{\lambda_{1}}\otimes\pi_{\lambda_{2}}\big{)}\Delta_{Y}({\mathrm{Q}}_{ab}^{(1)})=\Delta_{Y}(e_{a,b}),\qquad\big{(}\pi_{\lambda_{1}}\otimes\pi_{\lambda_{2}}\big{)}\Delta_{Y}({\mathrm{Q}}_{ab}^{(2)})=\Delta_{Y}(f_{a,b};\lambda_{1},\lambda_{2}),$
(2.17)

also,
$\big{(}\pi_{\lambda_{1}}\otimes\pi_{\lambda_{2}}\big{)}\Delta_{Y}^{(op)}({\mathrm{Q}}_{ab}^{(2)})=\Delta_{Y}^{(op)}(f_{a,b};\lambda_{1},\lambda_{2})$
and $\Delta_{Y}^{(op)}(f_{a,b};\lambda_{1},\lambda_{2})={\cal
P}\Delta_{Y}(f_{a,b}:\lambda_{2},\lambda_{1}){\cal P},$ then we explicitly
express the coproducts as

$\displaystyle\Delta_{Y}(e_{i,j})=\Delta^{(op)}(e_{i,j})=e_{i,j}\otimes
I+I\otimes e_{i,j},$ (2.18)
$\displaystyle\Delta_{Y}(f_{i,j};\lambda_{1},\lambda_{2})=\lambda_{1}e_{i,j}\otimes
I+\lambda_{2}I\otimes e_{i,j}+{1\over 2}\big{(}e_{i,k}\otimes
e_{k,j}-e_{k,j}\otimes e_{i,j}\big{)}.$ (2.19)

The Yangian $R$-matrix satisfies the following intertwining relations:

$\displaystyle
R_{Y}(\lambda_{1}-\lambda_{2})\Delta_{Y}(e_{i,j})=\Delta_{Y}(e_{i,j})R_{Y}(\lambda_{1}-\lambda_{2})$
(2.20) $\displaystyle
R_{Y}(\lambda_{1}-\lambda_{2})\Delta_{Y}(f_{i,j};\lambda_{1},\lambda_{2})=\Delta_{Y}^{(op)}(f_{i,j};\lambda_{1},\lambda_{2})R_{Y}(\lambda_{1}-\lambda_{2}).$
(2.21)

Specifically, $\check{R}_{Y}={\cal P}R_{Y}$ is $\mathfrak{gl}_{n}$ invariant,
i.e.$\big{[}\check{R}_{Y}(\lambda),\ \Delta_{Y}(e_{x,y})\big{]}=0.$

And with this we conclude our short discussion on the $\mathfrak{gl}_{n}$
Yangian, which will be useful for the findings of the next subsection.

###  2.3. Set-theoretic solutions of the YBE and quasi-bialgebras

After the brief review on the $\mathfrak{gl}_{n}$ Yangian we may now move on
to our main aim which is the study of set-theoretic solutions of the Yang-
Baxter equation associated to quasi-bialgebras. We first review some
fundamental results on the admissible Drinfeld twist for involutive set-
theoretic solution of the YBE derived in [11] and we use these admissible
twists to produce quasi-bialgebras associated to the Yangian.

From Proposition 3.3 in [10] we can extract explicit forms for the twist
$F\in\mbox{End}({\mathbb{C}}^{n})\otimes\mbox{End}({\mathbb{C}}^{n})$ and
state the following Proposition which is Proposition 3.10 in [11].

######  Proposition 2.4.

([10, 11]) Let $\check{r}=\sum_{x,y\in X}e_{x,\sigma_{x}(y)}\otimes
e_{y,\tau_{y}(x)}$ be the set-theoretic solution of the braid YBE, ${\cal P}$
is the permutation operator and $\hat{V}_{k},\ V_{k}$ are their respective
eigenvectors. Let $F^{-1}=\sum_{k=1}^{n^{2}}\hat{V}_{k}\ V_{k}^{T}$ be the
similarity transformation (twist), such that $\check{r}=F^{-1}{\cal P}F.$ Then
the twist can be explicitly expressed as $F=\sum_{x\in
X}e_{x,x}\otimes{\mathbb{V}}_{x}$, where we define
${\mathbb{V}}_{x}=\sum_{y\in X}e_{\sigma_{x}(y),y}$.

For a detailed proof of the Proposition we refer the interested reader to [10]
and [11]. However, by recalling that $r={\cal P}\check{r},$ and using the fact
that $\sigma_{x},\ \tau_{y}$ are bijections, we confirm by direct computation
that $(F^{(op)})^{-1}F=\sum_{x,\in X}e_{y,\sigma_{x}(y)}\otimes
e_{x,\tau_{y}(x)}=r$. This is an admissible twist as was shown in [11] and as
will be discussed later in the text.

Let the Baxterized solution of the YBE be $R(\lambda)=\lambda r+{\cal P}.$ If
$r$ satisfies the YBE and $r_{12}r_{21}=I$ then the Baxterized $R(\lambda)$
matrix also satisfies the YBE. If $r={\cal P}\check{r}$ is the set-theoretic
solution of the YBE then,
$R_{12}(\lambda)=F^{-1}_{21}(R_{Y})_{12}(\lambda)F_{12}$, where
$R_{Y}(\lambda)=\lambda I+{\cal P}$ is the Yangian $R$-matrix. This
immediately follows from the form $R_{Y}(\lambda)=\lambda I+{\cal P}$, and the
property $F_{21}^{-1}{\cal P}_{12}{\cal F}_{12}={\cal P}_{12}$. Note also that
the twist is not uniquely defined, for instance an alternative twist is of the
form $G=\sum_{x,y\in X}e_{\tau_{y}(x),x}\otimes e_{y,y}$, and $\sum_{x,\in
X}e_{y,\sigma_{x}(y)}\otimes e_{x,\tau_{y}(x)}=G_{21}^{-1}G_{12},$ see [11].

Before we introduce the next fundamental quantities it is useful to prove the
following Proposition.

######  Proposition 2.5.

Let
$R_{Y}:{\mathbb{C}}^{n}\otimes{\mathbb{C}}^{n}\to{\mathbb{C}}^{n}\otimes{\mathbb{C}}^{n}$
be the Yangian $R$-matrix. Let also ${\mathbb{V}}_{\eta}=\sum_{x\in
X}e_{\sigma_{\eta}(x),x},$ $\forall\eta\in X$ consist a family of group-like
elements, i.e.
$\Delta_{Y}({\mathbb{V}}_{\eta})={\mathbb{V}}_{\eta}\otimes{\mathbb{V}}_{\eta}$
and $R(\lambda)=\lambda r+{\cal P},$ where $r$ is the set-theoretic solution
$r=\sum_{x,y\in X}e_{y,\sigma_{x}(y)}\otimes e_{x,\tau_{y}(x)}.$ Then

$\Delta^{(op)}({\mathbb{V}}_{\eta})R(\lambda)=R(\lambda)\Delta({\mathbb{V}}_{\eta}),$
(2.22)

where $\Delta^{(op)}({\mathbb{V}}_{\eta})={\cal
P}\Delta({\mathbb{V}}_{\eta}){\cal P},$

$\Delta({\mathbb{V}}_{\eta})=\sum_{x\in
X}e_{\sigma_{\eta}(x),x}\otimes\sum_{y\in
X}e_{\sigma_{\tau_{x}(\eta)}(y),y}\big{|}_{C_{1}=0},$ (2.23)

and
$C_{1}=\sigma_{\sigma_{\eta}(x)}(\sigma_{\tau_{x}(y)}(y))-\sigma_{\eta}(\sigma_{x}(y)).$

###### Proof.

By Proposition 2.4 we have $R(\lambda)=(F^{(op)})^{-1}R_{Y}(\lambda)F,$ where
we recall $F=\sum_{x\in X}e_{x,x}\otimes{\mathbb{V}}_{x}$. Recall also
$\Delta_{Y}({\mathbb{V}}_{\eta})={\mathbb{V}}_{\eta}\otimes{\mathbb{V}}_{\eta},$
then

$\displaystyle\Delta_{Y}({\mathbb{V}}_{\eta})R_{Y}(\lambda)=R_{Y}(\lambda)\Delta_{Y}({\mathbb{V}}_{\eta})\
\Rightarrow\ $ (2.24)
$\displaystyle(F^{(op)})^{-1}\big{(}\Delta_{Y}({\mathbb{V}}_{\eta})R_{Y}(\lambda)\big{)}F=(F^{(op)})^{-1}\big{(}R_{Y}(\lambda)\Delta_{Y}({\mathbb{V}}_{\eta})\big{)}F\
\Rightarrow$ (2.25)
$\displaystyle\Delta^{(op)}({\mathbb{V}}_{\eta})R(\lambda)=R(\lambda)\Delta({\mathbb{V}}_{\eta}),$
(2.26)

where $\Delta({\mathbb{V}}_{\eta})=F^{-1}\Delta_{Y}({\mathbb{V}}_{\eta})F.$ By
recalling that $F=\sum_{\eta}e_{\eta,\eta}\otimes{\mathbb{V}}_{\eta}$ and
${\mathbb{V}}_{\eta}=\sum_{x\in X}e_{\sigma_{\eta}(x),x}$ we obtain

$\displaystyle\Delta({\mathbb{V}}_{\eta})$ $\displaystyle=$
$\displaystyle\big{(}\sum_{\bar{\eta},\bar{x}\in
X}e_{\bar{\eta},\bar{\eta}}\otimes
e_{\sigma_{\bar{\eta}}(\bar{x}),\bar{x}}\big{)}\big{(}\sum_{\xi\in
X}e_{\sigma_{\eta}(\xi),\xi}\otimes\sum_{\zeta\in
X}e_{\sigma_{\eta}(\zeta),\zeta}\big{)}\big{(}\sum_{\hat{\eta},\hat{x}\in
X}e_{\hat{\eta},\hat{\eta}}\otimes
e_{\hat{x},\sigma_{\hat{\eta}}(\hat{x})}\big{)}$ (2.27) $\displaystyle=$
$\displaystyle\sum_{\xi,\bar{x},\hat{x}\in X}e_{\sigma_{\eta}(\xi),\xi}\otimes
e_{\bar{x},\hat{x}}$ (2.28)

subject to the following constraints:
$\sigma_{\sigma_{\eta}(\xi)}(\bar{x})=\sigma_{\eta}(\zeta)$ and
$\zeta=\sigma_{\xi}(\hat{x}),$ which lead to

$\sigma_{\sigma_{\eta}(\xi)}(\bar{x})=\sigma_{\eta}(\sigma_{\xi}(\hat{x})).$
(2.29)

Also, the constraint $C_{1}=0$ holds for the twist $F$ to be admissible, i.e.
$\sigma_{\sigma_{\eta}(\xi)}(\sigma_{\tau_{\xi}(\eta)}(\hat{x}))=\sigma_{\eta}(\sigma_{\xi}(\hat{x}))$,
which combined with (2.29) leads to
$\bar{x}=\sigma_{\tau_{\xi}(\eta)}(\hat{x}),$ and thus

$\Delta({\mathbb{V}}_{\eta})=\sum_{x\in
X}e_{\sigma_{\eta}(x),x}\otimes\sum_{y\in
X}e_{\sigma_{\tau_{x}(\eta)}(y),y}\big{|}_{C_{1}=0}.$ (2.30)

Note that the constraint $C_{1}=0$ holds so that the set-theoretic $r$-matrix
satisfies the YBE (details on the constraints due to YBE in the form used
here, see also e.g. [11] and relevant references therein). ∎

We recall that in Proposition 3.13 in [11] the following quantities were
introduced:

$\displaystyle F_{1,23}=\sum_{x,y,\eta\in
X}e_{\sigma_{\eta}(x),\sigma_{\eta}(x)}\otimes e_{\eta,\tau_{x}(\eta)}\
\otimes e_{\sigma_{x}(y),y}\big{|}_{C_{1}=0}$ (2.31) $\displaystyle
F^{*}_{12,3}=\sum_{x,y,\eta\in X}e_{\sigma_{\eta}(x),\sigma_{\eta}(x)}\otimes
e_{\tau_{x}(\eta),\tau_{x}(\eta)}\ \otimes
e_{\sigma_{\eta}(\sigma_{x}(y)),y}\big{|}_{C_{1}=0}$ (2.32)

where
$C_{1}=\sigma_{\sigma_{\eta}(x)}(\sigma_{\tau_{x}(y)}(y))-\sigma_{\eta}(\sigma_{x}(y))$.
Let also $\check{r}=\sum_{x,y\in X}e_{x,\sigma_{x}(y)}\otimes
e_{y,\tau_{y}(x)}$, then

$\check{r}_{12}F^{*}_{12,3}=F^{*}_{12,3}\check{r}_{12},\
\quad\check{r}_{23}F_{1,23}=F_{1,23}\check{r}_{23}.$ (2.33)

The detailed proof of (2.33) is given in [11].

This is a straightforward, but useful comment. Recall, $r={\cal P}\check{r}$,
where ${\cal P}$ is the permutation operator. If
$F^{*}_{12,3}\check{r}_{12}=\check{r}_{12}F^{*}_{12,3},$ and
$F_{1,23}\check{r}_{23}=\check{r}_{23}F_{1,23},$ then by multiplying the
latter two equalities with ${\cal P}$ from the left we conclude:
$F^{*}_{21,3}r_{12}=r_{12}F^{*}_{12,3},$ and $F_{1,32}r_{23}=r_{23}F_{1,23}.$

A useful Corollary follows which is key in formulating our main Conjecture
later.

######  Corollary 2.6.

Consider $F_{1,23}$ and $F^{*}_{12,3}$ defined in (2.31) and (2.32)
respectively. Then $F_{1,23}=(\mbox{id}\otimes\Delta)F,$ but
$F^{*}_{12,3}\neq(\Delta\otimes\mbox{id})F.$ Also, if
$\epsilon({\mathbb{V}}_{\eta})=1,$ $\forall\eta\in X,$ then
$(\epsilon\otimes\mbox{id})F\neq I$ and $(\mbox{id}\otimes\epsilon)F=I.$

###### Proof.

We recall that $F=\sum_{\eta\in X}e_{\eta,\eta}\otimes{\mathbb{V}}_{\eta}$,
then $(\mbox{id}\otimes\Delta)F=\sum_{\eta\in
X}e_{\eta,\eta}\otimes\Delta({\mathbb{V}}_{\eta}).$ Recalling also the form of
$\Delta({\mathbb{V}}_{\eta})$ in (2.23) and the defintion (2.31) we conclude
that $F_{1,23}=(\mbox{id}\otimes\Delta)F.$ In a similar fashion,
$(\Delta\otimes\mbox{id})F=\sum_{\eta\in
X}\Delta(e_{\eta,\eta})\otimes\sum_{x\in X}e_{\sigma_{\eta}(x),x}\neq
F^{*}_{12,3}$ (2.32).

Also, we can immediately deduce that
$(\mbox{id}\otimes\epsilon)F=\sum_{\eta\in X}e_{\eta,\eta}=I$ and
$(\epsilon\otimes\mbox{id})F=\sum_{\eta\in
X}\epsilon(e_{\eta,\eta}){\mathbb{V}}_{\eta}\neq I,$ which is compatible with
Proposition 1.7. ∎

The admissibility of the twist is proven in Proposition 3.15 in [11], and is
stated below:

######  Proposition 2.7.

([11]) Let $F_{12}=F\otimes I$ and $F_{23}=I\otimes F$, where
$F=\sum_{\eta,x,y\in X}e_{\eta,\eta}\otimes e_{\sigma_{\eta}(x),x}$. Let also
$F^{*}_{12,3}$ and $F_{1,23}$ defined in (2.31) and (2.32). Then

$F_{123}:=F_{12}F^{*}_{12,3}=F_{23}F_{1,23}.$ (2.34)

The proof is straightforward although tedious. By substituting the expressions
for $F_{12},\ F_{23}$, $F^{*}_{12,3}$ and $F_{1,23}$ (recall $C_{1}=0$ holds
for $F^{*}_{12,3},$ $F_{1,23}$) we obtain by direct computation:
$F_{12}F^{*}_{12,3}=F_{23}F_{1,23}.$ The explicit form the 3-twist is given
from the expressions above as
$F_{123}=\sum_{\eta,x,y\in X}e_{\sigma_{\eta}(x),\sigma_{\eta}(x)}\otimes
e_{\eta,\tau_{x}(\eta)}\otimes e_{\sigma_{\eta}(\sigma_{x}(y)),y}|_{C_{1}=0},$
[11].

Given the findings of the first section based on Propositions 1.5, 1.7, Remark
1.9 and Corollary 2.6 we formulate the following conjecture:

######  Conjecture 2.8.

The element $F^{*}_{12,3}$ can be expressed as
$F_{12,3}^{*}=\big{(}(\Delta\otimes\mbox{id})F\big{)}\Phi^{-1},$ where
$\Phi\in\mbox{End}(({\mathbb{C}}^{n})^{\otimes 3})$ is an invertible element
such that $\Phi_{213}r_{12}=r_{12}\Phi_{123}.$

This conjecture suggests that the quantum group emerging from set-theoretic
solutions of the YBE is a quasi-bialgebra and is compatible with Propositions
1.5 and 1.7. A few comments are in order at this point: note that the explicit
expression for $F^{*}_{12,3}$ is given in (2.32), then following the statement
in Conjecture 2.8 and recalling that $\Delta$ is an algebra homomorphism we
can formally write
$\Phi^{-1}=\big{(}(\Delta\otimes\mbox{id})F^{-1}\big{)}F_{12,3}^{*}.$ Also, it
was shown in [11] that $\big{[}F_{12,3}^{*},\check{r}_{12}\big{]}=0$ and by
definition $\big{[}(\Delta\otimes\mbox{id})F,\ \check{r}_{12}\big{]}=0,$ so we
conclude that $\Phi_{123}$ should also commute with $\check{r}_{12}$.

We will be discussing below the symmetries of the set-theoretic $r$-matrix and
the corresponding Baxterized solutions.

######  Corollary 2.9.

Let
$R_{Y}:{\mathbb{C}}^{n}\otimes{\mathbb{C}}^{n}\to{\mathbb{C}}^{n}\otimes{\mathbb{C}}^{n}$
be the Yangian $R-$matrix and $R(\lambda)=\lambda r+{\cal P},$ where $r$ is
the set-theoretic solution $r=\sum_{x,y\in X}e_{y,\sigma_{x}(y)}\otimes
e_{x,\tau_{y}(x)}.$ Then

$\Delta^{(op)}(e_{z,w})R(\delta)=R(\delta)\Delta(e_{z,w}),\leavevmode\nobreak\
\leavevmode\nobreak\
\Delta^{(op)}(f_{z,w};\lambda_{1},\lambda_{2})R(\lambda)=R(\lambda)\Delta(f_{z,w};\lambda_{1},\lambda_{2})$
(2.35)

where $\delta:=\lambda_{1}-\lambda_{2},$ and

$\displaystyle\Delta(e_{z,w})=\sum_{\xi,\zeta\in X}\big{(}e_{z,w}\otimes
e_{\xi,\zeta}+e_{\sigma_{z}(\xi),\sigma_{w}(\zeta)}\otimes
e_{\tau_{\xi}(z),\tau_{\zeta(w)}}\big{)}_{\sigma_{z}(\xi)=\sigma_{w}(\zeta)}$
(2.36)
$\displaystyle\Delta(f_{z,w};\lambda_{1},\lambda_{2})=\sum_{\xi,\zeta\in
X}\big{(}\lambda_{1}e_{z,w}\otimes
e_{\xi,\zeta}+\lambda_{2}e_{\sigma_{z}(\xi),\sigma_{w}(\zeta)}\otimes
e_{\tau_{\xi}(z),\tau_{\zeta(w)}}\big{)}_{\sigma_{z}(\xi)=\sigma_{w}(\zeta)}.$
$\displaystyle\qquad\qquad\qquad\quad{1\over 2}\sum_{y,\hat{y}\in
X}\big{(}e_{z,\sigma_{z}(y)}\otimes
e_{y,\hat{y}}|_{w=\sigma_{\sigma_{z}(y)}(\hat{y})}-e_{\sigma_{w}(y),w}\otimes
e_{\hat{y},y}|_{z=\sigma_{\sigma_{w}(y)}(\hat{y})}\big{)}$ (2.37)

Moreover, the matrix $\check{R}(\lambda)={\cal P}R(\lambda)$ is
$\mathfrak{gl}_{n}$ symmetric.

###### Proof.

The Yangian $R$-matrix satisfies relations (2.21), (2.19), then
$R(\delta)\Delta(e_{z,w})=\Delta^{(op)}(e_{z,w})R(\delta)$ and
$R(\delta)\Delta(f_{z,w};\lambda_{1},\lambda_{2})=\Delta^{(op)}(f_{z,w};\lambda_{1},\lambda_{2})R(\delta),$
where $R(\lambda)=\lambda r+{\cal P}$ and $r$ is the set-theoretic solution
$r=\sum_{x,y\in X}e_{y\sigma_{x}(y)}\otimes e_{x,\tau_{y}(x)}$, also

$\displaystyle\Delta(e_{z,w})=F^{-1}\big{(}e_{z,w}\otimes I+I\otimes
e_{z,w}\big{)}F,$
$\displaystyle\Delta(f_{z,w},\lambda_{1},\lambda_{2})=F^{-1}\big{(}\lambda_{1}e_{z,w}\otimes
I+\lambda_{2}I\otimes e_{z,w}+{1\over 2}\sum_{y\in X}(e_{z,y}\otimes
e_{y,w}-e_{y,w}\otimes e_{z,y})\big{)}F,$ (2.38)

where we recall $F=\sum_{\eta,x\in X}e_{x,x}\otimes{\mathbb{V}}_{x}.$ Explicit
computation of the latter leads to (2.37).

Also, $\Delta$ is an algebra homomorphism, hence $\Delta(e_{x,y})$ also
satisfy the $\mathfrak{gl}_{n}$ algebra relations, thus $\check{R}={\cal P}R$
is $\mathfrak{gl}_{n}$ symmetric ie. $\big{[}\check{R}(\lambda),\
\Delta(e_{x,y})\big{]}=0$. ∎

### The Lyubashenko solution

We finish this section with a discussion of a special set-theoretic solution
known as Lyubashenko’s solution that supports Conjecture 2.8. The analysis for
this special class of set-theoretic solutions is compatible with Remark 1.6
and Example 1.11.

The Lyubashenko solution is defined as

$\check{r}=\sum_{x,y\in X}e_{x,\sigma(y)}\otimes e_{y,\tau(x)},$ (2.39)

where $\tau,\ \sigma:X\to X$ are bijective functions, such that
$\sigma(\tau(x))=\tau(\sigma(x))=x.$ Let ${\mathbb{V}}=\sum_{x\in
X}e_{x,\tau(x)},$ then as was shown in [10] the special solution (2.39) can be
obtained from the permutation operator as
$\check{r}=(I\otimes{\mathbb{V}}^{-1}){\cal P}(I\otimes{\mathbb{V}}),$ which
leads to (2.39). Also, $r={\cal P}\check{r}$ and the Baxterized solution
$R(\lambda)$ have a simple form for this class of solutions:

$r={\mathbb{V}}^{-1}\otimes{\mathbb{V}}\ \Rightarrow\
R(\lambda)=\lambda{\mathbb{V}}^{-1}\otimes{\mathbb{V}}+{\cal P}.$ (2.40)

Some examples of the above construction are given below:

1\. $\sigma(x)=x+1,\ \tau(x)=x-1$, where addition is modulo $n$ (see also
[39]).

2\. $\sigma(x)=n+1-x,\ \tau(x)=n+1-x$.

In both examples $x\in\\{1,\ldots,n\\}.$

In the case of Lyubashenko solution, we consider Example 1.11 (special case),
where we set ${\cal A}$ to be the Yangian ${\cal Y}(\mathfrak{gl}_{n})$, i.e.
$\big{(}{\cal Y}(\mathfrak{gl}_{n}),\Delta,\epsilon,\Phi,{\cal R}\big{)}$
($\Phi=1\otimes 1\otimes u^{-1}$) is a quasi-triangular quasi-bialgebra. We
recall the evaluation representation $\pi_{\lambda}:{\cal
Y}(\mathfrak{gl}_{n})\to\mbox{End}({\mathbb{C}}^{n}),$ such that
$u\mapsto{\mathbb{V}},$ ${\mathrm{Q}}_{xy}^{(1)}\mapsto e_{x,y},$
${\mathrm{Q}}_{xy}^{(2)}\mapsto\lambda e_{x,y}$ etc. Then the twist becomes
$F=I\otimes{\mathbb{V}}$ and the twisted coproducts are

$\displaystyle\Delta(e_{z,w})=e_{z,w}\otimes I+I\otimes e_{\tau(z),\tau(w)},$
$\displaystyle\Delta(f_{x,y};\lambda_{1},\lambda_{2})=\lambda_{1}e_{z,w}\otimes
I+\lambda_{2}I\otimes e_{\tau(z),\tau(w)}\ +{1\over 2}\sum_{y\in
X}\big{(}e_{z,\sigma(y)}\otimes e_{y,\tau(w)}-e_{\sigma(y),w}\otimes
e_{\tau(z),y}\big{)}.$

Recall that in this case $F=I\otimes{\mathbb{V}}$, and $\Phi=I\otimes
I\otimes{\mathbb{V}}^{-1},$ then we have:

$\displaystyle
F_{1,23}:=(\mbox{id}\otimes\Delta)F=I\otimes{\mathbb{V}}\otimes{\mathbb{V}}$
$\displaystyle
F_{12,3}^{*}:=\big{(}(\Delta\otimes\mbox{id})F\big{)}\Phi^{-1}=I\otimes
I\otimes{\mathbb{V}}^{2},$

in accordance to the Conjecture 2.8. Also, the $N$-fold twist is
$F_{12...N}=I\otimes{\mathbb{V}}\otimes{\mathbb{V}}^{2}\otimes\ldots\otimes{\mathbb{V}}^{(N-1)}$
(see also relevant findings in [11]).

##  3\. Quasi-bialgebras from
${\mathfrak{U}}_{q}(\widehat{\mathfrak{gl}_{n}})$

We will discuss in this section the $q$-generalizations of set-theoretic
solutions. Although the situations we are going to address here strictly
speaking are not set-theoretic solutions of the Yang-Baxter equation they are
certainly inspired by the results of the preceding section. After a brief
review on ${\mathfrak{U}}_{q}(\widehat{\mathfrak{gl}_{n}})$ we will consider
the $q$-deformed analogues of the set-theoretic solutions via the twists
discussed in the previous section and in [11], subject to certain extra
constraints. The findings of this section greatly generalize the preliminary
results of [10].

###  3.1. The algebra ${\mathfrak{U}}_{q}(\widehat{\mathfrak{gl}_{n}})$

It will be useful in what follows to recall the basic definitions regarding
the algebra ${\mathfrak{U}}_{q}(\widehat{\mathfrak{gl}_{n}})$ [13, 27, 28].
Let

$a_{ij}=2\delta_{ij}-(\delta_{i\ j+1}+\delta_{i\ j-1}+\delta_{i1}\
\delta_{jn}+\delta_{in}\ \delta_{j1}),\leavevmode\nobreak\
\leavevmode\nobreak\ i,\ j\in\\{1,\ldots,n\\}$ (3.1)

be the Cartan matrix of the affine Lie algebra
${\widehat{\mathfrak{sl}_{n}}}$222For the $\widehat{\mathfrak{sl}_{2}}$ case
in particular  $a_{ij}=2\delta_{ij}-2(\delta_{i1}\ \delta_{j2}+\delta_{i2}\
\delta_{j1}),\leavevmode\nobreak\ \leavevmode\nobreak\ i,\ j\in\\{1,2\\}$
(3.2) . Also define:

$\displaystyle[m]_{q}={q^{m}-q^{-m}\over q-q^{-1}},\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ [m]_{q}!=\prod_{k=1}^{m}\
[k]_{q},\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
[0]_{q}!=1$ $\displaystyle\left[\begin{array}[]{c}m\\\ n\\\
\end{array}\right]_{q}={[m]_{q}!\over[n]_{q}!\
[m-n]_{q}!},\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
m>n>0.$ (3.5)

######  Definition 3.1.

The quantum affine enveloping algebra
${\mathfrak{U}}_{q}(\widehat{\mathfrak{sl}_{n}})$ has the Chevalley-Serre
generators $e_{i}$, $f_{i}$, $q^{\pm{h_{i}\over 2}}$, $i\in\\{1,\ldots,n\\}$
obeying the defining relations:

$\displaystyle\Big{[}q^{\pm{h_{i}\over 2}},\ q^{\pm{h_{j}\over
2}}\Big{]}=0\,\qquad q^{{h_{i}\over 2}}\ e_{j}=q^{{1\over 2}a_{ij}}e_{j}\
q^{{h_{i}\over 2}}\,\qquad q^{{h_{i}\over 2}}\ f_{j}=q^{-{1\over
2}a_{ij}}f_{j}\ q^{{h_{i}\over 2}},$ $\displaystyle\Big{[}e_{i},\
f_{j}\Big{]}=\delta_{ij}{q^{h_{i}}-q^{-h_{i}}\over
q-q^{-1}},\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ i,j\in\\{1,\ldots,n\\}$ (3.6)

and the $q$ deformed Serre relations

$\displaystyle\sum_{n=0}^{1-a_{ij}}(-1)^{n}\left[\begin{array}[]{c}1-a_{ij}\\\
n\\\ \end{array}\right]_{q}\chi_{i}^{1-a_{ij}-n}\ \chi_{j}\
\chi_{i}^{n}=0,\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\chi_{i}\in\\{e_{i},\ f_{i}\\},\leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ i\neq j.$ (3.9)

######  Remark 3.2.

The generators $e_{i}$, $f_{i}$, $q^{\pm h_{i}}$ for $i\in\\{1,\ldots,n\\}$
form the ${\mathfrak{U}}_{q}(\widehat{\mathfrak{sl}_{n}})$ algebra. Also,
$q^{\pm h_{i}}=q^{\pm(\varepsilon_{i}-\varepsilon_{i+1})}$,
$i\in\\{1,\ldots,n-1\\}$ and $q^{\pm
h_{n}}=q^{\pm(\varepsilon_{n}-\varepsilon_{1})},$ where the elements
$q^{\pm\varepsilon_{i}}$ belong to
${\mathfrak{U}}_{q}(\widehat{{\mathfrak{gl}}_{n}})$. Recall that
${\mathfrak{U}}_{q}(\widehat{{\mathfrak{gl}_{n}}})$ is obtained by adding to
${\mathfrak{U}}_{q}(\widehat{{\mathfrak{sl}_{n}}})$ the elements
$q^{\pm\varepsilon_{i}}$ $i\in\\{1,\ldots,n\\}$ so that
$q^{\sum_{i=1}^{n}\varepsilon_{i}}$ belongs to the center (for more details
see [27]).

We also note that
$\big{(}{\mathfrak{U}}_{q}(\widehat{\mathfrak{gl}_{n}}),\Delta_{q},\epsilon,S_{q},{\cal
R}_{q}\big{)}$ is a quasi-triangular Hopf algebra over ${\mathbb{C}}$ equipped
with [13]:

  * •

A coproduct $\Delta_{q}:\
{\mathfrak{U}}_{q}(\widehat{\mathfrak{gl}_{n}})\to{\mathfrak{U}}_{q}(\widehat{\mathfrak{gl}_{n}})\otimes{\mathfrak{U}}_{q}(\widehat{\mathfrak{gl}_{n}})$
such that

$\displaystyle\Delta_{q}(\xi_{i})=q^{-{h_{i}\over
2}}\otimes\xi_{i}+\xi_{i}\otimes q^{{h_{i}\over 2}},\leavevmode\nobreak\
\leavevmode\nobreak\ \xi_{i}\in\Big{\\{}e_{i},\ f_{i}\Big{\\}}$ (3.10)
$\displaystyle\Delta_{q}(q^{\pm{\varepsilon_{i}\over
2}})=q^{\pm{\varepsilon_{i}\over 2}}\otimes q^{\pm{\varepsilon_{i}\over
2}},\quad i\in\\{1,\ldots,n\\}.$ (3.11)

The $l$-fold co-product $\Delta_{q}^{(l)}:\
{\mathfrak{U}}_{q}(\widehat{\mathfrak{gl}_{n}})\to{\mathfrak{U}}_{q}(\widehat{\mathfrak{gl}_{n}})^{\otimes(l)}$
is defined as
$\Delta_{q}^{(l)}=(\mbox{id}\otimes\Delta_{q}^{(l-1)})\Delta_{q}=(\Delta_{q}^{(l-1)}\otimes\mbox{id})\Delta_{q}.$

  * •

A co-unit
$\epsilon:{\mathfrak{U}}_{q}(\widehat{\mathfrak{gl}_{n}})\to{\mathbb{C}}$ such
that

$\epsilon(e_{j})=\epsilon(f_{j})=0,\leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \epsilon(q^{\varepsilon_{j}})=1.$
(3.12)

  * •

An antipode
$S_{q}:{\mathfrak{U}}_{q}(\widehat{\mathfrak{gl}_{n}})\to{\mathfrak{U}}_{q}(\widehat{\mathfrak{gl}_{n}})$
such that

$S_{q}(q^{\varepsilon_{i}})=q^{-\varepsilon_{i}},\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ S_{q}(\xi_{i})=-q^{h_{i}\over
2}\xi_{i}q^{-h_{i}\over 2}.$ (3.13)

  * •

There exists an invertible element ${\cal
R}_{q}\in{\mathfrak{U}}_{q}(\widehat{\mathfrak{gl}_{n}})\otimes{\mathfrak{U}}_{q}(\widehat{\mathfrak{gl}_{n}}),$
such that it satisfies the axioms of Definition 1.4 for $\Phi=1\otimes
1\otimes 1.$

We recall now Example 1.11 and we briefly discuss the notion of the antipode
for ${\mathfrak{U}}_{q}(\widehat{\mathfrak{gl}_{n}}).$ We fist introduce a
shorthand notation for the unit element of
${\mathfrak{U}}_{q}(\widehat{\mathfrak{gl}_{n}})$:
$1_{q}:=1_{{\mathfrak{U}}_{q}(\widehat{\mathfrak{gl}_{n}})}.$

######  Remark 3.3.

We specialize Example 1.11 to the case of
${\mathfrak{U}}_{q}(\widehat{\mathfrak{gl}_{n}}),$ i.e.
$\big{(}{\mathfrak{U}}_{q}(\widehat{\mathfrak{gl}_{n}}),\Delta,\epsilon,\Phi,{\cal
R}\big{)},$ where $\Phi=1\otimes 1\otimes u^{-1}$ for some group-like element
$u$, is a quasi-triangular quasi-bialgebra. We now want to test whether
$\big{(}{\mathfrak{U}}_{q}(\widehat{\mathfrak{gl}_{n}}),\Delta,\epsilon,\Phi,{\cal
R}\big{)}$ is a quasi-Hopf algebra, i.e. if there exist $S$, $\alpha$, $\beta$
such that axioms (1) and (2) in Definition 2.2 hold. Suppose that the group-
like element $u$ exists, then $\Delta(u)=u\otimes u$ and:

  1. (1)

$S(u)\alpha u=\alpha$ and $u\beta S(u)=\beta,$ which lead to $S(u)=\alpha
u^{-1}\alpha^{-1}=\beta^{-1}u^{-1}\beta.$

  2. (2)

$\beta\alpha u^{-1}=1_{q}$ and $\alpha\beta S(u)=1_{q},$ which lead to
$u=\beta\alpha$ and $S(u)=\beta^{-1}\alpha^{-1}.$

All the above equations are self-consistent. Similarly, we can show that
$S(u^{-1})=\alpha\beta$

We now check the axioms of Definition 2.2 for the elements $e_{j},\ f_{j},\
q^{{\varepsilon_{j}}},$ with coproducts, after twisting, given as
$\Delta(\xi_{j})=q^{-{h_{j}\over 2}}\otimes u^{-1}\xi_{j}u+\xi_{j}\otimes
u^{-1}q^{{h_{j}\over 2}}u,$ $\leavevmode\nobreak\ \xi_{j}\in\\{e_{j},\
f_{j}\\}$ and $\Delta(q^{\varepsilon_{j}})=q^{\varepsilon_{j}}\otimes
u^{-1}q^{\varepsilon_{j}}u$:

  1. (1)

We consider $S(q^{\varepsilon_{j}})\alpha u^{-1}q^{\varepsilon_{j}}u=\alpha$
and $q^{\varepsilon_{j}}\beta S(u^{-1}q^{\varepsilon_{j}}u)=\beta,$ which lead
to (by requiring that $S$ is an algebra anti-homomorphism):
$S(q^{\varepsilon_{j}})=\alpha
q^{-\varepsilon_{j}}\alpha^{-1}=\beta^{-1}q^{-\varepsilon_{j}}\beta.$ The
latter expression leads to $\big{[}q^{\varepsilon_{j}},\ u\big{]}=0.$ Similar
conclusions hold for $S(\xi_{j}).$ Indeed, from axiom (1) of Definition 2.2 we
conclude that $S(\xi_{j})=-\alpha q^{h_{j}\over 2}\xi_{j}\ q^{-{h_{j}\over
2}}\alpha^{-1}=-\beta^{-1}q^{h_{j}\over 2}\xi_{j}\ q^{-{h_{j}\over 2}}\beta$,
which leads to $\big{[}\xi_{j},\ u\big{]}=0.$

As in the case of the Yangian studied in the previous section, the axioms for
the antipode restrict $u$ to be in the center of the algebra, which in general
is not true. Indeed, simple examples of such group-like elements that are not
central are given by $u=q^{\pm\varepsilon_{j}}$. Hence, we can not
consistently define $S(w),\
w\in{\mathfrak{U}}_{q}(\widehat{\mathfrak{gl}_{n}})$ for a generic $u$ by
strictly following the axioms of Definition 2.2.

It will be useful for the findings of the next subsection to recall the
evaluation representation of ${\mathfrak{U}}_{q}\widehat{(\mathfrak{gl}_{n}})$
[27] (homogeneous gradation),
$\pi_{\lambda}:{\mathfrak{U}}_{q}(\widehat{\mathfrak{gl}_{n}})\to\mbox{End}({\mathbb{C}}^{n}),$
$\lambda\in{\mathbb{C}}$:

$\displaystyle\pi_{\lambda}(e_{i})=e_{i,i+1},\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\
\pi_{\lambda}(f_{i})=e_{i+1,i},\leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \pi_{\lambda}(q^{{\varepsilon_{i}\over
2}})=q^{{e_{i,i}\over 2}},\leavevmode\nobreak\ \leavevmode\nobreak\
i\in\\{1,\ldots,n-1\\}$ (3.14)
$\displaystyle\pi_{\lambda}(e_{n})=e^{-2\lambda}e_{n,1},\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\
\pi_{\lambda}(f_{n})=e^{2\lambda}e_{1,n},\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\pi_{\lambda}(q^{{h_{n}\over 2}})=e^{{e_{n,n}-e_{1,1}\over 2}}$ (3.15)

and we also introduce some useful notation:

$\displaystyle(\pi_{\lambda_{1}}\otimes\pi_{\lambda_{2}})\Delta_{q}(e_{j})=\Delta_{q}(e_{j,j+1}),\leavevmode\nobreak\
\leavevmode\nobreak\
(\pi_{\lambda_{1}}\otimes\pi_{\lambda_{2}})\Delta_{q}(f_{j})=\Delta_{q}(e_{j+1,j}),$
(3.16) $\displaystyle j\in\\{1,\ldots,n-1\\},$ (3.17)
$\displaystyle(\pi_{\lambda_{1}}\otimes\pi_{\lambda_{2}})\Delta_{q}(e_{n})=\Delta_{q}(e_{n,1};\lambda_{1},\lambda_{2}),\leavevmode\nobreak\
\leavevmode\nobreak\
(\pi_{\lambda_{1}}\otimes\pi_{\lambda_{2}})\Delta_{q}(f_{n})=\Delta_{q}(e_{1,n};\lambda_{1},\lambda_{2})$
(3.18)
$\displaystyle(\pi_{\lambda_{1}}\otimes\pi_{\lambda_{1}})\Delta_{q}(q^{\epsilon_{j}})=\Delta_{q}(q^{e_{j,j}}),\leavevmode\nobreak\
j\in\\{1,\ldots,n\\}.$ (3.19)

We recall also the ${\mathfrak{U}}_{q}(\mathfrak{gl}_{n})$-invariant
representation of the $A$-type Hecke algebra [28]:

${\mathrm{g}}=\sum_{x\neq y\in X}\Big{(}e_{x,y}\otimes
e_{y,x}-q^{-sgn(x-y)}e_{x,x}\otimes e_{y,y}\Big{)}+qI.$ (3.20)

Indeed, the above element satisfies the braid relation $(I\otimes g)(g\otimes
I)(I\otimes g)=(g\otimes I)(I\otimes g)(g\otimes I)$ as well as the Hecke
constraint $(g-q)(g+q^{-1})=0.$ The Baxterized
${\mathfrak{U}}_{q}(\widehat{\mathfrak{gl}_{n}})$ solution of the Yang Baxter
equation is the
$R_{q}(\lambda)=e^{\lambda}{\mathrm{g}}^{+}-e^{-\lambda}{\mathrm{g}}^{-},$
where ${\mathrm{g}}^{+}={\cal P}{\mathrm{g}},$ ${\mathrm{g}}^{-}={\cal
P}{\mathrm{g}}^{-1}$ (${\cal P}$ the permutation operator). We also define
$\Delta_{q}^{(op)}(w;\lambda_{1},\lambda_{2}):={\cal
P}\Delta_{q}(w;\lambda_{2},\lambda_{1}){\cal P},$ the
${\mathfrak{U}}_{q}(\widehat{\mathfrak{gl}_{n}})$ $R_{q}$-matrix satisfies the
intertwining relations:

$\displaystyle
R_{q}(\lambda_{1}-\lambda_{2})\Delta_{q}(\zeta)=\Delta_{q}^{(op)}(\zeta)R_{q}(\lambda_{1}-\lambda_{2})\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\zeta\in\\{e_{j,j+1},e_{j+1,j},q^{e_{j,j}},q^{e_{n,n}}\\},$ (3.21)
$\displaystyle j\in\\{1,\ldots,n-1\\}$ (3.22) $\displaystyle
R_{q}(\lambda_{1}-\lambda_{2})\Delta_{q}(\zeta_{n};\lambda_{1},\lambda_{2})=\Delta_{q}^{(op)}(\zeta_{n};\lambda_{1},\lambda_{2})R_{q}(\lambda_{1}-\lambda_{2}),\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \zeta_{n}\in\\{e_{n,1},e_{1,n,}\\}.$
(3.23)

This brief review on ${\mathfrak{U}}_{q}(\widehat{\mathfrak{gl}_{n}})$ will be
particularly useful for the findings of the subsequent subsection.

###  3.2. The $q$-analogues of set-theoretic solutions of the YBE $\&$ quasi-
bialgebras

Inspired by the set-theoretic solutions and the associated twists [11], as
discussed in the previous section, we generalize in what follows results
regarding the twist of the ${\mathfrak{U}}(\widehat{\mathfrak{gl}_{n}})$
$R$-matrix. Note that strictly speaking this solution is not a set-theoretic
solution of the braid equation. Nevertheless, the admissible twists found for
the set-theoretic solutions can be still exploited to yield generalized
solutions based on (3.20).

We state below a basic Lemma that will lead to the main Proposition of this
section associated to admissible twists of the
${\mathfrak{U}}_{q}(\widehat{\mathfrak{gl}_{n}})$ $R$-matrix. This
construction provides the $q$-analogue of the $R$-matrices coming from set-
theoretic solutions of the YBE and greatly generalizes the preliminary results
of [10].

######  Lemma 3.4.

Let ${\mathbb{V}}_{\eta}=\sum_{x\in X}e_{\sigma_{\eta}(x),x},$ $\forall\eta\in
X$ consist a family of group-like elements, i.e.
$\Delta_{q}({\mathbb{V}}_{\eta})={\mathbb{V}}_{\eta}\otimes{\mathbb{V}}_{\eta},$
and let ${\mathrm{g}}$ be the ${\mathfrak{U}}({\mathfrak{gl}_{n}})$-invariant
element (3.20). Then $\big{[}{\mathrm{g}},\
\Delta_{q}({\mathbb{V}}_{\eta})\big{]}=0,$ subject to the constraint
$sgn(x-y)=sgn\big{(}\sigma_{\eta}(x)-\sigma_{\eta}(y)\big{)},$
$\forall\eta,x,y\in X.$

###### Proof.

It is convenient to re-express the element ${\mathrm{g}}$ as

${\mathrm{g}}={\cal P}-\sum_{x,y\in X}q^{-sgn(x-y)}e_{x,x}\otimes e_{y,y}+qI,$
(3.24)

where ${\cal P}$ is the permutation operator and $I$ is the $n\times n$
identity matrix. It is obvious that $\Delta_{q}({\mathbb{V}}_{\eta})$ commutes
with ${\cal P}$, so it suffices to show that $\Delta_{q}({\mathbb{V}}_{\eta})$
commutes with the second term of (3.24), indeed we compute

$\big{(}\sum_{x,y\in X}q^{-sgn(x-y)}e_{x,x}\otimes
e_{y,y}\big{)}\big{(}\sum_{\xi,\zeta\in X}e_{\sigma_{\eta}(\xi),\xi}\otimes
e_{\sigma_{\eta}(\zeta),\zeta}\big{)}=\sum_{\zeta,\xi\in
X}q^{-sgn\big{(}\sigma_{\eta}(\xi)-\sigma_{\eta}(\zeta)\big{)}}e_{\sigma_{\eta}(\xi),\xi}\otimes
e_{\sigma_{\eta}(\zeta),\zeta}$ (3.25) $\big{(}\sum_{\xi,\zeta\in
X}e_{\sigma_{\eta}(\xi),\xi}\otimes
e_{\sigma_{\eta}(\zeta),\zeta}\big{)}\big{(}\sum_{x,y\in
X}q^{-sgn(x-y)}e_{x,x}\otimes e_{y,y}\big{)}=\sum_{\zeta,\xi\in
X}q^{-sgn(\xi-\zeta)}e_{\sigma_{\eta}(\xi),\xi}\otimes
e_{\sigma_{\eta}(\zeta),\zeta}.$ (3.26)

Requiring expressions (3.25) and (3.26) to be equal we conclude that
$[g,\Delta_{q}(V_{\eta})]=0,$ $\forall\eta\in X$, if and only if
$\sigma_{\eta}$ is an order-preserving permutation of $X$, i.e.
$sgn(x-y)=sgn\big{(}\sigma_{\eta}(x)-\sigma_{\eta}(y)\big{)},$
$\forall\eta,x,y\in X.$ ∎

If $X$ is finite then the identity map is the only order-preserving
permutation. It is important however to note that the condition
$sgn(x-y)=sgn\big{(}\sigma_{\eta}(x)-\sigma_{\eta}(y)\big{)}$ possibly holds
when considering countably infinite sets. A characteristic example is
demonstrated at the end of the section, where the $q$-analogue of the
Lyubaschenko solution is presented. A further detailed analysis of this
construction and possible generalizations will be presented in future works.

We come now to the main proposition of this section.

######  Proposition 3.5.

Let ${\mathrm{g}}$ be the ${\mathfrak{U}}({\mathfrak{gl}_{n}})$-invariant
element (3.20) and $F=\sum_{\eta\in X}e_{\eta,\eta}\otimes{\mathbb{V}}_{\eta}$
(${\mathbb{V}}_{\eta}=\sum_{x\in X}e_{\sigma_{\eta}(x),x}$) be the set-
theoretic twist. Let also $F_{1,23}$ and $F_{12,3}^{*}$ be the quantities
defined in (2.31), (2.32). We also define ${\mathrm{G}}=F^{-1}{\mathrm{g}}F,$
then provided that $C_{1}=0,$ $C=0$
($C_{1}=\sigma_{\eta}(\sigma_{x}(y))-\sigma_{\sigma_{\eta}(x)}(\sigma_{\tau_{x}(\eta)}(y)),$
$C=sgn(x-y)-sgn\big{(}\sigma_{\eta}(x)-\sigma_{\eta}(y)\big{)}$):

  1. (1)

$\big{[}{\mathrm{G}}_{12},F_{12,3}^{*}\big{]}=\big{[}{\mathrm{G}}_{23},F_{1,23}\big{]}=0.$

  2. (2)

The twisted element ${\mathrm{G}}$ satisfies the braid relation and the Hecke
constraint (i.e. it provides a representation of the $A$-type Hecke algebra).

###### Proof.

We showed in Lemma 3.4 that $\big{[}{\mathrm{g}},\
\Delta_{q}({\mathbb{V}}_{\eta})\big{]}=0,$ subject to the constraint $C=0.$
This leads to $\big{[}{\mathrm{G}},\ \Delta({\mathbb{V}}_{\eta})\big{]}=0,$
$\forall\eta\in X,$ where ${\mathrm{G}}=F^{-1}{\mathrm{g}}F$ and
$\Delta({\mathbb{V}}_{\eta})=F^{-1}\Delta_{q}({\mathbb{V}}_{\eta})F,$ where
$\Delta_{q}({\mathbb{V}}_{\eta})$ is derived subject to $C_{1}=0$ (see Lemma
3.4).

  1. (1)

We recall that $F_{1,23}=(\mbox{id}\otimes\Delta)F,$ i.e.
$F_{1,23}=\sum_{\eta\in X}e_{\eta,\eta}\otimes\Delta({\mathbb{V}}_{\eta}),$
then using $\big{[}{\mathrm{G}},\ \Delta({\mathbb{V}}_{\eta})\big{]}=0$ we
obtain, $\big{[}{\mathrm{G}}_{23},F_{1,23}\big{]}=0,$ subject to $C_{1}=C=0.$

We show now that $\big{[}{\mathrm{G}}_{12},F_{12,3}^{*}\big{]}=0,$ to achieve
this we explicitly derive the form of ${\mathrm{G}}:$

$\displaystyle{\mathrm{G}}=F^{-1}{\mathrm{g}}F=\check{r}-\sum_{x,y\in
X}q^{-sgn\big{(}x-\sigma_{x}(y)\big{)}}e_{x,x}\otimes e_{y,y}+qI,$ (3.27)

where $\check{r}=\sum_{x,y\in X}e_{x,\sigma_{x}(y)}\otimes e_{y,\tau_{y}(x)}.$
We recall also the explicit expression $F^{*}_{12,3}=\sum_{\eta,x,y\in
X}e_{\eta,\eta}\otimes e_{x,x}\otimes
e_{\sigma_{\eta}(\sigma_{x}(y))}|_{C_{1}=0}$ and we compute:

$\displaystyle\leavevmode\nobreak\
{\mathrm{G}}_{12}F_{12,3}^{*}=F_{12,3}^{*}{\mathrm{G}}_{12}=$ (3.28)
$\displaystyle\sum_{x,y,z\in X}\big{(}e_{x,\sigma_{x}(y)}\otimes
e_{y,\tau_{y}(x)}-q^{-sgn\big{(}x-\sigma_{x}(y)\big{)}}e_{x,x}\otimes
e_{y,y}\big{)}\otimes e_{\sigma_{x}(\sigma_{y}(z)),z}+qF^{*}_{12,3}.$

  2. (2)

Due to the fact that: (1) $F$ is an admissible Drinfeld twist, (2)
$\big{[}{\mathrm{G}}_{12},F_{12,3}^{*}\big{]}=\big{[}{\mathrm{G}}_{23},F_{1,23}\big{]}=0,$
and (3) ${\mathrm{g}}$ satisfies the braid relation, we conclude that
${\mathrm{G}}$ also satisfies the braid equation. Similarly,
$({\mathrm{g}}-q)({\mathrm{g}}+q^{-1})=0\Rightarrow({\mathrm{G}}-q)({\mathrm{G}}+q^{-1})=0.$
∎

######  Remark 3.6.

The Baxterized solution
$\check{R}(\lambda)=F^{-1}\check{R}_{q}(\lambda)F=e^{\lambda}{\mathrm{G}}-e^{-\lambda}{\mathrm{G}}^{-1}$
(recall
$\check{R}_{q}(\lambda)=e^{\lambda}{\mathrm{g}}-e^{-\lambda}{\mathrm{g}}^{-1}$),
and hence $R(\lambda)={\cal
P}\check{R}(\lambda)=e^{\lambda}{\mathrm{G}}^{+}-e^{-\lambda}{\mathrm{G}}^{-},$
which is a solution of the Yang-Baxter equation. Moreover, by means of (3.23)
and the $R$-matrix twist we conclude that

$\displaystyle
R(\lambda_{1}-\lambda_{2})\Delta(\zeta)=\Delta^{(op)}(\zeta)R_{q}(\lambda_{1}-\lambda_{2})\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\zeta\in\\{e_{j,j+1},e_{j+1,j},q^{e_{j,j}},q^{e_{n,n}}\\},$ (3.29)
$\displaystyle j\in\\{1,\ldots n-1\\}$ (3.30) $\displaystyle
R(\lambda_{1}-\lambda_{2})\Delta(\zeta_{n};\lambda_{1},\lambda_{2})=\Delta^{(op)}(\zeta_{n};\lambda_{1},\lambda_{2})R(\lambda_{1}-\lambda_{2}),\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \zeta_{n}\in\\{e_{n,1},e_{1,n}\\},$
(3.31)

where $\Delta(a)=F^{-1}\Delta_{q}(a)F,$ $\leavevmode\nobreak\ \forall
a\in{\mathfrak{U}}(\widehat{\mathfrak{gl}_{n}}).$ From (3.29) and recalling
that $\check{R}={\cal P}R$ we deduce

$\displaystyle\check{R}(\lambda_{1}-\lambda_{2})\Delta(\zeta)=\Delta(\zeta)\check{R}(\lambda_{1}-\lambda_{2})\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\zeta\in\\{e_{j,j+1},e_{j+1,j},q^{e_{j,j}},q^{e_{n,n}}\\},$ (3.32)
$\displaystyle j\in\\{1,\ldots,n-1\\},$

i.e. the $\check{R}$-matrix is
${\mathfrak{U}}_{q}({\mathfrak{gl}_{n}})$-invariant as is the
$\check{R}_{q}$-matrix.

### The $q$-Lyubashenko solution

We focus in the end of this section on a special example, the $q$-deformed
analogue of Lyubashenko’s solution introduced in [10]. The analysis for this
special class of $q$ set-theoretic like solutions is compatible with Remark
1.6 and Example 1.11.

Let $\tau,\sigma:X\to X$ be isomorprhisms, such that
$\sigma(\tau(x))=\tau(\sigma(x))=x$ and let ${\mathbb{V}}=\sum_{x\in
X}e_{x,\tau(x)},$ then by direct computation it follows that,
$\big{[}{\mathbb{V}}\otimes{\mathbb{V}},\ {\mathrm{g}}\big{]}=0$ provided that
$sgn(x-y)=sgn\big{(}\sigma(x)-\sigma(y)\big{)}$. We then define
${\mathrm{G}}=(I\otimes{\mathbb{V}}^{-1})\ {\mathrm{g}}\
(I\otimes{\mathbb{V}}),$ which leads to

$\displaystyle{\mathrm{G}}=\sum_{x,y\in X}\Big{(}e_{x,\sigma(y)}\otimes
e_{y,\tau(x)}-q^{-sgn(x-\sigma(y))}e_{x,x}\otimes e_{y,y}\Big{)}+qI$ (3.33)

i.e. the element ${\mathrm{G}}$ is obtained from the
${\mathfrak{U}}_{q}(\mathfrak{gl}_{n})$-invariant braid solution (3.20),
provided that $sgn(x-y)=sgn\big{(}\sigma(x)-\sigma(y)\big{)}$ [10].

An example compatible with the construction above is: $\sigma(x)=x+1,\
\tau(x)=x-1$, $x\in{\mathbb{Z}}$ (see also [39]),

We recall Example 1.11, where we set ${\cal A}$ to be
${\mathfrak{U}}_{q}(\widehat{\mathfrak{gl}_{n}})$, i.e.
$\big{(}{\mathfrak{U}}_{q}(\widehat{\mathfrak{gl}_{n}}),\Delta,\epsilon,\Phi,{\cal
R}\big{)}$ ($\Phi=1\otimes 1\otimes u^{-1}$) is a quasi-triangular quasi-
bialgebra. We recall the evaluation representation
$\pi_{\lambda}:{\mathfrak{U}}_{q}(\widehat{\mathfrak{gl}_{n}})\to\mbox{End}({\mathbb{C}}^{n}),$
such that $u\mapsto{\mathbb{V}},$ $e_{j}\mapsto e_{i,j+1},$ $f_{j}\mapsto
e_{j+1,j}$ and $q^{\varepsilon_{j}}\mapsto^{e_{j,j}}.$ Then the twist becomes
$F=I\otimes{\mathbb{V}}$ and the twisted coproducts are

$\displaystyle\Delta(q^{e_{i,i}})=q^{e_{i,i}}\otimes
q^{e_{\tau(i),\tau(i)}},\leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \Delta(\xi_{j})=\xi_{j}\otimes
q^{{h_{\tau(j)}\over 2}}+q^{-{h_{j}\over 2}}\otimes\xi_{\tau(j)}.$ (3.34)

$h_{j}=\big{(}e_{j,j}-e_{j+1,j+1}\big{)}$,
$h_{\tau(j)}=\big{(}e_{\tau(j),\tau(j)}-e_{\tau(j+1),\tau(j+1)}\big{)}$, for
$\xi_{j}\in\Big{\\{}e_{j,j+1},\ e_{j+1,j}\Big{\\}}$, we define respectively:
$\xi_{\tau(j)}\in\Big{\\{}e_{\tau(j),\tau(j+1)},\
e_{\tau(j+1),\tau(j)}\Big{\\}}$. We also recall that ${\mathrm{g}}$ (3.20) is
$\mathfrak{U}_{q}(\mathfrak{gl}_{n})$-invariant, i.e. $\big{[}{\mathrm{g}},\
\Delta_{q}(Y)\big{]}=0,$ $\leavevmode\nobreak\ Y\in\Big{\\{}e_{j,j+1},\
e_{j+1,j},\ q^{e_{j,j}}\Big{\\}}$ and the co-products $\Delta_{q}$ of the
algebra elements are given in (3.10), (3.11) and (3.19). Then it follows that
the element ${\mathrm{G}}$ (3.33) is also
${\mathfrak{U}}_{q}(\mathfrak{gl}_{n})$ symmetric [10], i.e.
$\big{[}{\mathrm{G}},\ \Delta(Y)\big{]}=0,$ where the modified co-products are
given in (3.34). Naturally the Baxterized $\check{R}$-matrix is also
$\mathfrak{U}_{q}(\mathfrak{gl}_{n})$-invariant (3.32), (3.34).

As in the case of the Yangian the admissible twist is $F=I\otimes{\mathbb{V}}$
and $\Phi=I\otimes I\otimes{\mathbb{V}}^{-1},$ then we have:
$F_{1,.23}=(\mbox{id}\otimes\Delta)F=I\otimes{\mathbb{V}}\otimes{\mathbb{V}}$
and $F_{12,3}^{*}=\big{(}(\Delta\otimes\mbox{id})F\big{)}\Phi^{-1}=I\otimes
I\otimes{\mathbb{V}}^{2},$ in accordance to the Conjecture 2.8. Also, the
$N$-fold twist is
$F_{12...N}=I\otimes{\mathbb{V}}\otimes{\mathbb{V}}^{2}\otimes\ldots\otimes{\mathbb{V}}^{(N-1)},$
(see also [11]).

### Acknowledgments

We are grateful to A. Smoktunowicz and B. Rybolowicz for useful discussions.
Support from the EPSRC research grants EP/V008129/1 and EP/R009465/1 is
acknowledged. A.G. acknowledges support from Heriot-Watt University via a
James Watt scholarship.

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