# Freidel-Maillet type presentations of $U_{q}(sl_{2})$

Pascal Baseilhac  Institut Denis-Poisson CNRS/UMR 7013 - Université de Tours -
Université d’Orléans Parc de Grammont, 37200 Tours, FRANCE
<EMAIL_ADDRESS>
###### Abstract.

A unified framework for the Chevalley and equitable presentation of
$U_{q}(sl_{2})$ is introduced. It is given in terms of a system of Freidel-
Maillet type equations satisfied by a pair of quantum K-operators
${\cal{K}}^{\pm}$, whose entries are expressed in terms of either Chevalley or
equitable generators. The Hopf algebra structure is reconsidered in light of
this presentation, and interwining relations for K-operators are obtained. A
K-operator solving a spectral parameter dependent Freidel-Maillet equation is
also considered. Specializations to $U_{q}(sl_{2})$ admit a decomposition in
terms of ${\cal{K}}^{\pm}$. Explicit examples of K-matrices are constructed.

MSC: 16T25; 17B37; 81R50.

Keywords: $U_{q}(sl_{2})$; Equitable presentation; FRT presentation; Freidel-
Maillet algebras

##  1\. Introduction

The Faddeev-Reshetikhin-Takhtajan (FRT) presentation of quantum algebras
[FRT87, FRT89] has played a central role in the development of integrable
quantum field theory and statistical mechanics, providing a powerful framework
for studying the properties of the system under consideration (spectrum,
scattering data, correlation functions,…). For any quantum algebra $U_{q}(g)$,
given a R-matrix satisfying the Yang-Baxter equation

(1.1) $\displaystyle R_{12}R_{13}R_{23}=R_{23}R_{13}R_{12}\ ,$

where the standard notation $R_{ij}\in\mathrm{End}({\cal V}_{i}\otimes{\cal
V}_{j})$ is used, the FRT presentation gives a matrix realization of
$U_{q}(g)$. It consists of a pair of quantum L-operators with entries
$({\cal{L}}^{\pm}_{i,j})_{1\leq i,j\leq n}\in U_{q}(g)$ obeying a system of
exchange relations called the Yang-Baxter algebra. The defining relations take
the form:

(1.2) $\displaystyle{\cal{L}}^{\pm}_{i,i}{\cal{L}}^{\mp}_{i,i}=1\ ,\qquad
R{\cal{L}}_{1}^{\pm}{\cal{L}}_{2}^{\pm}={\cal{L}}_{2}^{\pm}{\cal{L}}_{1}^{\pm}R\
,\qquad
R{\cal{L}}_{1}^{+}{\cal{L}}_{2}^{-}={\cal{L}}_{2}^{-}{\cal{L}}_{1}^{+}R\ ,$

where the shorthand notation ${\cal{L}}_{1}={\cal{L}}\otimes{\mathbb{I}}$,
${\cal{L}}_{2}={\mathbb{I}}\otimes{\cal{L}}$ is used. The extension of (1.2)
to any affine Kac-Moody algebra $U_{q}(\widehat{g})$ has been extensively
studied, starting from [RS90, DF93], until the recent achievements [JLM19,
JLM20, LP21]. In this case, the corresponding Yang-Baxter equation (1.1)
depends on an indeterminate $u$, called the spectral parameter in the
mathematical physics literature. In this latter context, corresponding
L-operators and R-matrix solutions are the key ingredients for the
construction of transfer matrices that provide generating functions for
mutually commuting quantities in quantum integrable systems. Furthermore,
powerful standard techniques such as the Bethe ansatz and separation of
variables essentially depend on the presentation (1.2). For a recent review,
see e.g. [Sl18]. Solutions of (1.1) also arise in the context of knot theory
and quotients of the braid group, see e.g. [CP94, Chapter 15].

In all these works, it is important to point out that besides the presentation
(1.2), the quantum algebra $U_{q}(g)$ always arises in the so-called Chevalley
(or Drinfeld-Jimbo) presentation with generators $\\{E_{i},F_{i},K^{\pm
1}_{i},i=1,...,n\\}$ [J85, D86]. However, in recent years a different
presentation for $U_{q}(g)$ has been introduced, called equitable, with
generators $\\{X^{\pm 1}_{i},Y_{i},Z_{i},i=1,...,n\\}$ [ITW05, T05]. Since
[ITW05], the equitable presentation of $U_{q}(sl_{2})$ has been studied from
various perspectives: algebra [T11, T15b, GVZ15, FJ16, LXJ19], combinatorics
and representation theory [A11, MR12, Fu13, HG13, IRT13, BCT14, T14, Y15, N15,
T15a, SG16, T20] for instance. Also, note that irreducible finite dimensional
representations have been studied in details in [ITW05, T09]. In addition, let
us mention that the subalgebra generated by $\\{Y,Z\\}$ of the equitable
$U_{q}(sl_{2})$ is isomorphic to the non-homogeneous Borel subalgebra
considered in [LV17, V18, LV19]. Importantly, contrary to the Chevalley
presentation which fits with the FRT framework, up to now a similar set up for
the equitable presentation has remained unknown.

In this paper, a unified framework for the Chevalley and equitable
presentation of $U_{q}(sl_{2})$ is proposed. The basic data consists of two
R-matrices $R$, $R^{(0)}$, and a pair of K-operators with
$({\cal{K}}^{\pm}_{i,j})_{1\leq i,j\leq 2}\in U_{q}(sl_{2})$ \- written in
terms of the Chevalley or equitable generators - satisfying a system of
Freidel-Maillet type equations111See also [NC92, Bab92, KS92]. Note that
equations of the form (1.3) arise naturally in the context of braided Yang-
Baxter algebras. See e.g. [P94]. [FM91]. This unified presentation of
$U_{q}(sl_{2})$ takes the form (see Theorem 3.5):

(1.3) $\displaystyle({\cal{K}}^{\pm}_{1,1})^{\pm
1}({\cal{K}}^{\pm}_{1,1})^{\mp 1}=1\ ,\qquad
R{\cal{K}}_{1}^{\pm}R^{(0)}{\cal{K}}_{2}^{\pm}={\cal{K}}_{2}^{\pm}R^{(0)}{\cal{K}}_{1}^{\pm}R\
,\qquad
R{\cal{K}}_{1}^{+}R^{(0)}{\cal{K}}_{2}^{-}={\cal{K}}_{2}^{-}R^{(0)}{\cal{K}}_{1}^{+}R\
.$

The Hopf algebra structure is investigated in light of this presentation, and
an interpretation of K-operators as intertwiners of certain
$U_{q}(sl_{2})$-modules is discussed. Also, it is found that the simplest
example of non-homogeneous Borel subalgebra of $U_{q}(sl_{2})$ considered in
[LV17, V18, LV19] fits in this framework. Some consequences for the spectral
parameter dependent Freidel-Maillet equation (4.16) and corresponding
K-operator solutions are studied.

The text is organized as follows. Section 2 collects the basic material for
our purpose. The quantum algebra $U_{q}(gl_{2})$ is recalled, as well as the
Chevalley and equitable presentations of $U_{q}(sl_{2})$. Section 3 is devoted
to the Freidel-Maillet type presentation of $U_{q}(sl_{2})$. It is naturally
suggested from a detailed analysis of K-operator solutions built from
$U_{q}(gl_{2})$ L-operators introduced at the begining of the same section.
The Hopf algebra structure is discussed, and a set of intertwining relations
satisfied by the K-operators are obtained. A Freidel-Maillet type presentation
for the non-homogeneous Borel subalgebra of $U_{q}(sl_{2})$ follows. In
Section 4, the K-operator (4.15) solution of the spectral parameter dependent
Freidel-Maillet equation (4.16) recently obtained in [B20] is considered in
light of (1.3). This K-operator characterizes a quotient of the alternating
presentation for $U_{q}(\widehat{sl_{2}})$ introduced in [T18]. It is shown
that the class of solutions generated from ${\cal{K}}^{\pm}$ follow from
specializations to $U_{q}(sl_{2})$ of this K-operator. For the simplest
irreducible finite dimensional representations in the Chevalley or equitable
presentation of $U_{q}(sl_{2})$, explicit examples of spectral parameter
dependent K-matrices are obtained. They satisfy a Freidel-Maillet equation of
the form:

(1.4) $\displaystyle
R_{12}{\cal{K}}_{13}R^{(0)}_{12}{\cal{K}}_{23}={\cal{K}}_{23}R^{(0)}_{12}{\cal{K}}_{13}R_{12}\
,$

and naturally generate quantum integrable models associated with
$U_{q}(sl_{2})$. For the Chevalley case, some examples have already been
studied in the literature, see e.g. [H94, Ku95, FR02].

Let us briefly comment on some perspectives and related subjects. Besides
potential applications to quantum integrable systems, it is expected that the
presentation of Freidel-Maillet type for $U_{q}(sl_{2})$ (1.3) generalizes to
any higher rank Lie algebra, thus prodiving a unified framework for the
Chevalley and equitable presentation of $U_{q}(g)$. Also, let us point out
that the equitable presentation of $U_{q}(\widehat{sl_{2}})$ is known [IT07].
So, it is natural to ask for the corresponding Freidel-Maillet type
presentation of $U_{q}(\widehat{sl_{2}})$. This is closely related with the
subject of [B20, B21]. Note that the present work is also motivated by the
theory of quantum symmetric pairs (see e.g. [Le99, MRS03, Ko12, BW13, LV19]),
reflection algebras [Sk88, KS92, DKM03, Ko08, KoS08, BalKo15, RV16, Ko17,
AV20], the q-Onsager algebra [T99, B04] and its alternating central extension,
see [T21] and references therein. As a last comment, for a Hopf algebra recall
that the quasi-cocommutativity property is characterized by the universal
$R$-matrix and the concept of quasi-triangularity [D86]. In view of the
results of subsection 3.4 it seems desirable to investigate the universal
K-matrix of equitable type associated with a universal version of (1.4), and
clarify its interpretation with respect to $U_{q}(sl_{2})$-modules. Some of
these problems will be considered elsewhere.

Notation 1.1. Let ${\mathbb{K}}$ denote an algebraically closed field of
characteristic $0$. ${\mathbb{K}}(q)$ denotes the field of rational functions
in an indeterminate $q$. The $q$-commutator
$\big{[}X,Y\big{]}_{q}=qXY-q^{-1}YX$ is introduced. We denote
$[X,Y]=[X,Y]_{q=1}$. We also denote the $q$-number
$[n]_{q}=(q^{n}-q^{-n})/(q-q^{-1})$.

##  2\. The quantum algebras $U_{q}(gl_{2})$ and $U_{q}(sl_{2})$

In this section, we recall the definitions and basic properties of the
Chevalley presentations of $U_{q}(gl_{2})$ and $U_{q}(sl_{2})$, as well as the
equitable presentation of $U_{q}(sl_{2})$ recently introduced in [ITW05]. For
each presentation, the comultiplication, counit and antipode that ensure the
Hopf algebra structure are given. Central elements are also displayed. For
further discussion, irreducible finite dimensional representations [CP94,
ITW05, T09] are recalled.

###  2.1. The Chevalley presentation of $U_{q}(gl_{2})$

We refer the reader to [J85].

######  Definition 2.1.

$U_{q}(gl_{2})$ is the unital associative ${\mathbb{K}}(q)$-algebra with
Chevalley generators $E,F,K^{\pm 1}_{1},K_{2}^{\pm 1}$ satisfying the
following relations:

(2.1) $\displaystyle K_{i}K_{i}^{-1}$ $\displaystyle=$ $\displaystyle
K_{i}^{-1}K_{i}=1\ ,\quad i=1,2\ ,$ (2.2) $\displaystyle K_{i}K_{j}$
$\displaystyle=$ $\displaystyle K_{j}K_{i}\ ,$ (2.3) $\displaystyle
K_{1}EK_{1}^{-1}$ $\displaystyle=$ $\displaystyle qE,\qquad\quad
K_{1}FK_{1}^{-1}=q^{-1}F\ ,$ (2.4) $\displaystyle K_{2}EK_{2}^{-1}$
$\displaystyle=$ $\displaystyle q^{-1}E,\qquad K_{2}FK_{2}^{-1}=qF\ ,$ (2.5)
$\displaystyle\big{[}E,F\big{]}$ $\displaystyle=$
$\displaystyle\frac{K_{1}K_{2}^{-1}-K_{1}^{-1}K_{2}}{q-q^{-1}}\ .$

In the text, the multiplication and unit map of $U_{q}(gl_{2})$ will be
denoted $\mu:U_{q}(gl_{2})\otimes U_{q}(gl_{2})\rightarrow U_{q}(gl_{2})$ and
$\eta:{\mathbb{K}}\rightarrow U_{q}(gl_{2})$, respectively.

The quantum algebra $U_{q}(gl_{2})$ is endowed with the following Hopf algebra
structure. As a coalgebra, it is equipped with the comultiplication
$\Delta:U_{q}(gl_{2})\rightarrow U_{q}(gl_{2})\otimes U_{q}(gl_{2})$ such that

(2.6) $\displaystyle\Delta(E)=E\otimes 1+K_{1}K_{2}^{-1}\otimes E\
,\quad\qquad\Delta(F)=F\otimes K_{1}^{-1}K_{2}+1\otimes F\
,\qquad\quad\Delta(K_{i})=K_{i}\otimes K_{i}\ ,$

and the counit $\varepsilon:U_{q}(gl_{2})\rightarrow{\mathbb{K}}$ such that

(2.7) $\displaystyle\varepsilon(E)=0\ ,\qquad\qquad\varepsilon(F)=0\
,\qquad\qquad\varepsilon(K_{i})=1\ .$

As a Hopf algebra, in addition it is equipped with the antipode
$S:U_{q}(gl_{2})\rightarrow U_{q}(gl_{2})$ such that

(2.8) $\displaystyle S(E)=-K_{1}^{-1}K_{2}E\ ,\qquad\qquad
S(F)=-FK_{1}K_{2}^{-1}\ ,\qquad\qquad S(K_{i})=K_{i}^{-1}\ .$

The center of $U_{q}(gl_{2})$ is generated by two central elements, given by:

(2.9) $\displaystyle\Omega_{1,c}$ $\displaystyle=$ $\displaystyle K_{1}K_{2}\
,$ (2.10) $\displaystyle\Omega_{2,c}$ $\displaystyle=$
$\displaystyle\frac{q^{-1}K_{1}K_{2}^{-1}+qK_{1}^{-1}K_{2}}{(q-q^{-1})^{2}}+EF=\frac{qK_{1}K_{2}^{-1}+q^{-1}K_{1}^{-1}K_{2}}{(q-q^{-1})^{2}}+FE.$

###  2.2. The Chevalley and equitable presentations of $U_{q}(sl_{2})$

Two different presentations of the quantum algebra $U_{q}(sl_{2})$ in terms of
generators and relations have been introduced in the literature. The Chevalley
presentation of $U_{q}(sl_{2})$ is first recalled [KR83, Sk85]. For the
notations, see e.g. [Ja96, p. 9].

######  Definition 2.2.

$U_{q}(sl_{2})$ is the unital associative ${\mathbb{K}}(q)$-algebra with
Chevalley generators $E,F,K,K^{-1}$ satisfying the following relations:

(2.11) $\displaystyle KK^{-1}$ $\displaystyle=$ $\displaystyle K^{-1}K=1,$
(2.12) $\displaystyle KEK^{-1}$ $\displaystyle=$ $\displaystyle q^{2}E,\qquad
KFK^{-1}=q^{-2}F,$ (2.13) $\displaystyle\big{[}E,F\big{]}$ $\displaystyle=$
$\displaystyle\frac{K-K^{-1}}{q-q^{-1}}.$

The quantum algebra $U_{q}(sl_{2})$ has the following Hopf algebra structure.
The comultiplication $\Delta:U_{q}(sl_{2})\rightarrow U_{q}(sl_{2})\otimes
U_{q}(sl_{2})$ is such that

(2.14) $\displaystyle\Delta(E)=E\otimes 1+K\otimes E\ ,\quad\Delta(F)=F\otimes
K^{-1}+1\otimes F\ ,\quad\Delta(K)=K\otimes K\ ,$

The counit $\varepsilon:U_{q}(sl_{2})\rightarrow{\mathbb{K}}(q)$ is such that

(2.15) $\displaystyle\varepsilon(E)=0\ ,\qquad\qquad\varepsilon(F)=0\
,\qquad\qquad\varepsilon(K)=1\ $

and the antipode $S:U_{q}(sl_{2})\rightarrow U_{q}(sl_{2})$ is such that

(2.16) $\displaystyle S(E)=-K^{-1}E\ ,\qquad\qquad S(F)=-FK\ ,\qquad\qquad
S(K)=K^{-1}\ .$

Also, note the automorphism:

(2.17) $\displaystyle\theta:\qquad E\leftrightarrow F\ ,\qquad
K\leftrightarrow K^{-1}\ .$

The central element of $U_{q}(sl_{2})$ is the so-called Casimir operator,
given by:

(2.18)
$\displaystyle\Omega_{c}=\frac{q^{-1}K+qK^{-1}}{(q-q^{-1})^{2}}+EF=\frac{qK+q^{-1}K^{-1}}{(q-q^{-1})^{2}}+FE.$

Note that in the Chevalley presentation, $U_{q}(sl_{2})$ is the quotient of
$U_{q}(gl_{2})$ by the relation $K_{1}K_{2}=1$ (see (2.9)). It is is generated
by

(2.19) $\displaystyle E\ ,\qquad F\ ,\qquad K=K_{1}K_{2}^{-1}\ ,\qquad
K^{-1}=K_{1}^{-1}K_{2}\ .$

In further sections, we will consider examples of irreducible finite
dimensional representations of $U_{q}(sl_{2})$. For the Chevalley
presentation, see e.g. [CP94]. Choose ${\mathbb{K}}={\mathbb{C}}$. Define the
vector space $V^{(s)}$ of dimension $2s+1$ with basis
$\\{v_{0},...,v_{2s}\\}$. In this basis, the Chevalley generators act as
matrices with non-vanishing entries $(i,j)$, $0\leq i,j\leq 2s$, such that

$\displaystyle(\rho_{V^{(s)}}(K))_{i,i}=q^{2s-2i}\
,\qquad(\rho_{V^{(s)}}(E))_{i-1,i}=[2s-i+1]_{q}\
,\qquad(\rho_{V^{(s)}}(F))_{i,i-1}=[i]_{q}\ .$

For instance, for the spin$-1/2$ representation of $U_{q}(sl_{2})$ one has:

(2.20) $\displaystyle\rho_{V^{(1/2)}}(K)=\begin{pmatrix}q&0\\\
0&q^{-1}\end{pmatrix}\ ,\qquad\rho_{V^{(1/2)}}(E)=\begin{pmatrix}0&1\\\
0&0\end{pmatrix}\ ,\qquad\rho_{V^{(1/2)}}(F)=\begin{pmatrix}0&0\\\
1&0\end{pmatrix}\ .$

More recently, the equitable presentation of the quantum algebra
$U_{q}(sl_{2})$ has been introduced.

######  Theorem 2.3.

[ITW05, Theorem 2.1] The algebra $U_{q}(sl_{2})$ is isomorphic to the unital
associative ${\mathbb{K}}(q)$-algebra with equitable generators $X$, $X^{-1}$,
$Y$, $Z$ satisfying the following relations:

(2.21) $\displaystyle XX^{-1}=X^{-1}X$ $\displaystyle=$ $\displaystyle 1\ ,$
(2.22) $\displaystyle\frac{qXY-q^{-1}YX}{q-q^{-1}}$ $\displaystyle=$
$\displaystyle 1\ ,\qquad\frac{qYZ-q^{-1}ZY}{q-q^{-1}}=1\
,\qquad\frac{qZX-q^{-1}XZ}{q-q^{-1}}=1\ .$

The isomorphism with the Chevalley presentation of Definition 2.2 is given by:

(2.23) $\displaystyle\phi:\quad X^{{\pm}1}$ $\displaystyle\rightarrow$
$\displaystyle K^{{\pm}1}\ ,\qquad Y\rightarrow K^{-1}+(q-q^{-1})F\ ,\qquad
Z\rightarrow K^{-1}-q(q-q^{-1})K^{-1}E\ .$

Using the automorphism $\theta$, one gets another isomorphism:

(2.24) $\displaystyle X^{{\pm}1}$ $\displaystyle\rightarrow$ $\displaystyle
K^{\mp 1}\ ,\qquad Y\rightarrow K+(q-q^{-1})E\ ,\qquad Z\rightarrow
K-q(q-q^{-1})KF\ .$

Note that the above isomorphisms can be generalized combining automorphisms of
the Chevalley presentation of $U_{q}(sl_{2})$. For instance [ITW05, Lemma
2.3]:

(2.25) $\displaystyle E\rightarrow aEK^{\ell}\ ,\qquad F\rightarrow
a^{-1}K^{-\ell}F\ ,\qquad K\rightarrow K\ ,\vspace{1mm}$

for any integer $\ell$ and $a\in{\mathbb{K}}$.

The Hopf algebra structure is ensured by the comultiplication $\Delta$ such
that

(2.26) $\displaystyle\Delta(X)=X\otimes X\ ,\qquad\Delta(Y)=(Y-1)\otimes
X^{-1}+1\otimes Y\ ,\qquad\Delta(Z)=(Z-1)\otimes X^{-1}+1\otimes Z\ .$

The counit $\varepsilon$ is such that:

(2.27) $\displaystyle\varepsilon(X)=1\ ,\qquad\qquad\varepsilon(Y)=1\
,\qquad\qquad\varepsilon(Z)=1$

and the antipode $S$ is such that:

(2.28) $\displaystyle S(X)=X^{-1}\ ,\qquad\qquad S(Y)=1+X-YX\ ,\qquad\qquad
S(Z)=1+X-ZX\ .$

Note that the subalgebra $\cal B$ generated by $\\{Y,Z\\}$ gives an example of
non-homogeneous Borel subalgebra, see [LV19, Example 3.6]. There is no analog
of $\cal B$ in $sl_{2}$ (specialization $q\rightarrow 1$). As can be seen from
(2.26), $\cal B$ is a right coideal subalgebra of $U_{q}(sl_{2})$:
$\Delta(\cal B)\subset\cal B\otimes U_{q}(sl_{2})$. On this subject, see
[HKo11, LV17, V18, LV19].

In the equitable presentation of $U_{q}(sl_{2})$, the central element is given
by:

(2.29) $\displaystyle\Omega_{e}=qX+q^{-1}Y+qZ-qXYZ.$

Several equivalent expressions can be derived using (2.22). Note that using
the inverse of (2.23), one finds
$\Omega_{c}\rightarrow\Omega_{e}/(q-q^{-1})^{2}$.

For the equitable presentation, irreducible finite dimensional representations
have been studied in details in [ITW05, T09]. Adapting the notations of [T09]
(see Theorem 10.12), choose ${\mathbb{K}}={\mathbb{C}}$ and consider the
vector space $V^{(s)}$. Among the twelve known bases, choose the one denoted
$\\{u_{0},...,u_{2s}\\}$ of type ‘$[y]_{col}$’ in [T09, Theorem 10.12]. In
this basis, the equitable generators act as matrices with non-vanishing
entries $(i,j)$, $0\leq i,j\leq 2s$ such that

$\displaystyle(\rho_{V^{(s)}}(X))_{i,i}$ $\displaystyle=$ $\displaystyle
q^{2s-2i}\ ,$ $\displaystyle(\rho_{V^{(s)}}(Y))_{i,i}$ $\displaystyle=$
$\displaystyle q^{2i-2s}\
,\qquad(\rho_{V^{(s)}}(Y))_{i,i-1}=q^{2s}-q^{2i-2-2s}\ ,$
$\displaystyle(\rho_{V^{(s)}}(Z))_{i,i}$ $\displaystyle=$ $\displaystyle
q^{2i-2s}\ ,\qquad(\rho_{V^{(s)}}(Z))_{i-1,i}=q^{-2s}-q^{2i-2s}\ .$

For instance, for the spin$-1/2$ representation of $U_{q}(sl_{2})$ one has:

(2.30) $\displaystyle\rho_{V^{(1/2)}}(X)=\begin{pmatrix}q&0\\\
0&q^{-1}\end{pmatrix}\ ,\qquad\rho_{V^{(1/2)}}(Y)=\begin{pmatrix}q^{-1}&0\\\
q-q^{-1}&q\end{pmatrix}\
,\qquad\rho_{V^{(1/2)}}(Z)=\begin{pmatrix}q^{-1}&q^{-1}-q\\\ 0&q\end{pmatrix}\
.$

##  3\. Freidel-Maillet type presentations for $U_{q}(sl_{2})$

In this section, we construct quantum K-operators that satisfy a system of
Freidel-Maillet equations associated with $U_{q}(gl_{2})$. Then, a
presentation of Freidel-Maillet type for $U_{q}(sl_{2})$ in terms of
K-operators of Chevalley or equitable type is proposed. The comultiplication,
counit and antipode of $U_{q}(sl_{2})$ are identified, and central elements
are given. In particular, the non-homogeneous Borel subalgebra of
$U_{q}(sl_{2})$ fits in this framework. A set of intertwining relations
satisfied by the K-operators is obtained. Some basic examples of constant
K-matrices of Chevalley and equitable type are also displayed.

As a preliminary, we first recall the well-known FRT presentation of
$U_{q}(gl_{2})$ that will be used to construct a one-parameter family of
K-operators whose entries are in a subalgebra of $U_{q}(gl_{2})$.

###  3.1. The FRT presentation of $U_{q}(gl_{2})$

We refer the reader to [FRT87, FRT89, DF93] for details. Define the quantum
R-matrix

(3.5) $\displaystyle R=\left(\begin{array}[]{cccc}q&0&0&0\\\ 0&1&q-q^{-1}&0\\\
0&0&1&0\\\ 0&0&0&q\end{array}\right)\ $

with deformation parameter $q$. It is known that $R$ satisfies the quantum
Yang-Baxter equation (1.1) in the space ${\cal V}_{1}\otimes{\cal
V}_{2}\otimes{\cal V}_{3}$. In addition, $R$ satisfies
$R-PR^{-1}P=(q-q^{-1})P$, where $P$ is the permutation operator
$P(v_{1}\otimes v_{2})=v_{2}\otimes v_{1}$, $v_{1},v_{2}\in{\mathbb{C}}^{2}$.
Note that $R^{-1}=R_{q\leftrightarrow q^{-1}}$ and
$R^{t_{1}t_{2}}=PRP=R_{21}$. Recall the notation
${\cal{L}}_{1}={\cal{L}}\otimes{\mathbb{I}}$,
${\cal{L}}_{2}={\mathbb{I}}\otimes{\cal{L}}$.

######  Theorem 3.1.

[FRT89, DF93] $U_{q}(gl_{2})$ admits a presentation of FRT type. Define the
quantum L-operators with entries ${\cal{L}}^{\pm}_{i,j}$, $i,j=1,2$:

(3.6) $\displaystyle{\cal{L}}^{+}=\begin{pmatrix}K_{1}&(q-q^{-1})K_{1}F\\\
0&K_{2}\end{pmatrix}\ ,\qquad{\cal{L}}^{-}=\begin{pmatrix}K_{1}^{-1}&0\\\
-(q-q^{-1})EK_{1}^{-1}&K_{2}^{-1}\end{pmatrix}\ .$

The defining relations are given in a matrix form as follows:

(3.7) $\displaystyle{\cal{L}}^{+}_{i,i}{\cal{L}}^{-}_{i,i}$ $\displaystyle=$
$\displaystyle{\cal{L}}^{-}_{i,i}{\cal{L}}^{+}_{i,i}=1\ ,$ (3.8)
$\displaystyle R{\cal{L}}_{1}^{\pm}{\cal{L}}_{2}^{\pm}$ $\displaystyle=$
$\displaystyle{\cal{L}}_{2}^{\pm}{\cal{L}}_{1}^{\pm}R\ ,$ (3.9) $\displaystyle
R{\cal{L}}_{1}^{+}{\cal{L}}_{2}^{-}$ $\displaystyle=$
$\displaystyle{\cal{L}}_{2}^{-}{\cal{L}}_{1}^{+}R\ .$

From (3.8), (3.9), one shows:

(3.10) $\displaystyle
R_{21}^{-1}{\cal{L}}_{1}^{\pm}{\cal{L}}_{2}^{\pm}={\cal{L}}_{2}^{\pm}{\cal{L}}_{1}^{\pm}R_{21}^{-1}\
,\qquad
R_{21}^{-1}{\cal{L}}_{1}^{-}{\cal{L}}_{2}^{+}={\cal{L}}_{2}^{+}{\cal{L}}_{1}^{-}R_{21}^{-1}\
.$

######  Remark 3.2.

The inverse L-operators are given by:

(3.11)
$\displaystyle({\cal{L}}^{+})^{-1}=\begin{pmatrix}K_{1}^{-1}&-(q-q^{-1})FK_{2}^{-1}\\\
0&K_{2}^{-1}\end{pmatrix}\
,\qquad({\cal{L}}^{-})^{-1}=\begin{pmatrix}K_{1}&0\\\
(q-q^{-1})K_{2}E&K_{2}\end{pmatrix}\ .$

The relations satisfied by the inverse L-operators are easily derived from
(3.8)-(3.9), see [DF93, Proposition 2.1]. For instance,

(3.12) $\displaystyle
R_{21}({\cal{L}}^{\pm})^{-1}_{1}({\cal{L}}^{\pm})^{-1}_{2}$ $\displaystyle=$
$\displaystyle({\cal{L}}^{\pm})^{-1}_{2}({\cal{L}}^{\pm})^{-1}_{1}R_{21}\ ,$
(3.13) $\displaystyle R_{21}({\cal{L}}^{-})^{-1}_{1}({\cal{L}}^{+})^{-1}_{2}$
$\displaystyle=$
$\displaystyle({\cal{L}}^{+})^{-1}_{2}({\cal{L}}^{-})^{-1}_{1}R_{21}\ .$

In the FRT presentation of Theorem 3.1, the Hopf algebra structure of
$U_{q}(gl_{2})$ is characterized as follows. The corresponding expressions for
the comultiplication (2.6), counit (2.7) and antipode (2.8) are defined
by222$((T)_{[\textsf{1}]}(T^{\prime})_{[\textsf{2}]})_{ij}=\sum_{k=1}^{2}(T)_{ik}\otimes(T^{\prime})_{kj}$
[Sk88].

(3.14)
$\displaystyle\Delta({\cal{L}}^{\pm})=({\cal{L}}^{\pm})_{[1]}({\cal{L}}^{\pm})_{[2]}\
,\qquad\varepsilon({\cal{L}}^{\pm})={\mathbb{I}}\ ,\qquad
S({\cal{L}}^{\pm})=({\cal{L}}^{\pm})^{-1}\ ,$

where ${\mathbb{I}}$ denotes the $2\times 2$ identity matrix. Note that due to
the comultiplication rule, the last expressions easily follow from the
defining property of the antipode $\mu\circ(id\otimes
S)\circ\Delta=\mu\circ(S\otimes id)\circ\Delta=\eta\circ\varepsilon$, which
gives
${\cal{L}}^{\pm}S({\cal{L}}^{\pm})=S({\cal{L}}^{\pm}){\cal{L}}^{\pm}={\mathbb{I}}$.

Central elements of $U_{q}(gl_{2})$ are derived from the following quantum
determinants. Define $U=PR-q({\mathbb{I}}\otimes{\mathbb{I}})$ which satisfies
the Hecke braid relation $U_{12}U_{23}U_{12}=U_{12}$ and $U^{2}=-(q+q^{-1})U$.
As usual, below ‘$\rm tr_{12}$’ stands for the trace over $\cal
V_{1}\otimes\cal V_{2}$. One has:

(3.15) $\displaystyle{\rm det}_{q}^{(1)}$ $\displaystyle=$ $\displaystyle{\rm
tr}_{12}(U_{12}{\cal{L}}_{1}^{+}{\cal{L}}_{2}^{+})=-(q+q^{-1})\Omega_{1,c}\ ,$
(3.16) $\displaystyle{\rm det}_{q}^{(2)}$ $\displaystyle=$ $\displaystyle{\rm
tr}_{12}(U_{12}{\cal{L}}_{1}^{+}{\cal{L}}_{2}^{-})=-(q-q^{-1})^{2}\Omega_{2,c}\
,$

with (2.9), (2.10).

For further discussion, let us introduce $U^{\prime}_{q}(sl_{2})$ the
extension of the Chevalley presentation of $U_{q}(sl_{2})$ by the elements
$K^{\pm 1/2}$ [FMu02]. The defining relations of $U^{\prime}_{q}(sl_{2})$ are
given by (2.1)-(2.5) with the substitution

(3.17) $\displaystyle K_{1}\rightarrow K^{1/2}\ ,\qquad K_{2}\rightarrow
K^{-1/2}\ .$

A FRT presentation for $U^{\prime}_{q}(sl_{2})$ is obtained as a corollary of
Theorem 3.1: it is defined as the quotient of (3.7)-(3.9) by $\rm
det({\cal{L}}^{\pm})=1$. See e.g. [DF93, FMu02] for details. The corresponding
quantum L-operators are given by (3.6) with (3.17).

###  3.2. K-operators and $U_{q}(gl_{2})$

Explicit examples of ‘dressed’ solutions of Freidel-Maillet type equations can
be constructed using the FRT presentation of $U_{q}(gl_{2})$ given in Theorem
3.1. The analysis below is done by analogy with [Sk88, Proposition 2], so we
only sketch the main steps of the proofs. Consider the R-matrix (3.5) and
introduce the diagonal matrix:

(3.18) $\displaystyle R^{(0)}=diag(1,q^{-1},q^{-1},1)\ .$

Consider the Freidel-Maillet equations:

(3.19) $\displaystyle R\ ({\cal{K}}\otimes{\mathbb{I}})\ R^{(0)}\
({\mathbb{I}}\otimes{\cal{K}})\ $ $\displaystyle=$ $\displaystyle\
({\mathbb{I}}\otimes{\cal{K}})\ R^{(0)}\ ({\cal{K}}\otimes{\mathbb{I}})\ R\ ,$
(3.20) $\displaystyle R\ ({\cal{K}}\otimes{\mathbb{I}})\ R^{(0)}\
({\mathbb{I}}\otimes{\cal{K}}^{\prime})\ $ $\displaystyle=$ $\displaystyle\
({\mathbb{I}}\otimes{\cal{K}}^{\prime})\ R^{(0)}\
({\cal{K}}\otimes{\mathbb{I}})\ R\ .$

For the following analysis, we need the ‘reduced’ quantum Lax operators:

(3.21) $\displaystyle{\cal{L}}^{0,+}=diag(K_{1},K_{2})\
,\qquad\bar{\cal{L}}^{0,-}=diag(K_{2}^{-1},K_{1}^{-1})\ $

and the elementary K-matrices

(3.22) $\displaystyle{\cal{K}}^{0,\alpha}=\begin{pmatrix}1&0\\\
\alpha&1\end{pmatrix}\ ,$

where the scalar $\alpha\in{\mathbb{K}}$ is introduced. Combining (3.6),
(3.21) and (3.22), ‘dressed’ K-operators satisfying the system (3.19),(3.20),
are easily obtained.

######  Lemma 3.3.

For any $\alpha\in{\mathbb{K}}$, the K-operators

(3.23)
$\displaystyle{\cal{K}}^{+,\alpha}=\bar{\cal{L}}^{0,-}{\cal{K}}^{0,\alpha}{\cal{L}}^{+}\qquad\mbox{and}\qquad{\cal{K}}^{-,\alpha}={\cal{L}}^{0,+}{\cal{K}}^{0,\alpha}{\cal{L}}^{-}\
$

satisfy the Freidel-Maillet equations:

(i) (3.19) for ${\cal{K}}\rightarrow{\cal{K}}^{+,\alpha}$ or
${\cal{K}}\rightarrow{\cal{K}}^{-,\alpha}$;

(ii) (3.20) for ${\cal{K}}\rightarrow{\cal{K}}^{+,\alpha}$ and
${\cal{K}}^{\prime}\rightarrow{\cal{K}}^{-,\alpha}$.

###### Proof.

Consider (i). Assume there exists a matrix ${\cal{K}}^{0}$, two quantum Lax
operators ${\cal{L}},{\cal{L}}^{0}$ and a R-matrix $R^{\prime}$ such that the
following relations hold:

(3.24) $\displaystyle R\ {\cal{K}}^{0}_{1}\ R^{\prime}\ {\cal{K}}^{0}_{2}\ $
$\displaystyle=$ $\displaystyle\ {\cal{K}}^{0}_{2}\ R^{\prime}\
{\cal{K}}^{0}_{1}\ R\ ,$ (3.25) $\displaystyle R{\cal{L}}_{1}{\cal{L}}_{2}$
$\displaystyle=$ $\displaystyle{\cal{L}}_{2}{\cal{L}}_{1}R\ ,$ (3.26)
$\displaystyle R({\cal{L}}^{0})_{1}({\cal{L}}^{0})_{2}$ $\displaystyle=$
$\displaystyle({\cal{L}}^{0})_{2}({\cal{L}}^{0})_{1}R\ ,$ (3.27)
$\displaystyle({\cal{L}}^{0})_{1}R^{\prime}{\cal{L}}_{2}$ $\displaystyle=$
$\displaystyle{\cal{L}}_{2}R^{\prime}({\cal{L}}^{0})_{1}\ ,$ (3.28)
$\displaystyle{\cal{L}}_{1}R^{\prime}({\cal{L}}^{0})_{2}$ $\displaystyle=$
$\displaystyle({\cal{L}}^{0})_{2}R^{\prime}{\cal{L}}_{1}\ $

and

(3.29) $\displaystyle\big{[}{\cal{K}}^{0}_{i},{\cal{L}}_{j}\big{]}=0\
,\qquad\big{[}{\cal{K}}^{0}_{i},({\cal{L}}^{0})_{j}\big{]}=0\ ,\qquad i\neq j\
.$

Recall (3.6), (3.21), (3.22). For the choices

(3.30) $\displaystyle R^{\prime}=R^{(0)}\
,\quad{\cal{L}}\rightarrow{\cal{L}}^{+}\
,\quad{\cal{L}}^{0}\rightarrow\bar{\cal{L}}^{0,-}\quad\mbox{and}\quad{\cal{K}}^{0}\rightarrow{\cal{K}}^{0,\alpha}\
,$

one finds that all above relations are satisfied. Adapting [Sk88, Proposition
2], using those relations one finds
$\bar{\cal{L}}^{0,-}{\cal{K}}^{0,\alpha}{\cal{L}}^{+}$ satisfies (3.19). For
the choices

(3.31) $\displaystyle R^{\prime}=R^{(0)}\
,\quad{\cal{L}}\rightarrow{\cal{L}}^{-}\
,\quad{\cal{L}}^{0}\rightarrow{\cal{L}}^{0,+}\quad\mbox{and}\quad{\cal{K}}^{0}\rightarrow{\cal{K}}^{0,\alpha}\
,$

one finds ${\cal{L}}^{0,+}{\cal{K}}^{0,\alpha}{\cal{L}}^{-}$ also satisfies
(3.19). This completes the proof of (i).

To show (ii), it is sufficient to check that the basic equations (3.24)-(3.28)
and (3.29) hold for the substitutions $R^{\prime}=R^{(0)}$ and

(3.32)
$\displaystyle({\cal{L}}^{0})_{1}\rightarrow(\bar{\cal{L}}^{0,-})_{1}\quad\mbox{and}\quad({\cal{L}}^{0})_{2}\rightarrow({\cal{L}}^{0,+})_{2}\
.$

∎

Explicitly, the K-operators (3.23) read

(3.33)
$\displaystyle\quad{\cal{K}}^{+,\alpha}=\begin{pmatrix}K_{1}K_{2}^{-1}&(q-q^{-1})K_{1}K_{2}^{-1}F\\\
\alpha&K_{2}K_{1}^{-1}+\alpha(q-q^{-1})F\end{pmatrix}\quad\mbox{and}\quad{\cal{K}}^{-,\alpha}=\begin{pmatrix}1&0\\\
\alpha K_{2}K_{1}^{-1}-q(q-q^{-1})K_{2}K_{1}^{-1}E&1\end{pmatrix}\ .\ $

Note that although the K-operators are built from $U_{q}(gl_{2})$, it is seen
from the explicit form (3.33) that their entries belong to the subalgebra
generated by (2.19).

To prepare the analysis in the next subsection, let us first observe that more
general ‘dressed’ solutions of (3.19) and (3.20) can be easily derived by
induction. The proof of the following Lemma333The notation
$((T)_{[\textsf{2}]}(T^{\prime})_{[\textsf{1}]}(T^{\prime\prime})_{[\textsf{2}]})_{ij}=\sum_{k,\ell=1}^{2}(T^{\prime})_{k\ell}\otimes(T)_{ik}(T^{\prime\prime})_{\ell
j}$ is used. can be done similarly to Lemma 3.3, thus we skip it.

######  Lemma 3.4.

The assertions (i), (ii), in Lemma 3.3 hold for the substitution

(3.34)
$\displaystyle{\cal{K}}^{+,\alpha}\rightarrow\bar{\cal{K}}^{+,\alpha}=(\bar{\cal{L}}^{0,-})_{[2]}({\cal{K}}^{+,\alpha})_{[1]}({\cal{L}}^{+})_{[2]}\
,\qquad{\cal{K}}^{-,\alpha}\rightarrow\bar{\cal{K}}^{-,\alpha}=({\cal{L}}^{0,+})_{[2]}({\cal{K}}^{-,\alpha})_{[1]}({\cal{L}}^{-})_{[2]}\
.$

Explicitly, the entries of the K-operators read as follows:

$\displaystyle(\bar{\cal{K}}^{+,\alpha})_{1,1}$ $\displaystyle=$
$\displaystyle K_{1}K_{2}^{-1}\otimes K_{1}K_{2}^{-1}\ ,$
$\displaystyle(\bar{\cal{K}}^{+,\alpha})_{2,2}$ $\displaystyle=$
$\displaystyle\left(K_{2}K_{1}^{-1}+\alpha(q-q^{-1})F-\alpha\right)\otimes
K_{2}K_{1}^{-1}+1\otimes\left(\alpha K_{2}K_{1}^{-1}+\alpha(q-q^{-1})F\right)\
,$ $\displaystyle(\bar{\cal{K}}^{+,\alpha})_{1,2}$ $\displaystyle=$
$\displaystyle
K_{1}K_{2}^{-1}\otimes(q-q^{-1})K_{1}K_{2}^{-1}F+(q-q^{-1})K_{1}K_{2}^{-1}F\otimes
1\ ,$ $\displaystyle(\bar{\cal{K}}^{+,\alpha})_{2,1}$ $\displaystyle=$
$\displaystyle\alpha(1\otimes 1)\ $

and

$\displaystyle(\bar{\cal{K}}^{-,\alpha})_{1,1}$ $\displaystyle=$
$\displaystyle((\bar{\cal{K}}^{-,\alpha}))_{2,2}=1\otimes 1\ ,$
$\displaystyle(\bar{\cal{K}}^{-,\alpha})_{1,2}$ $\displaystyle=$
$\displaystyle 0\ ,$ $\displaystyle(\bar{\cal{K}}^{-,\alpha})_{2,1}$
$\displaystyle=$
$\displaystyle\left(\alpha(K_{2}K_{1}^{-1}-1)-q(q-q^{-1})K_{2}K_{1}^{-1}E\right)\otimes
K_{2}K_{1}^{-1}+1\otimes\left(\alpha
K_{2}K_{1}^{-1}-q(q-q^{-1})K_{2}K_{1}^{-1}E\right)\ .$

The action of Hopf algebra homomorphisms $\Delta,\varepsilon$ and
antiautomorphism $S$ of $U_{q}(gl_{2})$ on the K-operators is finally
considered. Acting with (2.6) (resp. (2.7)) on the K-operators in (3.33), one
compares the resulting expressions with the entries
$(\tilde{{\cal{K}}}^{\pm,\alpha})_{i,j}$ (resp. $({\cal{K}}^{0,\alpha})_{i,j}$
with (3.22)). One concludes:

(3.35) $\displaystyle\Delta({\cal{K}}^{+,\alpha})$ $\displaystyle=$
$\displaystyle(\bar{\cal{L}}^{0,-})_{[2]}({\cal{K}}^{+,\alpha})_{[1]}({\cal{L}}^{+})_{[2]}\
,\qquad\Delta({\cal{K}}^{-,\alpha})=({\cal{L}}^{0,+})_{[2]}({\cal{K}}^{-,\alpha})_{[1]}({\cal{L}}^{-})_{[2]}\
,$ (3.36) $\displaystyle\varepsilon({\cal{K}}^{\pm,\alpha})$ $\displaystyle=$
$\displaystyle{\cal{K}}^{0,\alpha}\ $

Furthermore, using (2.8) one checks the compatibility property:

(3.37) $\displaystyle\mu\circ(id\otimes
S)\circ\Delta({\cal{K}}^{\pm,\alpha})=\mu\circ(S\otimes
id)\circ\Delta({\cal{K}}^{\pm,\alpha})=\eta\circ\varepsilon({\cal{K}}^{\pm,\alpha})\
.$

###  3.3. Freidel-Maillet type presentations of $U_{q}(sl_{2})$

Consider the restriction in $U_{q}(sl_{2})$ of the one-parameter family of
K-operators (3.33), using (2.19). They are denoted
$({\cal{K}}^{\pm,\alpha})|_{U_{q}(sl_{2})}$. For the specialization
$\alpha=0$, the entries of $({\cal{K}}^{\pm,0})|_{U_{q}(sl_{2})}$ are
expressed in terms of the Chevalley generators $E,F,K^{\pm 1}$. For
$\alpha=1$, applying $\phi^{-1}$ with (2.23) the entries are expressed in
terms of the equitable generators $X^{\pm 1},Y,Z$. Accordingly, introduce the
notation:

(3.38)
$\displaystyle{\cal{K}}_{c}^{\pm}\equiv({\cal{K}}^{\pm,0})|_{U_{q}(sl_{2})}\qquad\mbox{and}\qquad{\cal{K}}_{e}^{\pm}\equiv\phi^{-1}(({\cal{K}}^{\pm,1})|_{U_{q}(sl_{2})})\
.$

By Lemma 3.3, they satisfy the Freidel-Maillet equations (3.19), (3.20).
Furthermore:

######  Theorem 3.5.

$U_{q}(sl_{2})$ admits a presentation of Freidel-Maillet type. The defining
relations are given in a matrix form by:

(3.39) $\displaystyle{\cal{K}}^{\pm}_{1,1}({\cal{K}}^{\pm}_{1,1})^{-1}$
$\displaystyle=$
$\displaystyle({\cal{K}}^{\pm}_{1,1})^{-1}{\cal{K}}^{\pm}_{1,1}=1\ ,$ (3.40)
$\displaystyle R\ ({\cal{K}}^{\pm}\otimes{\mathbb{I}})\ R^{(0)}\
({\mathbb{I}}\otimes{\cal{K}}^{\pm})\ $ $\displaystyle=$ $\displaystyle\
({\mathbb{I}}\otimes{\cal{K}}^{\pm})\ R^{(0)}\
({\cal{K}}^{\pm}\otimes{\mathbb{I}})\ R\ ,$ (3.41) $\displaystyle R\
({\cal{K}}^{+}\otimes{\mathbb{I}})\ R^{(0)}\
({\mathbb{I}}\otimes{\cal{K}}^{-})\ $ $\displaystyle=$ $\displaystyle\
({\mathbb{I}}\otimes{\cal{K}}^{-})\ R^{(0)}\
({\cal{K}}^{+}\otimes{\mathbb{I}})\ R\ ,$

where ${\cal{K}}^{\pm}\equiv{\cal{K}}^{\pm}_{c}$ or
${\cal{K}}^{\pm}\equiv{\cal{K}}^{\pm}_{e}$ is a square matrix with entries
${\cal{K}}^{\pm}_{i,j}$, $i,j=1,2$, such that

(3.42) $\displaystyle{\cal{K}}^{+}_{c}=\begin{pmatrix}K&(q-q^{-1})KF\\\
0&K^{-1}\end{pmatrix}\ ,\qquad{\cal{K}}^{-}_{c}=\begin{pmatrix}1&0\\\
-q(q-q^{-1})K^{-1}E&1\end{pmatrix}\ $

and

(3.43) $\displaystyle{\cal{K}}^{+}_{e}=\begin{pmatrix}X&XY-1\\\
1&Y\end{pmatrix}\ ,\qquad{\cal{K}}^{-}_{e}=\begin{pmatrix}1&0\\\
Z&1\end{pmatrix}\ .$

We call ${\cal{K}}^{\pm}_{c}$ and ${\cal{K}}^{\pm}_{e}$ the Chevalley and
equitable K-operators of $U_{q}(sl_{2})$, respectively.

###### Proof.

Eq. (3.39) is equivalent to (2.11) or (2.21). Insert (3.42) or (3.43) into
(3.39)-(3.41). It is checked that the resulting system of equations is
equivalent to the remaining relations given in Definition 2.2 or Theorem 2.3.
∎

Note that (3.40), (3.41) imply

(3.44) $\displaystyle R_{21}^{-1}\ ({\cal{K}}^{\pm}\otimes{\mathbb{I}})\
R^{(0)}\ ({\mathbb{I}}\otimes{\cal{K}}^{\pm})\ =\
({\mathbb{I}}\otimes{\cal{K}}^{+})\ R^{(0)}\
({\cal{K}}^{-}\otimes{\mathbb{I}})\ R_{21}^{-1}\ ,$ (3.45) $\displaystyle
R_{21}^{-1}\ ({\cal{K}}^{-}\otimes{\mathbb{I}})\ R^{(0)}\
({\mathbb{I}}\otimes{\cal{K}}^{+})\ =\ ({\mathbb{I}}\otimes{\cal{K}}^{+})\
R^{(0)}\ ({\cal{K}}^{-}\otimes{\mathbb{I}})\ R_{21}^{-1}\ .$

######  Remark 3.6.

The inverse K-operators are given by:

(3.46)
$\displaystyle({\cal{K}}_{c}^{+})^{-1}=\begin{pmatrix}K^{-1}&-(q-q^{-1})FK\\\
0&K\end{pmatrix}\ ,\qquad({\cal{K}}_{c}^{-})^{-1}=\begin{pmatrix}1&0\\\
q^{-1}(q-q^{-1})EK^{-1}&1\end{pmatrix}\ $

and

(3.47) $\displaystyle({\cal{K}}_{e}^{+})^{-1}=\begin{pmatrix}Y&1-YX\\\
-1&X\end{pmatrix}\ ,\qquad({\cal{K}}_{e}^{-})^{-1}=\begin{pmatrix}1&0\\\
-Z&1\end{pmatrix}\ .$

By analogy with (3.12), (3.13), other examples of Freidel-Maillet equations
can be derived from (3.40)-(3.41).

######  Proposition 3.7.

The following relations hold:

(3.48) $\displaystyle
R_{21}({\cal{K}}^{\pm})^{-1}_{1}(R^{(0)})^{-1}({\cal{K}}^{\pm})^{-1}_{2}$
$\displaystyle=$
$\displaystyle({\cal{K}}^{\pm})^{-1}_{2}(R^{(0)})^{-1}({\cal{K}}^{\pm})^{-1}_{1}R_{21}\
,$ (3.49) $\displaystyle
R_{21}({\cal{K}}^{-})^{-1}_{1}(R^{(0)})^{-1}({\cal{K}}^{+})^{-1}_{2}$
$\displaystyle=$
$\displaystyle({\cal{K}}^{+})^{-1}_{2}(R^{(0)})^{-1}({\cal{K}}^{-})^{-1}_{1}R_{21}\
.$

Note that the Chevalley K-operators have a triangular structure, contrary to
one of the two equitable K-operators.

The Freidel-Maillet type presentation of Theorem 3.5 is not unique. For
instance, the relation (2.21) is not necessary to show (3.40), (3.41), given
the K-operators (3.43): only the relations (2.22) are needed. Because these
latter relations remain unchanged by the rotation $r:\ X\rightarrow Y$,
$Y\rightarrow Z$, $Z\rightarrow X$, other expressions for K-operators obtained
from (3.43) can be considered as well. For instance, denote
$r({\cal{K}}_{e}^{+})\equiv{\cal{K}}_{\cal B}$ and
$r({\cal{K}}_{e}^{-})\equiv{\cal{K}}_{X}$. It follows:

######  Remark 3.8.

A Freidel-Maillet type presentation of $U_{q}(sl_{2})$ is given by:

(3.50) $\displaystyle({\cal{K}}_{a})_{2,1}({\cal{K}}_{a})_{2,1}^{-1}$
$\displaystyle=$
$\displaystyle({\cal{K}}_{a})_{2,1}^{-1}({\cal{K}}_{a})_{2,1}=1\ ,$ (3.51)
$\displaystyle R\ ({\cal{K}}_{a}\otimes{\mathbb{I}})\ R^{(0)}\
({\mathbb{I}}\otimes{\cal{K}}_{a})\ $ $\displaystyle=$ $\displaystyle\
({\mathbb{I}}\otimes{\cal{K}}_{a})\ R^{(0)}\
({\cal{K}}_{a}\otimes{\mathbb{I}})\ R\ \quad\mbox{for}\qquad a=X,{\cal B},$
(3.52) $\displaystyle R\ ({\cal{K}}_{\cal B}\otimes{\mathbb{I}})\ R^{(0)}\
({\mathbb{I}}\otimes{\cal{K}}_{X})\ $ $\displaystyle=$ $\displaystyle\
({\mathbb{I}}\otimes{\cal{K}}_{X})\ R^{(0)}\ ({\cal{K}}_{\cal
B}\otimes{\mathbb{I}})\ R\ ,$

where ${\cal{K}}_{\cal B},{\cal{K}}_{X}$ are square matrices such that

(3.53) $\displaystyle{\cal{K}}_{\cal B}=\begin{pmatrix}Y&YZ-1\\\
1&Z\end{pmatrix}\ ,\qquad{\cal{K}}_{X}=\begin{pmatrix}1&0\\\ X&1\end{pmatrix}\
.$

As mentionned in the introduction, the subalgebra $\cal B$ generated by
$\\{Y,Z\\}$ in the equitable presentation of $U_{q}(sl_{2})$ provides the
simplest example of non-homogeneous Borel subalgebras [LV17, V18, LV19]. This
is seen from a direct comparison between (2.23) and [LV19, Example 3.6].
According to previous remark, $\cal B$ is isomorphic to the Freidel-Maillet
algebra:

(3.54) $\displaystyle R\ ({\cal{K}}_{\cal B}\otimes{\mathbb{I}})\ R^{(0)}\
({\mathbb{I}}\otimes{\cal{K}}_{\cal B})\ $ $\displaystyle=$ $\displaystyle\
({\mathbb{I}}\otimes{\cal{K}}_{\cal B})\ R^{(0)}\ ({\cal{K}}_{\cal
B}\otimes{\mathbb{I}})\ R\ .$

Another Freidel-Maillet type presentation is obtained by modifying the
R-matrix $R^{(0)}$:

######  Remark 3.9.

A Freidel-Maillet type presentation of $U_{q}(sl_{2})$ is obtained from
(3.39)-(3.41) with the substitution $R^{(0)}\rightarrow(R^{(0)})^{-1}$,
${\cal{K}}^{\pm}_{1,1}\rightarrow{\cal{K}}^{\pm}_{2,2}$ and the Chevalley and
equitable K-operators given, respectively, by:

(3.55)
$\displaystyle{\cal{K}}_{c}^{+}\rightarrow\begin{pmatrix}1&(q-q^{-1})F\\\
0&1\end{pmatrix}\
,\qquad{\cal{K}}_{c}^{-}\rightarrow\begin{pmatrix}K^{-1}&0\\\
-q(q-q^{-1})KE&K\end{pmatrix}\ $

and

(3.56) $\displaystyle{\cal{K}}_{e}^{+}\rightarrow\begin{pmatrix}1&Y\\\
0&1\end{pmatrix}\ ,\qquad{\cal{K}}_{e}^{-}\rightarrow\begin{pmatrix}Z&1\\\
XZ-1&X\end{pmatrix}\ .$

For the FRT presentation of $U^{\prime}_{q}(sl_{2})$ introduced at the end of
Subsection 3.1, the Hopf algebra structure is characterized by a
comultiplication, counit and antipode that follow from the restriction of
(3.14) to $U^{\prime}_{q}(sl_{2})$. A characterization of the Hopf algebra
structure of $U_{q}(sl_{2})$ is obtained by combining the Freidel-Maillet type
presentation of Theorem 3.5 and the FRT presentation of
$U^{\prime}_{q}(sl_{2})$ extended by the ‘reduced’ L-operators (3.21). The
comultiplication (2.14) or (2.26) and the counit (2.15), (2.27) are obtained
as follows. Denote by $M|_{U_{q}(sl_{2})}$ the restriction of the operator $M$
to $U_{q}(sl_{2})$.

######  Proposition 3.10.

The comultiplication $\Delta:U_{q}(sl_{2})\rightarrow U_{q}(sl_{2})\otimes
U_{q}(sl_{2})$ is such that

(3.57)
$\displaystyle\Delta({\cal{K}}_{c}^{+})=\left((\bar{\cal{L}}^{0,-})_{[2]}({\cal{K}}_{c}^{+})_{[1]}({\cal{L}}^{+})_{[2]}\right)|_{U_{q}(sl_{2})_{[2]}}\
,\qquad\Delta({\cal{K}}_{c}^{-})=\left(({\cal{L}}^{0,+})_{[2]}({\cal{K}}_{c}^{-})_{[1]}({\cal{L}}^{-})_{[2]}\right)|_{U_{q}(sl_{2})_{[2]}}\
,$ (3.58) $\displaystyle\Delta({\cal{K}}_{e}^{+})$ $\displaystyle=$
$\displaystyle\left(id\otimes\phi^{-1})\circ(\bar{\cal{L}}^{0,-})_{[2]}({\cal{K}}_{e}^{+})_{[1]}({\cal{L}}^{+})_{[2]}\right)|_{U_{q}(sl_{2})_{[2]}}\
,$ $\displaystyle\Delta({\cal{K}}_{e}^{-})$ $\displaystyle=$
$\displaystyle(id\otimes\phi^{-1})\circ\left(({\cal{L}}^{0,+})_{[2]}({\cal{K}}_{e}^{-})_{[1]}({\cal{L}}^{-})_{[2]}\right)|_{U_{q}(sl_{2})_{[2]}}\
.$

The counit $\varepsilon:U_{q}(sl_{2})\rightarrow{\mathbb{K}}$ is such that

(3.59)
$\displaystyle\varepsilon({\cal{K}}_{c}^{\pm})={\cal{K}}^{0,0}\qquad\mbox{and}\qquad\varepsilon({\cal{K}}_{e}^{\pm})={\cal{K}}^{0,1}\
$

with (3.22).

###### Proof.

Firstly, we show that (3.57), (3.58) are equivalent to the comultiplication
rules (2.14), (2.26). For the specialization $\alpha=0$, one takes the
restriction of (3.35) to $U_{q}(sl_{2})\otimes U_{q}(sl_{2})$ and uses (3.38).
For the specialization $\alpha=1$, in addition one acts on the restriction of
(3.35) with $\phi^{-1}\otimes\phi^{-1}$. Secondly, (3.59) produces the counit
rules (2.15), (2.27): this follows from (3.36) and (3.38). ∎

In addition, the compatibility property satisfied by the antipode
$S:U_{q}(sl_{2})\rightarrow U_{q}(sl_{2})$ holds:

(3.60) $\displaystyle\mu\circ(id\otimes
S)\circ\Delta({\cal{K}}^{\pm})=\mu\circ(S\otimes
id)\circ\Delta({\cal{K}}^{\pm})=\eta\circ\varepsilon({\cal{K}}^{\pm})\
\quad\mbox{for}\quad\mbox{${\cal{K}}\equiv{\cal{K}}_{c}$ or ${\cal{K}}_{e}$}.$

It follows from the restriction of (3.37) to $U_{q}(sl_{2})$. Note that
contrary to the FRT presentation with (3.14), the comultiplication rules
(3.57), (3.58) have no group-like structure. So, the antipode of a Chevalley
or equitable K-operator (i.e. $S({\cal{K}}_{c}^{\pm})$ and
$S({\cal{K}}_{e}^{\pm})$) computed using (2.16), (2.28), doesn’t take a simple
form in terms of L and K-operators.

To conclude this subsection, let us mention that the central element of
$U_{q}(sl_{2})$ given by (2.18) or equivalently (2.29) can be derived from the
Freidel-Maillet quantum determinant analog of (3.16). Namely,

(3.61) $\displaystyle{\rm det}_{q}^{FM}$ $\displaystyle=$ $\displaystyle{\rm
tr}_{12}(U_{12}{\cal{K}}_{1}^{+}R^{(0)}{\cal{K}}_{2}^{-})=-q^{-1}(q-q^{-1})^{2}\Omega_{c}=-q^{-1}\Omega_{e}\
.$

Note that the central element can be alternatively derived from the so-called
quantum trace [F96, p. 46] with (3.55), (3.56):

(3.62) $\displaystyle{\rm
tr}_{q}({\cal{K}}^{+}({\cal{K}}^{-})^{-1})=tr(D{\cal{K}}^{+}({\cal{K}}^{-})^{-1})=(q-q^{-1})^{2}\Omega_{c}=\Omega_{e}\qquad\mbox{with}\qquad
D=diag(q,q^{-1})\ .$

###  3.4. K-operators of $U_{q}(sl_{2})$ as intertwiners

The quasi-cocommutativity property of $U_{q}(sl_{2})$ according to the
Chevalley presentation is characterized by the universal R-matrix
$\mathfrak{R}\in U_{q}(sl_{2})\otimes U_{q}(sl_{2})$ [D86]. With respect to
the comultiplication $\Delta$ and opposite comultiplication
$\Delta^{\prime}=\sigma\circ\Delta$, $\sigma(x\otimes y)=y\otimes x$,
$\mathfrak{R}$ is the isomorphism such that
$\Delta(x)=\mathfrak{R}\Delta^{\prime}(x)\mathfrak{R}^{-1}$ for $x\in
E,F,K^{\pm 1}$. It implies that the L-operators of $U^{\prime}_{q}(sl_{2})$
satisfy the intertwining relations (see e.g. [DEFK09]):

(3.63) $\displaystyle(\rho_{V^{(1/2)}}\times
id)\Delta(x){\cal{L}}^{\pm}|_{U^{\prime}_{q}(sl_{2})}={\cal{L}}^{\pm}|_{U^{\prime}_{q}(sl_{2})}(\rho_{V^{(1/2)}}\otimes
id)\Delta^{\prime}(x)\quad\mbox{for any}\quad x\in K^{\pm 1/2},E,F\ ,$

as can be easily checked using (2.14), (3.6) with (3.17) and (2.20). In
particular, R-matrices that solve the Yang-Baxter equation (1.1) can be
derived from $R_{12}=(\rho_{1}\otimes\rho_{2})(\mathfrak{R})$, where
$\rho_{i}:U_{q}(sl_{2})\rightarrow\text{\rm End}({\cal V}_{i})$.

By analogy, it is expected that K-operators of $U_{q}(sl_{2})$ satisfy certain
intertwining relations. Assume there exists a map
$\tilde{\delta}:U_{q}(sl_{2})\rightarrow U_{q}(sl_{2})\otimes U_{q}(sl_{2})$
such that:

(3.64) $\displaystyle(\rho_{V^{(1/2)}}\otimes id)\tilde{\delta}(K^{\pm 1})$
$\displaystyle=$ $\displaystyle\begin{pmatrix}q^{\pm 1}K^{\pm 1}&0\\\ 0&q^{\mp
1}K^{\pm 1}\end{pmatrix}\ ,$ (3.65) $\displaystyle(\rho_{V^{(1/2)}}\otimes
id)\tilde{\delta}(E)$ $\displaystyle=$
$\displaystyle\begin{pmatrix}q^{2}E&K\\\ 0&q^{-2}E\end{pmatrix}\
,\quad(\rho_{V^{(1/2)}}\otimes
id)\tilde{\delta}(F)=\begin{pmatrix}q^{-1}F&0\\\ K^{-2}&qF\end{pmatrix}\ ,$

for the Chevalley presentation, and

(3.66) $\displaystyle(\rho_{V^{(1/2)}}\otimes
id)\tilde{\delta}(X)=\begin{pmatrix}qX&0\\\ q-q^{-1}&q^{-1}X\end{pmatrix}\
,\quad\ \ \ (\rho_{V^{(1/2)}}\otimes
id)\tilde{\delta}(X^{-1})=\begin{pmatrix}q^{-1}X^{-1}&0\\\
-(q-q^{-1})X^{-2}&qX^{-1}\end{pmatrix}\ ,$ (3.67)
$\displaystyle(\rho_{V^{(1/2)}}\otimes
id)\tilde{\delta}(Y)=\begin{pmatrix}q^{-1}Y&0\\\ 0&qY\end{pmatrix}\
,\quad\quad\quad\quad(\rho_{V^{(1/2)}}\otimes
id)\tilde{\delta}(Z)=\begin{pmatrix}qZ&q^{-1}-q\\\ 0&q^{-1}Z\end{pmatrix}\ ,$

for the equitable presentation. By straightforward calculations, it is found
that the defining relations of $U_{q}(sl_{2})$ hold for the substitution
$x\rightarrow(\rho_{V^{(1/2)}}\otimes id)\tilde{\delta}(x)\in\text{\rm
End}({\mathbb{C}}^{2})\otimes U_{q}(sl_{2})$. Furthermore, one finds that the
following intertwining relations are satisfied by the K-operators (3.42),
(3.43):

(3.68) $\displaystyle(\rho_{V^{(1/2)}}\otimes
id)\tilde{\delta}(x){\cal{K}}_{c}^{\pm}={\cal{K}}_{c}^{\pm}(\rho_{V^{(1/2)}}\otimes
id)\Delta^{\prime}(x)\quad\mbox{for}\quad x\in K^{\pm 1},E,F\ ,$

and

(3.69) $\displaystyle(\rho_{V^{(1/2)}}\otimes
id)\tilde{\delta}(x){\cal{K}}_{e}^{\pm}={\cal{K}}_{e}^{\pm}(\rho_{V^{(1/2)}}\otimes
id)\Delta^{\prime}(x)\quad\mbox{for}\quad x\in X^{\pm 1},Y,Z\ ,$

where (2.20), (2.30) are used.

The above results suggest to introduce a universal K-matrix for
$U_{q}(sl_{2})$, characterizing the relation between $U_{q}(sl_{2})$-modules
associated with $\Delta^{\prime}$ and $\tilde{\delta}$. From this perspective,
the universal K-matrix ${\mathfrak{K}}\in U_{q}(sl_{2})\otimes U_{q}(sl_{2})$
would be an invertible element such that
$\tilde{\delta}(x)=\mathfrak{K}\Delta^{\prime}(x)\mathfrak{K}^{-1}$ for all
$x\in U_{q}(sl_{2})$. For the Chevalley presentation, this problem is solved
via the embedding $U_{q}(sl_{2})\hookrightarrow U^{\prime}_{q}(sl_{2})$.
Indeed, observe that ${\cal{K}}^{\pm}=(\rho_{V^{(1/2)}}\otimes id)({\cal
F}){\cal{L}}^{\pm}|_{U^{\prime}_{q}(sl_{2})}$ where $\cal F=q^{(H\otimes
H)/2}$ is a Drinfeld twist [D86, R90] such that $K\equiv q^{H}$. Then,
${\cal{K}}^{\pm}$ are obtained by specializing $\mathfrak{K}$, where
$\mathfrak{K}$ gives an example of ‘modified’ universal R-matrix [P94,
Proposition 3.7]. It is easily checked that the map $\tilde{\delta}$ with
(3.64), (3.65), follows from the twisted comultiplication. For the equitable
case, finding the universal K-matrix is an open problem that will be
considered elsewhere. In any case, for finite dimensional tensor product
representations K-matrices satisfying the Freidel-Maillet equation (1.4) would
follow from the universal K-matrix. Examples of K-matrices are given in the
next subsection.

To conclude this subsection, let us mention that similar results hold for the
non-homogeneous Borel subalgebra ${\cal B}$ with defining relations (3.54) and
K-operator ${\cal{K}}_{\cal B}$ in (3.53). Assume there are two maps
$\tilde{\delta}_{\cal B}$, $\delta^{\prime}_{\cal B}:\cal B\rightarrow
U_{q}(sl_{2})\otimes{\cal B}$ such that:

$\displaystyle(\rho_{V^{(1/2)}}\otimes id)\delta^{\prime}_{\cal B}(Y)$
$\displaystyle=$ $\displaystyle\begin{pmatrix}qY&0\\\ 0&q^{-1}Y\end{pmatrix}\
,\quad\quad\quad(\rho_{V^{(1/2)}}\otimes id)\delta^{\prime}_{\cal
B}(Z)=\begin{pmatrix}q^{-1}Z&0\\\ q-q^{-1}&qZ\end{pmatrix}\ ,$
$\displaystyle(\rho_{V^{(1/2)}}\otimes id)\tilde{\delta}_{\cal B}(Y)$
$\displaystyle=$ $\displaystyle\begin{pmatrix}qY&0\\\
q-q^{-1}&q^{-1}Y\end{pmatrix}\ ,\quad(\rho_{V^{(1/2)}}\otimes
id)\tilde{\delta}_{\cal B}(Z)=\begin{pmatrix}q^{-1}Z&0\\\ 0&qZ\end{pmatrix}\
.$

It is straightforward to check that the image of the defining relation
$qYZ-q^{-1}ZY=q-q^{-1}$ by above maps holds. Furthermore, the following
intertwining relations are obtained:

(3.70) $\displaystyle(\rho_{V^{(1/2)}}\otimes id)\tilde{\delta}_{\cal
B}(x){\cal{K}}_{\cal B}={\cal{K}}_{\cal B}(\rho_{V^{(1/2)}}\otimes
id)\delta^{\prime}_{\cal B}(x)\quad\mbox{for}\quad x\in Y,Z\ .$

This suggests to investigate the universal version of (3.54) and its solutions
from the approach based on the universal analog of the intertwining relations
above.

###  3.5. Constant K-matrices of $U_{q}(sl_{2})$

It is well-known that the simplest (non-trivial) constant solutions of the
Yang-Baxter equation (1.1) can be derived from (3.6) by taking the fundamental
representation of $U^{\prime}_{q}(sl_{2})$. Using (2.20) one gets

(3.71) $\displaystyle
R=q^{1/2}\rho_{V^{(1/2)}}({\cal{L}}^{+}|_{U^{\prime}_{q}(sl_{2})})\qquad\mbox{and}\qquad
R_{21}^{-1}=q^{-1/2}\rho_{V^{(1/2)}}({\cal{L}}^{-}|_{U^{\prime}_{q}(sl_{2})})\
.$

By analogy, the simplest non-trivial examples of constant K-matrices of
$U_{q}(sl_{2})$ satisfying the Freidel-Maillet equation

(3.72) $\displaystyle
R_{12}{\cal{K}}_{13}R^{(0)}_{12}{\cal{K}}_{23}={\cal{K}}_{23}R^{(0)}_{12}{\cal{K}}_{13}R_{12}\
$

are derived from (3.42), (3.43) and (3.53). For the one-dimensional (spin
$s=0$) representation, the images of the Chevalley K-operators are trivial.
However, the image of the equitable K-operators (3.43) or (3.53) is not, given
by:

(3.73)
$\displaystyle\rho_{V^{(0)}}({\cal{K}}_{e}^{\pm})=\rho_{V^{(0)}}({\cal{K}}_{X})=\rho_{V^{(0)}}({\cal{K}}_{\cal
B})=\begin{pmatrix}1&0\\\ 1&1\end{pmatrix}\ .$

For the spin $s=1/2$ representation, for the Chevalley case using (2.20) one
gets $\rho_{V^{(1/2)}}({\cal{K}}_{c}^{+})=R^{(0)}R$ and
$\rho_{V^{(1/2)}}({\cal{K}}_{c}^{-})=qR^{(0)}R_{21}^{-1}$. In this case, the
K-matrices ${\cal{K}}_{c}^{\pm}$ are just examples of constant solutions to
braided Yang-Baxter equations. For the equitable case, from (2.30) one gets

(3.82)
$\displaystyle\rho_{V^{(1/2)}}({\cal{K}}_{e}^{+})=\left(\begin{array}[]{cccc}q&0&0&0\\\
0&q^{-1}&q^{-1}(q-q^{-1})&0\\\ 1&0&q^{-1}&0\\\
0&1&q-q^{-1}&q\end{array}\right)\
,\qquad\rho_{V^{(1/2)}}({\cal{K}}_{e}^{-})=\left(\begin{array}[]{cccc}1&0&0&0\\\
0&1&0&0\\\ q^{-1}&-(q-q^{-1})&1&0\\\ 0&q&0&1\end{array}\right)\ $

and

(3.91) $\displaystyle\rho_{V^{(1/2)}}({\cal{K}}_{\cal
B})=\left(\begin{array}[]{cccc}q^{-1}&0&q^{-2}-1&q^{-2}-1\\\
q-q^{-1}&q&1-q^{-2}&1-q^{-2}\\\ 1&0&q^{-1}&q^{-1}-q\\\
0&1&0&q\end{array}\right)\
,\qquad\rho_{V^{(1/2)}}({\cal{K}}_{X})=\left(\begin{array}[]{cccc}1&0&0&0\\\
0&1&0&0\\\ q&0&1&0\\\ 0&q^{-1}&0&1\end{array}\right)\ .$

As recalled in previous subsection, the constant R-matrices (3.71) can be
derived from the universal R-matrix of $U_{q}(sl_{2})$, see e.g. [CP94,
Example 6.4.12]. For the equitable case, it is expected that the constant
K-matrices above can be derived from a universal K-matrix of equitable type.

##  4\. Spectral parameter dependent K-operators for $U_{q}(sl_{2})$

In this section, we study a class of K-operators that solve the spectral
parameter dependent Freidel-Maillet equation (4.16) recently revisited444For
the class of solutions associated with ‘twisted’ Lax operators, (4.16) can be
mapped to a braided Yang-Baxter equation. See e.g. [Ku95]. in [B20]. We start
from a K-operator (4.15) with entries in $\cal
A\otimes{\mathbb{K}}[u,u^{-1}]$, where $\cal A$ is a quotient of the
alternating subalgebra of $U_{q}(\widehat{sl_{2}})$ introduced in [T18]. See
[B20, Section 4] for details. Certain specializations $\cal A\rightarrow
U_{q}(sl_{2})$ are considered. It is shown that corresponding images of the
K-operator admit a decomposition in terms of ${\cal{K}}^{\pm}$. Examples of
spectral parameter dependent K-matrices are given.

As a preliminary, let us recall that the FRT presentation associated with
$U^{\prime}_{q}(sl_{2})$ can be encoded into a single spectral parameter
dependent Yang-Baxter equation [F96]. Introduce the quantum R-matrix [Bax82]

(4.5) $\displaystyle R(u)=\left(\begin{array}[]{cccc}uq-u^{-1}q^{-1}&0&0&0\\\
0&u-u^{-1}&q-q^{-1}&0\\\ 0&q-q^{-1}&u-u^{-1}&0\\\
0&0&0&uq-u^{-1}q^{-1}\end{array}\right)\ .$

It is known that $R(u)$ satisfies the (spectral parameter dependent) Yang-
Baxter equation in the space ${\cal V}_{1}\otimes{\cal V}_{2}\otimes{\cal
V}_{3}$:

(4.6) $\displaystyle
R_{12}(u/v)R_{13}(u)R_{23}(v)=R_{23}(v)R_{13}(u)R_{12}(u/v)\ \qquad\forall
u,v\ .$

Note that the permutation operator $P=R(1)/(q-q^{-1})$, and
$R_{12}(u)=PR_{12}(u)P=R_{21}(u)$. The FRT presentation of
$U^{\prime}_{q}(sl_{2})$ is encoded as

(4.7) $\displaystyle K^{1/2}K^{-1/2}$ $\displaystyle=$ $\displaystyle
K^{-1/2}K^{1/2}=1\ ,$ (4.8) $\displaystyle R(u/v)\
({\cal{L}}(u)\otimes{\mathbb{I}})\ ({\mathbb{I}}\otimes{\cal{L}}(v))$
$\displaystyle=$ $\displaystyle({\mathbb{I}}\otimes{\cal{L}}(v))\
({\cal{L}}(u)\otimes{\mathbb{I}})\ R(u/v)\ .$

Up to automorphisms of $U^{\prime}_{q}(sl_{2})$, the so-called quantum Lax
operator $({\cal{L}}(u))_{i,j}\in
U^{\prime}_{q}(sl_{2})\otimes{\mathbb{K}}[u,u^{-1}]$ reads:

(4.9)
$\displaystyle{\cal{L}}(u)=\begin{pmatrix}uK^{1/2}-u^{-1}K^{-1/2}&(q-q^{-1})FK^{1/2}\\\
(q-q^{-1})K^{-1/2}E&uK^{-1/2}-u^{-1}K^{1/2}\end{pmatrix}\ .$

In the form (4.8), a generating function for central elements is given by the
quantum determinant ${\rm det}_{q}({\cal{L}}(u))$. For instance, for
$U^{\prime}_{q}(sl_{2})$ the Casimir element $\Omega_{c}$ given by (2.18)
follows from ${\rm det}_{q}({\cal{L}}(u))={\rm
tr}_{12}\big{(}P^{-}_{12}{\cal{L}}_{1}(u){\cal{L}}_{2}(uq))$ with
$P^{-}_{12}=(1-P)/2$:

(4.10) $\displaystyle{\rm
det}_{q}({\cal{L}}(u))=qu^{2}+q^{-1}u^{2}-(q-q^{-1})^{2}\Omega_{c}\ .$

Our purpose is now to investigate the simplest spectral parameter dependent
K-operator solutions of the Freidel-Maillet type equation (4.16) in light of
the results in previous section. To this end, we introduce a more general
setting.

Consider the $R-$matrices given by (4.5) and (3.18). Note that $R^{(0)}$ can
be viewed as a limiting case of (4.5). Introduce the associative algebra $\cal
A$ with generators $\\{\cal W_{0},\cal W_{1},\cal Z_{1},\tilde{\cal}Z_{1}\\}$
such that the following relations hold:

(4.11) $\displaystyle\big{[}\cal W_{0},\cal W_{1}\big{]}$ $\displaystyle=$
$\displaystyle k_{+}\tilde{\cal}Z_{1}-k_{-}\cal Z_{1}\ ,$ (4.12)
$\displaystyle\big{[}\cal W_{0},\cal Z_{1}\big{]}_{q}$ $\displaystyle=$
$\displaystyle-k_{+}\bar{\epsilon}_{+}\ ,\qquad\big{[}\tilde{\cal}Z_{1},\cal
W_{0}\big{]}_{q}=-k_{-}\bar{\epsilon}_{+}\ ,$ (4.13) $\displaystyle\big{[}\cal
W_{1},\tilde{\cal}Z_{1}\big{]}_{q}$ $\displaystyle=$ $\displaystyle-
k_{-}\bar{\epsilon}_{-}\ ,\qquad\big{[}\cal Z_{1},\cal
W_{1}\big{]}_{q}=-k_{+}\bar{\epsilon}_{-}\ ,$ (4.14) $\displaystyle\big{[}\cal
Z_{1},\tilde{\cal}Z_{1}\big{]}$ $\displaystyle=$
$\displaystyle(q-q^{-1})\big{(}\bar{\epsilon}_{+}\cal
W_{1}-\bar{\epsilon}_{-}\cal W_{0}\big{)}\ ,$

where $k_{\pm},\bar{\epsilon}_{\pm}$ are scalars in ${\mathbb{K}}(q)$. Note
that the last relation (4.14) can be derived from (4.11)-(4.13), or
alternatively the first relation (4.11) can be derived from (4.12)-(4.14).
Thus, in each case the defining relations for $\cal A$ correspond to a subset
of (4.11)-(4.14).

######  Lemma 4.1.

The K-operator

(4.15) $\displaystyle\qquad\qquad\tilde{\cal{K}}_{g}(u)$ $\displaystyle=$
$\displaystyle\begin{pmatrix}uq\cal W_{0}-u^{-1}\bar{\epsilon}_{+}&\cal
Z_{1}+\frac{k_{+}qu^{2}}{(q-q^{-1})}\\\
\tilde{\cal}Z_{1}+\frac{k_{-}qu^{2}}{(q-q^{-1})}&uq\cal
W_{1}-u^{-1}\bar{\epsilon}_{-}\end{pmatrix}\ $

satisfies the spectral parameter dependent Freidel-Maillet equation

(4.16) $\displaystyle R(u/v)\ ({\cal{K}}(u)\otimes{\mathbb{I}})\ R^{(0)}\
({\mathbb{I}}\otimes{\cal{K}}(v))\ =\ ({\mathbb{I}}\otimes{\cal{K}}(v))\
R^{(0)}\ ({\cal{K}}(u)\otimes{\mathbb{I}})\ R(u/v)\ \ .$

###### Proof.

Insert (4.15) in (4.16). One extracts the relations independent of $u$, which
are given by (4.11)-(4.14). ∎

Note that (4.16) for the symmetric R-matrix (4.5) can be viewed as a limiting
case of a reflection equation [C84, Sk88]. For a Freidel-Maillet equation of
the form (4.16), a generating function for central elements is given by the
so-called Sklyanin quantum determinant $\Gamma(u)$. By analogy with [Sk88,
Proposition 5], one finds

(4.17) $\displaystyle\Gamma(u)={\rm
tr}_{12}\big{(}P^{-}_{12}({\cal{K}}(u)\otimes{\mathbb{I}})\
R^{(0)}({\mathbb{I}}\otimes{\cal{K}}(uq))\big{)}\ \ $

is such that $\big{[}\Gamma(u),({\cal{K}}(v))_{i,j}\big{]}=0$. A detailed
proof can be found in [B20]. For the K-operator $\tilde{\cal{K}}_{g}(u)$,
computing the r.h.s. of (4.17) one gets:

(4.18)
$\displaystyle\Gamma(u)=u^{2}q^{4}\Gamma_{0}-(q-q^{-1})^{2}q^{2}\Gamma_{1}-k_{+}k_{-}u^{4}q^{6}+(q-q^{-1})^{2}\bar{\epsilon}_{+}\bar{\epsilon}_{-}u^{-2}\
,$

where

(4.19) $\displaystyle\Gamma_{0}$ $\displaystyle=$
$\displaystyle\frac{1}{2}\left((q-q^{-1})^{2}(\cal W_{0}\cal W_{1}+\cal
W_{1}\cal W_{0})-(q^{2}-q^{-2})(k_{+}\tilde{\cal}Z_{1}+k_{-}\cal
Z_{1})\right)\ ,$ (4.20) $\displaystyle\Gamma_{1}$ $\displaystyle=$
$\displaystyle\frac{1}{2}\left(\cal
Z_{1}\tilde{\cal}Z_{1}+\tilde{\cal}Z_{1}\cal
Z_{1}+(q+q^{-1})(\bar{\epsilon}_{+}\cal W_{1}+\bar{\epsilon}_{-}\cal
W_{0})\right)\ ,$

are central in $\cal A$.

Three different specializations
$\varphi_{c},\varphi_{e},\varphi^{\prime}_{c}:\cal A\rightarrow U_{q}(sl_{2})$
are now considered. In the table below, the corresponding images of $\cal
W_{0},\cal W_{1},\cal Z_{1},\tilde{\cal}Z_{1}$, structure constants
$\bar{\epsilon}_{\pm},k_{\pm}$ and central elements $\Gamma_{0},\Gamma_{1}$
are displayed.

${\cal A}$ | $\varphi_{c}({\cal A})$ | $\varphi_{e}({\cal A})$ | $\varphi^{\prime}_{c}({\cal A})$
---|---|---|---
$\cal W_{0}$ | $K$ | $X$ | $E$
$\cal W_{1}$ | $K^{-1}$ | $Y$ | $F$
$\cal Z_{1}$ | $(q-q^{-1})FK$ | $q^{-1}(YX-1)$ | $-(q-q^{-1})^{-1}K$
$\tilde{\cal}Z_{1}$ | $(q-q^{-1})K^{-1}E$ | $-Z$ | $-(q-q^{-1})^{-1}K^{-1}$
$\bar{\epsilon}_{\pm}$ | $1$ | $1$ | $0$
$(k_{+},k_{-})$ | $(0,0)$ | $(0,q-q^{-1})$ | $(1,1)$
$\Gamma_{0}$ | $(q-q^{-1})^{2}$ | $(q-q^{-1})^{2}$ | $\Omega_{c}(q-q^{-1})^{2}$
$\Gamma_{1}$ | $\Omega_{c}(q-q^{-1})^{2}$ | $\Omega_{e}$ | $(q-q^{-1})^{-2}$

Table 1. Three specializations ${\cal A}\rightarrow U_{q}(sl_{2})$

Note that other specializations can be obtained applying $r:\ X\rightarrow Y$,
$Y\rightarrow Z$, $Z\rightarrow X$ to $\varphi_{e}(\cal A)$.

The K-operator (4.15) is now specialized. From Lemma 4.1, Table 1, it follows:

(4.21)
$\displaystyle{\cal{K}}_{c}(u)=\begin{pmatrix}uqK-u^{-1}&(q-q^{-1})FK\\\
(q-q^{-1})K^{-1}E&uqK^{-1}-u^{-1}\end{pmatrix}\ $

or

(4.22)
$\displaystyle{\cal{K}}_{e}(u)=\begin{pmatrix}uqX-u^{-1}&q^{-1}(YX-1)\\\
qu^{2}-Z&uqY-u^{-1}\end{pmatrix}\ $

satisfy (4.16). Note that contrary to (4.9), in both cases (4.21), (4.22), the
entries

(4.23) $\displaystyle({\cal{K}}(u))_{i,j}\in
U_{q}(sl_{2})\otimes{\mathbb{K}}[u,u^{-1}]\ .$

Other examples of equitable K-operators with spectral parameter can be
obtained applying $r$ on (4.22). For instance, denote ${\cal{K}}_{{\cal
B}X}(u)=r(K_{e}(u))$. By cyclicity of (2.22) under the action of $r$, then

(4.24) $\displaystyle{\cal{K}}_{{\cal
B}X}(u)=\begin{pmatrix}uqY-u^{-1}&q^{-1}(ZY-1)\\\
qu^{2}-X&uqZ-u^{-1}\end{pmatrix}\ $

solves (4.16).

The isomorphism (2.23) can be derived using the connection between the
Freidel-Maillet algebra (4.16) and the Yang-Baxter algebra (4.8). Adaptating
[Sk88, Proposition 2], [FM91], to the Freidel-Maillet type equation (4.16),
let ${\cal{K}}_{0}(u)$ be a solution of (4.16). Assume there exists a pair of
quantum Lax operators ${\cal{L}}(u),{\cal{L}}_{0}(u)$ satisfying the exchange
relations:

(4.25) $\displaystyle R(u/v)\ ({\cal{L}}(u)\otimes{\mathbb{I}})\
({\mathbb{I}}\otimes{\cal{L}}(v))$ $\displaystyle=$
$\displaystyle({\mathbb{I}}\otimes{\cal{L}}(v))\
({\cal{L}}(u)\otimes{\mathbb{I}})\ R(u/v)\ ,$ (4.26) $\displaystyle R(u/v)\
({\cal{L}}_{0}(u)\otimes{\mathbb{I}})\ ({\mathbb{I}}\otimes{\cal{L}}_{0}(v))$
$\displaystyle=$ $\displaystyle({\mathbb{I}}\otimes{\cal{L}}_{0}(v))\
({\cal{L}}_{0}(u)\otimes{\mathbb{I}})\ R(u/v)\ ,$ (4.27) $\displaystyle
R^{(0)}\ ({\cal{L}}_{0}(u)\otimes{\mathbb{I}})\
({\mathbb{I}}\otimes{\cal{L}}(v))$ $\displaystyle=$
$\displaystyle({\mathbb{I}}\otimes{\cal{L}}(v))\
({\cal{L}}_{0}(u)\otimes{\mathbb{I}})\ R^{(0)}\qquad\forall u,v\ .$

Then, it is easy to show that the ‘dressed’ K-operator defined by

(4.28)
$\displaystyle\tilde{\cal{K}}(u)={\cal{L}}_{0}(u){\cal{K}}_{0}(u){\cal{L}}(u)$

is also a solution of (4.16). For instance, in addition to (4.9) define

(4.29) $\displaystyle{\cal{L}}_{0}(u)=diag(uK^{1/2},uK^{-1/2})$

and

(4.30) $\displaystyle{\cal{K}}_{0}^{e\rightarrow
c}(u)=\begin{pmatrix}u^{-1}&0\\\ q^{-1}&u^{-1}\end{pmatrix}\ .$

It is checked that ${\cal{K}}_{0}^{e\rightarrow c}(u)$ satisfies (4.16). Also,
by definition (4.25) holds. Then, the relations (4.26)-(4.27) follow as
limiting cases of (4.25). By previous comments,

(4.31)
$\displaystyle{\cal{K}}_{e}(u)\rightarrow{\cal{L}}_{0}(u){\cal{K}}_{0}^{e\rightarrow
c}(u){\cal{L}}(u)\ $

relates (4.16) to (4.25)-(4.27). Computing explicitely the r.h.s of (4.31),
one finds that the resulting expression coincides with the image of (4.22) by
the isomorphism (2.23).

###  4.1. Decomposition of K-operators

The spectral parameter dependent equations (4.7), (4.8), encode the defining
relations of the FRT presentation of $U^{\prime}_{q}(sl_{2})$, see [F96, p.
44] for details. Introduce the standard non-symmetric R-matrix $R^{ns}(u)$:

(4.32) $\displaystyle R^{ns}(u)=uR-u^{-1}R_{21}^{-1}\ .$

It is related with the symmetric R-matrix (4.5) through the similarity
transformation

(4.33) $\displaystyle R_{12}(u/v)$ $\displaystyle=$ $\displaystyle\cal
M(u)_{1}\cal M(v)_{2}R^{ns}_{12}(u/v)\cal M(v)_{2}^{-1}\cal M(u)_{1}^{-1}\
,\quad\mbox{with}\quad{\cal M}(u)=\begin{pmatrix}u^{1/2}&0\\\
0&u^{-1/2}\end{pmatrix}\ .$

Defining ${\cal{L}}^{ns}(u)=\cal M(u){\cal{L}}(u)\cal M(u)^{-1}$ with (4.9),
one has the decomposition
${\cal{L}}^{ns}(u)=uq^{1/2}{\cal{L}}^{+}-u^{-1/2}q^{-1/2}{\cal{L}}^{-}$. In
terms of the new L-operators, the Yang-Baxter relation reads
$R^{ns}(u/v){\cal{L}}^{ns}_{1}(u){\cal{L}}^{ns}_{2}(v)={\cal{L}}^{ns}_{2}(v){\cal{L}}^{ns}_{1}(u)R^{ns}(u/v)$.
Extracting the independent relations in
$R,R_{21}^{-1},{\cal{L}}_{1}^{\pm},{\cal{L}}_{2}^{\pm}$ from the latter, one
recovers the FRT presentation of $U^{\prime}_{q}(sl_{2})$ obtained from
(3.7)-(3.9).

A similar statement holds for the Freidel-Maillet equation (4.16) with
specialized K-operators (4.21) or (4.22). To show that, we apply the same
technique555In the context of non-ultralocal integrable models and braided
Yang-Baxter algebras, this technique has been introduced in [HK96].. For
${\cal{K}}\in\\{{\cal{K}}_{c},{\cal{K}}_{e}\\}$ with (4.21), (4.22), define
the new K-operators:

(4.34) $\displaystyle{\cal{K}}^{ns}(u)=\cal M(u){\cal{K}}(u)\cal M(u)^{-1}\ .$

It is readily checked that they admit the following simple decomposition:

(4.35) $\displaystyle{\cal{K}}^{ns}(u)=uq{\cal{K}}^{+}-u^{-1}{\cal{K}}^{-}\ .$

By (4.33), they satisfy the Freidel-Maillet equation for the non-symmetric
R-matrix:

(4.36) $\displaystyle
R^{ns}(u/v){\cal{K}}^{ns}_{1}(u)R^{(0)}{\cal{K}}^{ns}_{2}(v)\ =\
{\cal{K}}^{ns}_{2}(v)R^{(0)}{\cal{K}}^{ns}_{1}(u)R^{ns}(u/v)\ \ .$

Expanding this equation in $u,v$, one extracts seven equations. Six coincide
with (3.40), (3.41), (3.44), (3.45), but recall that (3.44), (3.45), follow
from the three equations (3.40), (3.41). The remaining equation to show reads

(4.37) $\displaystyle
R{\cal{K}}^{-}_{1}R^{(0)}{\cal{K}}^{+}_{2}-{\cal{K}}^{+}_{2}R^{(0)}{\cal{K}}^{-}_{1}R=R_{21}^{-1}{\cal{K}}^{+}_{1}R^{(0)}{\cal{K}}^{-}_{2}-{\cal{K}}^{-}_{2}R^{(0)}{\cal{K}}^{+}_{1}R_{21}^{-1}\
.$

Using $R-R_{21}^{-1}=(q-q^{-1})P$ and the Freidel-Maillet equations (3.41),
(3.45), by simple computation one finds that this equation is satisfied. Thus,
the independent relations are given by (3.39)-(3.41). We conclude that the
Freidel-Maillet type equation (4.16) encodes the relations (3.40), (3.41) of
Theorem 3.5.

###  4.2. K-matrices of $U_{q}(sl_{2})$ with spectral parameter

In the context of quantum integrable systems, R and K-matrices are the basic
ingredient for the construction of mutually commuting quantities. For the
class of quantum integrable systems generated from the Freidel-Maillet algebra
of Theorem 3.5, in general for irreducible finite dimensional representations
of $U_{q}(sl_{2})$ the corresponding spectral parameter dependent Freidel-
Maillet equation takes the form:

(4.38) $\displaystyle
R_{12}(u/v){\cal{K}}_{13}(u)R^{(0)}_{12}{\cal{K}}_{23}(v)={\cal{K}}_{23}(v)R^{(0)}_{12}{\cal{K}}_{13}(u)R_{12}(u/v)\
\in\ \mathrm{End}({\cal V}_{1}\otimes{\cal V}_{2}\otimes{\cal V}_{3}).$

According to the choice of the Chevalley or equitable presentation of
$U_{q}(sl_{2})$, the simplest examples of K-matrices of $U_{q}(sl_{2})$ with a
spectral parameter take a rather different form. For the one-dimensional (spin
$s=0$) representation, the image of (4.22) or (4.24) produces a non-trivial
solution of (4.38) given by:

(4.39)
$\displaystyle\rho_{V^{(0)}}({\cal{K}}_{e}(u))=\rho_{V^{(0)}}({\cal{K}}_{{\cal
B}X}(u))=\begin{pmatrix}uq-u^{-1}&0\\\ u^{2}q-1&uq-u^{-1}\end{pmatrix}\ .$

For the spin $s=1/2$ representation, from (4.21), (4.22) using (2.20), (2.30),
one gets:

(4.44)
$\displaystyle\rho_{V^{(1/2)}}({\cal{K}}_{c}(u))=\left(\begin{array}[]{cccc}uq^{2}-u^{-1}&0&0&0\\\
0&u-u^{-1}&q(q-q^{-1})&0\\\ 0&q^{-1}(q-q^{-1})&u-u^{-1}&0\\\
0&0&0&uq^{2}-u^{-1}\end{array}\right)\ ,$ (4.49)
$\displaystyle\rho_{V^{(1/2)}}({\cal{K}}_{e}(u))=\left(\begin{array}[]{cccc}uq^{2}-u^{-1}&0&0&0\\\
0&u-u^{-1}&q-q^{-1}&0\\\ u^{2}q-q^{-1}&q-q^{-1}&u-u^{-1}&0\\\
0&q(u^{2}-1)&uq(q-q^{-1})&uq^{2}-u^{-1}\end{array}\right)\ .$

From (4.24), one gets:

(4.54) $\displaystyle\rho_{V^{(1/2)}}({\cal{K}}_{{\cal
B}X}(u))=\left(\begin{array}[]{cccc}u-u^{-1}&0&q^{-1}-q&q^{-1}-q\\\
u(q^{2}-1)&uq^{2}-u^{-1}&q-q^{-1}&q-q^{-1}\\\
q(u^{2}-1)&0&u-u^{-1}&u(1-q^{2})\\\
0&qu^{2}-q^{-1}&0&uq^{2}-u^{-1}\end{array}\right)\ .$

Acknowledgments: I thank Nicolas Crampé for gratefully sharing a MAPLE code
and comments, and Azat Gainutdinov for discussions. I thank Paul Terwilliger
for comments on the manuscript and discussions, Hadewijch De Clercq and Bart
Vlaar for communications and interest in this work. P.B. is supported by
C.N.R.S.

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