# On the second realization for the positive part
of $U_{q}(\widehat{sl_{2}})$ of equitable type

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

The equitable presentation of the quantum algebra $U_{q}(\widehat{sl_{2}})$ is
considered. This presentation was originally introduced by T. Ito and P.
Terwilliger. In this paper, following Terwilliger’s recent works the
(nonstandard) positive part of $U_{q}(\widehat{sl_{2}})$ of equitable type
$U_{q}^{IT,+}$ and its second realization (current algebra) $U_{q}^{T,+}$ are
introduced and studied. A presentation for $U_{q}^{T,+}$ is given in terms of
a K-operator satisfying a Freidel-Maillet type equation and a condition on its
quantum determinant. Realizations of the K-operator in terms of Ding-Frenkel
L-operators are considered, from which an explicit injective homomorphism from
$U_{q}^{T,+}$ to a subalgebra of Drinfeld’s second realization (current
algebra) of $U_{q}(\widehat{sl_{2}})$ is derived, and the comodule algebra
structure of $U_{q}^{T,+}$ is characterized. The central extension of
$U_{q}^{T,+}$ and its relation with Drinfeld’s second realization of
$U_{q}(\widehat{gl_{2}})$ is also described using the framework of Freidel-
Maillet algebras.

MSC: 16T25; 17B37; 81R50.

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

##  1\. Introduction

Originally introduced in [J85, D86], the quantum affine algebra
$U_{q}(\widehat{sl_{2}})$ admits a presentation in terms of generators
$\\{E_{i},F_{i},K_{i}^{\pm 1}|i=0,1\\}$ and relations. In the literature, this
presentation is usually referred as the Drinfeld-Jimbo or Chevalley type
presentation of $U_{q}(\widehat{sl_{2}})$, denoted $U_{q}^{DJ}$. V. Drinfeld
gave also another presentation [D88], the so-called Drinfeld’s second
realization of $U_{q}(\widehat{sl_{2}})$ in terms generators
$\\{{{\textsf{x}}}_{k}^{\pm},{\textsf{h}}_{\ell},{\textsf{K}}^{\pm 1},C^{\pm
1/2}|k\in{\mathbb{Z}},\ell\in{\mathbb{Z}}\backslash\\{0\\}\\}$ and relations,
denoted $U_{q}^{Dr}$. For further analysis, both $U_{q}^{DJ}$ and $U_{q}^{Dr}$
are recalled in Appendix A. A third presentation, initiated by Reshetikhin-
Semenov-Tian-Shansky in [RS90] and denoted $U_{q}^{RS}$, takes the form of a
Faddeev-Reshetikhin-Takhtajan (FRT) type presentation [FRT89]. In this case,
generating functions for the generators of $U_{q}^{Dr}$ (and more generally
Drinfeld’s second realization of $U_{q}(\widehat{gl_{2}})$) are the entries of
the so-called L-operators, see [DF93] for details. In these definitions, note
that the derivation generator is ommited (see [CP94, Remark 2, page 393]). In
the context of mathematics and physics, the presentation $U_{q}^{DJ}$ and
especially $U_{q}^{Dr}$, $U_{q}^{RS}$, have played a crucial role in
developments of quantum affine algebras, conformal field theory and integrable
lattice systems.

In [IT03], T. Ito and P. Terwilliger obtained a fourth presentation of
$U_{q}(\widehat{sl_{2}})$ called ‘equitable’, here denoted $U_{q}^{IT}$, see
Theorem 4.3. It is generated by $\\{y_{i}^{\pm},k_{i}^{\pm}|i=0,1\\}$ subject
to the defining relations (2.1)-(2.3). An explicit isomorphism
$U_{q}^{IT}\rightarrow U_{q}^{DJ}$ is known [IT03], see (2.4)-(2.6). To our
knowledge, the relationship between $U_{q}^{IT}$, $U_{q}^{Dr}$ and
$U_{q}^{RS}$ has not been investigated. As a starting point, in this paper we
consider the subalgebra of $U_{q}^{IT}$ generated by
$\\{y_{0}^{+},y_{1}^{+}\\}$. We denote this subalgebra by $U_{q}^{IT,+}$ and
call it the (nonstandard) positive part of $U_{q}^{IT}$. It is known that
$U_{q}^{IT,+}$ has a presentation by generators $\\{y_{0}^{+},y_{1}^{+}\\}$
subject to the q-Serre relations; see (2.3). In a recent work [T19a], P.
Terwilliger gave a second realization - called ‘alternating’ - for an algebra
with q-Serre defining relations. Adapting the results and notations of [T19a]
to $U_{q}^{IT,+}$, we introduce Terwilliger’s second realization of
$U_{q}^{IT,+}$, denoted $U_{q}^{T,+}$. It has equitable generators
$\\{y_{-k}^{+},y_{k+1}^{+},z^{+}_{k+1},{\tilde{z}}^{+}_{k+1}|{k\in{\mathbb{N}}}\\}$
subject to a set of relations displayed in Theorem 2.7. A PBW basis for
$U_{q}^{T,+}$ is given in Proposition 2.9. For completeness, following [T19b]
the central extension of $U_{q}^{T,+}$, denoted ${\cal U}_{q}^{T,+}$, is
considered in the last section. See Definitions 4.1, 4.2.

The purpose of this paper is to study the relationship between $U_{q}^{T,+}$,
its central extension ${\cal U}_{q}^{T,+}$ and certain subalgebras of
$U_{q}^{RS}$, $U_{q}^{Dr}$ and $U_{q}(\widehat{gl_{2}})$’s counterparts. The
main result is a Freidel-Maillet type presentation [FM91] for $U_{q}^{T,+}$,
see Theorem 2.10. In this presentation, the generators of $U_{q}^{T,+}$ arise
as coefficients of generating functions characterizing the entries of a
K-operator that satisfies a Freidel-Maillet type equation and a quantum
determinant equation. A K-operator that reads as a quadratic combination of
L-operators of $U_{q}^{RS}$ (known in the literature as Sklyanin’s dressed
operators [Sk88]) is derived, see Lemma 3.5 and (3.37). Using this relation
between K and L-operators, the following results are obtained in a
straightforward manner: generating functions of equitable generators of
$U_{q}^{T,+}$ and Drinfeld generators of $U_{q}^{Dr}$ are related, see
Proposition 3.6 and Example 3.7; $U_{q}^{T,+}$ is interpreted as a comodule
algebra. See Proposition 3.10 and Lemma 3.11. Relaxing the condition on the
quantum determinant, the central extension ${\cal U}_{q}^{T,+}$ is studied
along the same lines using a Freidel-Maillet type presentation. See Theorem
4.3, Propositions 4.4, 4.9 and Corollary 4.10.

The text is organized as follows. In Section 2, the equitable presentation
$U_{q}^{IT}$, its nonstandard positive part $U_{q}^{IT,+}$ and Terwilliger’s
second realization $U_{q}^{T,+}$ are introduced. Then, following recent
results [B20] a Freidel-Maillet type presentation for $U_{q}^{T,+}$ is
proposed. In Section 3, the analysis of [B20, Subsection 5.2] is extended:
K-operator solutions of a Freidel-Maillet type equation are constructed, from
which an injective homomorphism
$\nu:U_{q}^{T,+}\rightarrow{U^{\prime}_{q}}^{Dr,\triangleright,+}$ is derived,
where ${U^{\prime}_{q}}^{Dr,\triangleright,+}$ is a subalgebra for
$U_{q}^{Dr}$. Using the Freidel-Maillet type presentation, it is also shown
that $U_{q}^{T,+}$ admits a (left) comodule algebra structure
$\delta:U_{q}^{T,+}\rightarrow{U^{\prime}_{q}}^{Dr,\triangleright,+}\otimes
U_{q}^{T,+}$. For the specialization $C=1$, the image of the equitable
generators by the corresponding (left) coaction map is given. In Section 4,
for completeness a Freidel-Maillet type presentation for ${\cal U}_{q}^{T,+}$
is given. An injective homomorphism $\mu:{\cal
U}_{q}^{T,+}\rightarrow{U^{\prime}_{q}}(\widehat{gl_{2}})^{\triangleright,+}$
is derived, where ${U^{\prime}_{q}}(\widehat{gl_{2}})^{\triangleright,+}$ is a
subalgebra of $U_{q}(\widehat{gl_{2}})$. In particular, in terms of the
Drinfeld’s generators of $U_{q}(\widehat{gl_{2}})$ the image of the quantum
determinant by $\mu$ enjoys a simple factorized structure, see (4.18) or
(4.20). In the last section, the results here presented together with [B20]
are summarized and some perspectives are given.

Nota bene. In a recent paper [Ter21], the relationship between the positive
part of $U_{q}(\widehat{sl_{2}})$ denoted $U_{q}^{+}$ (see comments around eq.
(2.7)) and its central extension ${\cal U}_{q}^{+}$ is studied in details
using the framework of generating functions. Explicit relations between
generating functions in terms of Damiani’s root vectors for $U_{q}^{+}$ and
generating functions for the alternating generators of ${\cal U}_{q}^{+}$ are
obtained. For the choice $\bar{\epsilon}_{\pm}=0$, fixing $\bar{k}_{\pm}$ and
$\lambda$ according to the normalizations chosen in [Ter21] and using Beck’s
correspondence between Drinfeld’s generators and root vectors [Be94] (see
(3.49)-(3.50)), it can be readily checked that the expressions given in
Proposition 4.9 with (3.39)-(3.42), and eq. (4.20) match with the expressions
given in [Ter21, Propositions 9.1, 9.3] and [Ter21, eq. (65)], respectively.

Notation 1. Recall the natural numbers ${\mathbb{N}}=\\{0,1,2,\cdots\\}$ and
integers ${\mathbb{Z}}=\\{0,\pm 1,\pm 2,\cdots\\}$. ${\mathbb{C}}(q)$ 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
$[x]=(q^{x}-q^{-x})/(q-q^{-1})$.

##  2\. The equitable subalgebra $U_{q}^{IT,+}$ and second realization
$U_{q}^{T,+}$

In this section, the equitable presentation of $U_{q}(\widehat{sl_{2}})$
introduced in [IT03] is recalled, and an isomorphism $U_{q}^{IT}\rightarrow
U_{q}^{DJ}$ is displayed. Then, the positive part $U_{q}^{IT,+}$ is
considered. For its second realization $U_{q}^{T,+}$ recently introduced in
[T18], some properties are recalled. Following [B20] a Freidel-Maillet type
presentation is given for $U_{q}^{T,+}$.

######  Theorem 2.1.

[IT03] The quantum affine algebra $U_{q}(\widehat{sl_{2}})$ is isomorphic to
the unital associative $\mathbb{C}(q)$-algebra with equitable generators
$\\{y^{\pm}_{i},k_{i}^{{\pm}1}|i=0,1\\}$ and the following relations:

(2.1) $\displaystyle k_{i}k^{-1}_{i}=k^{-1}_{i}k_{i}$ $\displaystyle=$
$\displaystyle 1,\qquad k_{0}k_{1}\ \ \mbox{central},$ (2.2)
$\displaystyle\frac{\big{[}y^{+}_{i},k_{i}\big{]}_{q}}{q-q^{-1}}$
$\displaystyle=$ $\displaystyle
1,\qquad\frac{\big{[}k_{i},y^{-}_{i}\big{]}_{q}}{q-q^{-1}}=1,\qquad\frac{\big{[}y^{-}_{i},y^{+}_{i}\big{]}_{q}}{q-q^{-1}}=1,\qquad\frac{\big{[}y^{+}_{i},y^{-}_{j}\big{]}_{q}}{q-q^{-1}}=k^{-1}_{0}k^{-1}_{1},\qquad
i\not=j,$ (2.3)
$\displaystyle(y^{\pm}_{i})^{3}y^{\pm}_{j}-[3]_{q}(y^{\pm}_{i})^{2}y^{\pm}_{j}y^{\pm}_{i}+[3]_{q}y^{\pm}_{i}y^{\pm}_{j}(y^{\pm}_{i})^{2}-y^{\pm}_{j}(y^{\pm}_{i})^{3}=0,\qquad
i\not=j.$

We call $U_{q}^{IT}$ the Ito-Terwilliger or equitable presentation of
$U_{q}(\widehat{sl_{2}})$.

An isomorphism $U_{q}^{IT}\rightarrow U_{q}^{DJ}$ is given in [IT03], where
$U_{q}^{DJ}$ is the Drinfeld-Jimbo presentation of $U_{q}(\widehat{sl_{2}})$
recalled in Appendix A. Namely,

(2.4) $\displaystyle k^{\pm 1}_{i}$ $\displaystyle\mapsto$ $\displaystyle
K^{\pm 1}_{i}\ ,$ (2.5) $\displaystyle y^{-}_{i}$ $\displaystyle\mapsto$
$\displaystyle K^{-1}_{i}+(q-q^{-1})F_{i}\ ,$ (2.6) $\displaystyle y^{+}_{i}$
$\displaystyle\mapsto$ $\displaystyle K^{-1}_{i}-q(q-q^{-1})K^{-1}_{i}E_{i}\
.$

In this paper, we focus on the following subalgebra.

######  Definition 2.2.

$U_{q}^{IT,+}$ is the subalgebra of $U_{q}^{IT}$ generated by
$\\{y^{+}_{0},y^{+}_{1}\\}$. We call $U_{q}^{IT,+}$ the positive part of
$U_{q}(\widehat{sl_{2}})$ of equitable type.

By (2.3), this subalgebra has a presentation by generators
$\\{y^{+}_{0},y^{+}_{1}\\}$ subject to the q-Serre relations. Let
$U_{q}^{DJ,+}$ (resp. $U_{q}^{DJ,-}$) denote the subalgebra of $U_{q}^{DJ}$
generated by $E_{0},E_{1}$ (resp. $F_{0},F_{1}$); See Appendix A. In the
literature, $U_{q}^{DJ,+}$ (resp. $U_{q}^{DJ,-}$) is usually called the
positive (resp. negative) part of $U_{q}(\widehat{sl_{2}})$. For this reason,
the definition above of positive part of $U_{q}(\widehat{sl_{2}})$ is
nonstandard. The negative part of $U_{q}(\widehat{sl_{2}})$ of equitable type
- denoted by $U_{q}^{IT,-}$ \- can be introduced similarly. It is generated by
$\\{y_{0}^{-},y_{1}^{-}\\}$. Another subalgebra is the ‘Cartan part’ denoted
$U_{q}^{IT,0}$, generated by $\\{k^{\pm 1}_{0},k^{\pm 1}_{1}\\}$.

######  Definition 2.3.

${U^{\prime}_{q}}^{DJ,+}$ (resp. ${U^{\prime}_{q}}^{DJ,-}$) denotes the
subalgebra of $U_{q}^{DJ}$ generated by $U_{q}^{DJ,+}$ (resp. $U_{q}^{DJ,-}$)
and $\\{K^{\pm 1}_{0},K^{\pm 1}_{1}\\}$.

By (A.1), (2.3), it follows:

######  Remark 2.4.

An injective homomorphism $U_{q}^{IT,+}\rightarrow{U^{\prime}}_{q}^{DJ,+}$ is
given by (2.6).

From the point of view of generators and relations, $U_{q}^{DJ,+}$ and
$U_{q}^{IT,+}$ are exactly the same, up to isomorphism. However, it is seen
that their embeddings into ${U_{q}}^{DJ}$ essentially differ. To avoid any
confusion in further discussions, let us introduce the algebra $U_{q}^{+}$
with fundamental generators $A,B$ and q-Serre defining relations:

(2.7)
$\displaystyle\big{[}A,\big{[}A,\big{[}A,B\big{]}_{q}\big{]}_{q^{-1}}\big{]}=0\
,\quad\big{[}B,\big{[}B,\big{[}B,A\big{]}_{q}\big{]}_{q^{-1}}\big{]}=0\ .$

According to previous definitions,

######  Lemma 2.5.

There exists an algebra isomorphism $U_{q}^{+}\rightarrow U_{q}^{DJ,+}$ that
sends $A\mapsto E_{0}$ and $B\mapsto E_{1}$.

######  Lemma 2.6.

There exists an algebra isomorphism $U_{q}^{+}\rightarrow U_{q}^{IT,+}$ that
sends $A\mapsto y^{+}_{0}$ and $B\mapsto y_{1}^{+}$.

For $U_{q}^{+}$, Terwilliger recently gave a new presentation called
alternating. The alternating presentation consists of infinitly many countable
alternating elements called the alternating generators satisfying certain
relations [T18, T19a].

For $A\mapsto E_{0}$ and $B\mapsto E_{1}$, the alternating presentation
produces a new ‘current’ realization for $U_{q}^{DJ,+}$ besides the known one
in terms of Drinfeld generators and relations [Be94]. In this case, an
explicit isomorphism between Terwilliger’s alternating algebra and certain
alternating subalgebras of $U_{q}^{Dr}$ is established in [B20]. For the
precise relation between the alternating and Drinfeld’s generators, see [B20,
Subsection 5.2.3]. Using the correspondence (A.13), in particular one finds
$A\mapsto{\textsf{x}}_{1}^{-}{\textsf{K}}^{-1}$ and
$B\mapsto{\textsf{x}}_{0}^{+}$.

For $A\mapsto y^{+}_{0}$ and $B\mapsto y_{1}^{+}$, the alternating
presentation produces similarly a new realization for $U_{q}^{IT,+}$.

######  Theorem 2.7.

(see [T19a]) $U_{q}^{IT,+}$ is isomorphic to the unital associative
$\mathbb{C}(q)$-algebra with equitable generators
$\\{y_{-k}^{+},y_{k+1}^{+},z_{k+1}^{+},{\tilde{z}}_{k+1}^{+}|k\in{\mathbb{N}}\\}$
subject to the following relations

(2.8)
$\displaystyle[{y}^{+}_{1},{y}^{+}_{-k}]=[{y}^{+}_{k+1},{y}^{+}_{0}]=\frac{({z}^{+}_{k+1}-\tilde{z}^{+}_{k+1})}{q+q^{-1}},$
(2.9)
$\displaystyle[{y}^{+}_{1},\tilde{z}^{+}_{k+1}]_{q}=[{z}^{+}_{k+1},{y}^{+}_{1}]_{q}=\bar{\rho}{y}^{+}_{k+2},$
(2.10)
$\displaystyle[\tilde{z}^{+}_{k+1},{y}^{+}_{0}]_{q}=[{y}^{+}_{0},{z}^{+}_{k+1}]_{q}=\bar{\rho}{y}^{+}_{-k-1},$
(2.11)
$\displaystyle[{y}^{+}_{k+1},{y}^{+}_{\ell+1}]=0,\qquad[{y}^{+}_{-k},{y}^{+}_{-\ell}]=0,$
(2.12)
$\displaystyle[{y}^{+}_{k+1},{y}^{+}_{-\ell}]+[{y}^{+}_{-k},{y}^{+}_{\ell+1}]=0,$
(2.13)
$\displaystyle[{y}^{+}_{k+1},\tilde{z}^{+}_{\ell+1}]+[\tilde{z}^{+}_{k+1},{y}^{+}_{\ell+1}]=0,$
(2.14)
$\displaystyle[{y}^{+}_{k+1},{z}^{+}_{\ell+1}]+[{z}^{+}_{k+1},{y}^{+}_{\ell+1}]=0,$
(2.15)
$\displaystyle[{y}^{+}_{-k},\tilde{z}^{+}_{\ell+1}]+[\tilde{z}^{+}_{k+1},{y}^{+}_{-\ell}]=0,$
(2.16)
$\displaystyle[{y}^{+}_{-k},{z}^{+}_{\ell+1}]+[{z}^{+}_{k+1},{y}^{+}_{-\ell}]=0,$
(2.17)
$\displaystyle[\tilde{z}^{+}_{k+1},\tilde{z}^{+}_{\ell+1}]=0,\qquad[{z}^{+}_{k+1},{z}^{+}_{\ell+1}]=0,$
(2.18)
$\displaystyle[{z}^{+}_{k+1},\tilde{z}^{+}_{\ell+1}]+[\tilde{z}^{+}_{k+1},{z}^{+}_{\ell+1}]=0\
,$

and the condition ($z^{+}_{0}=\tilde{z}^{+}_{0}=\bar{\rho}/(q-q^{-1})$):

(2.19)
$\displaystyle\bar{\rho}(q+q^{-1})\sum_{k=0}^{n}q^{-n+2k}y_{k+1}^{+}y_{-n+k}^{+}-\sum_{k=0}^{n+1}q^{2k-n-1}z_{k}^{+}\tilde{z}_{n+1-k}^{+}=0\
,\quad n\geq 0\ ,$

with

(2.20) $\displaystyle\bar{\rho}=q^{-1}(q^{2}-q^{-2})^{2}\ .$

This algebra is denoted $U_{q}^{T,+}$. We call $U_{q}^{T,+}$ Terwilliger’s
second realization of $U_{q}^{IT,+}$. For a proof of the above Theorem, we
refer the reader to [T19a, T19b] for all details. Compared with the
conventions in [T19a, T19b], the following substitutions are considered:

$\displaystyle y^{+}_{k+1}\rightarrow W_{-k}\ ,\quad y^{+}_{-k}\rightarrow
W_{k+1}\ ,$ $\displaystyle z^{+}_{k+1}\rightarrow q^{-1}(q^{2}-q^{-2})G_{k+1}\
,\quad\tilde{z}^{+}_{k+1}\rightarrow q^{-1}(q^{2}-q^{-2}){\tilde{G}}_{k+1}\ ,$
$\displaystyle\bar{\rho}\rightarrow q^{-1}(q^{2}-q^{-2})(q-q^{-1})\ .$

######  Remark 2.8.

The relations (2.8)-(2.18) coincide with the defining relations for the
alternating central extension of $U_{q}^{+}$, denoted ${\cal U}_{q}^{+}$, see
[T19b, Definition 3.1]. To get $U_{q}^{T,+}$ from ${\cal U}_{q}^{+}$, the
additional relation (2.19) is asserted, see [T19b, Lemma 2.8].

Note that there exists an automorphism $\sigma$ and an antiautomorphism $S$
(see [T19a, Proposition 5.3]) such that:

(2.21) $\displaystyle\sigma:$ $\displaystyle{y}^{+}_{-k}\mapsto{y}^{+}_{k+1}\
,\quad{y}^{+}_{k+1}\mapsto{y}^{+}_{-k}\
,\quad{z}^{+}_{k+1}\mapsto\tilde{z}^{+}_{k+1}\
,\quad\tilde{z}^{+}_{k+1}\mapsto{z}^{+}_{k+1}\ ,$ (2.22) $\displaystyle S:$
$\displaystyle{y}^{+}_{-k}\mapsto{y}^{+}_{-k}\
,\quad{y}^{+}_{k+1}\mapsto{y}^{+}_{k+1}\
,\quad{z}^{+}_{k+1}\mapsto\tilde{z}^{+}_{k+1}\
,\quad\tilde{z}^{+}_{k+1}\mapsto{z}^{+}_{k+1}\ .$

The following proposition is a straightforward adaptation of [T19b, Theorem
10.2].

######  Proposition 2.9.

A PBW basis for $U_{q}^{T,+}$ is obtained by its equitable generators

$\displaystyle\\{y^{+}_{-k}\\}_{k\in{\mathbb{N}}}\
,\quad\\{z^{+}_{n+1}\\}_{n\in{\mathbb{N}}}\
,\quad\\{y^{+}_{\ell+1}\\}_{\ell\in{\mathbb{N}}}$

in any linear order $<$ that satisfies

$\displaystyle y^{+}_{-k}<z^{+}_{n+1}<y^{+}_{\ell+1}\ ,\qquad
k,\ell,n\in{\mathbb{N}}\ .$

Combining $\sigma$, $S$, other examples of PBW bases can be obtained.

The equitable generators of $U_{q}^{T,+}$ are polynomials in
$y_{0}^{+},y_{1}^{+}$. Explicit expressions are obtained recursively adapting
[T19a, Lemma 2.9]. For instance, besides $y_{0}^{+},y_{1}^{+}$, the first
generators read:

(2.24) $\displaystyle\
z^{+}_{1}=q{y^{+}_{1}}y^{+}_{0}-q^{-1}{y^{+}_{0}}y^{+}_{1}\ ,$ $\displaystyle
y^{+}_{-1}=\frac{1}{\bar{\rho}}\left((q^{2}+q^{-2})y_{0}^{+}y_{1}^{+}y_{0}^{+}-(y_{0}^{+})^{2}y^{+}_{1}-y^{+}_{1}(y_{0}^{+})^{2}\right)\
$

and $y^{+}_{2}=\sigma(y^{+}_{-1})$, ${\tilde{z}}^{+}_{1}=\sigma(z^{+}_{1})$.
Thus, the algebra $U_{q}^{T,+}$ has a natural $\mathbb{N}^{2}$-grading. Define
${\rm deg}:U_{q}^{T,+}\rightarrow\mathbb{N}\times\mathbb{N}$. For instance,
${\rm deg}(y^{+}_{0})=(1,0)$ and ${\rm deg}(y^{+}_{1})=(0,1)$. More generally,
${\rm deg}(y^{+}_{-k})=(k+1,k)$, ${\rm deg}(y^{+}_{k+1})=(k,k+1)$, ${\rm
deg}(z^{+}_{k+1})={\rm deg}(\tilde{z}^{+}_{k+1})=(k+1,k+1)$.

The algebra $U_{q}^{T,+}$ admits a presentation in the form of a quadratic
algebra of Freidel-Maillet type [FM91], which can be viewed as a limiting case
of a reflection algebra introduced in the context of boundary quantum inverse
scattering theory [C84, Sk88]. Let $R(u)$ be the quantum $R-$matrix defined by
[Ba82]

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

where $u$ is an indeterminate, called ‘spectral parameter’ in the literature
on integrable systems, and deformation parameter $q$. It is known that $R(u)$
satisfies the quantum Yang-Baxter equation in the space ${\cal
V}_{1}\otimes{\cal V}_{2}\otimes{\cal V}_{3}$, with ${\cal
V}\equiv{\mathbb{C}}^{2}$. Using the standard notation

(2.30) $\displaystyle R_{ij}(u)\in\mathrm{End}({\cal V}_{i}\otimes{\cal
V}_{j}),$

the Yang-Baxter equation reads

(2.31) $\displaystyle
R_{12}(u/v)R_{13}(u)R_{23}(v)=R_{23}(v)R_{13}(u)R_{12}(u/v)\ .$

As usual, intoduce the permutation operator $P=R(1)/(q-q^{-1})$. Here, note
that $R_{12}(u)=PR_{12}(u)P=R_{21}(u)$. In addition to (2.29), define:

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

Define the generating functions:

(2.33)
$\displaystyle{{\cal{Y}}}_{+}(u)=\sum_{k\in{\mathbb{N}}}{y}^{+}_{k+1}U^{-k-1}\
,\quad{{\cal{Y}}}_{-}(u)=\sum_{k\in{\mathbb{N}}}{y}^{+}_{-k}U^{-k-1}\ ,$
(2.34)
$\displaystyle\quad{{\cal{Z}}}_{+}(u)=\sum_{k\in{\mathbb{N}}}{\tilde{z}}^{+}_{k+1}U^{-k-1}\
,\quad{{\cal{Z}}}_{-}(u)=\sum_{k\in{\mathbb{N}}}{z}^{+}_{k+1}U^{-k-1}\ ,$

where the shorthand notation $U=qu^{2}/(q+q^{-1})$ is used. Let
$\bar{k}_{\pm}\in{\mathbb{C}}(q)$ such that

(2.35) $\displaystyle\bar{\rho}={\bar{k}}_{+}{\bar{k}}_{-}(q+q^{-1})^{2}\ .$

For the alternating central extension of $U_{q}^{+}$, a Freidel-Maillet type
presentation has been proposed in [B20, Theorem 3.1]. It is given in terms of
a K-operator satisfying a Freidel-Maillet type equation. To get $U_{q}^{+}$, a
condition on the quantum determinant of the K-operator is required.

######  Theorem 2.10.

The algebra $U_{q}^{T,+}$ has a presentation of Freidel-Maillet type. Let
$K(u)$ be a square matrix such that

(2.36) $\displaystyle
K(u)=\begin{pmatrix}uq{\cal{Y}}_{+}(u)&\frac{1}{{\bar{k}}_{-}(q+q^{-1})}{\cal{Z}}_{+}(u)+\frac{{\bar{k}}_{+}(q+q^{-1})}{(q-q^{-1})}\\\
\frac{1}{{\bar{k}}_{+}(q+q^{-1})}{\cal{Z}}_{-}(u)+\frac{{\bar{k}}_{-}(q+q^{-1})}{(q-q^{-1})}&uq{\cal{Y}}_{-}(u)\end{pmatrix}$

with (2.33)-(2.34). The defining relations are given by:

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

and111As usual, ‘$\rm tr_{12}$’ stands for the trace over $\cal
V_{1}\otimes\cal V_{2}$. Also, we denote $P^{-}_{12}=(1-P)/2$.

(2.38) $\displaystyle\text{\rm
tr}_{12}\big{(}P^{-}_{12}(K(u)\otimes{\mathbb{I}})\
R^{(0)}({\mathbb{I}}\otimes K(uq))\big{)}=-\frac{\bar{\rho}}{(q-q^{-1})^{2}}\
.$

###### Proof.

By specializing some results of [B20], the proof follows. The first part of
the proof concerns the equivalence between (2.8)-(2.18) and (2.37). Recall the
defining relations of the alternating central extension of $U_{q}^{+}$ (i.e.
${\cal U}_{q}^{+}$) given in [B20, Definition 2.1]. Observe that they coincide
with the subset of relations (2.8)-(2.18) upon the substitution:

(2.39) $\displaystyle{\textsf{W}}_{-k}\rightarrow y^{+}_{k+1}\
,\qquad{\textsf{W}}_{k+1}\rightarrow y^{+}_{-k}\ ,$ (2.40)
$\displaystyle{\textsf{G}}_{k+1}\rightarrow{\tilde{z}}^{+}_{k+1},\qquad\tilde{{\textsf{G}}}_{k+1}\rightarrow
z^{+}_{k+1}\ .$

Now, by [B20, Theorem 3.1] it is known that ${\cal U}_{q}^{+}$ admits a
Freidel-Maillet type presentation given by a K-operator satisfying (2.37). So,
for the K-operator (2.36), the relations (2.8)-(2.18) are equivalent to
(2.37).

The second part of the proof concerns the equivalence between (2.19) and
(2.38). By [B20, Proposition 3.3], the l.h.s of (2.38) is the so-called
quantum determinant that generates the center of ${\cal U}_{q}^{+}$. For
convenience, define

(2.41) $\displaystyle{\cal
C}(u)=(q-q^{-1})u^{2}q^{2}{\cal{Y}}_{+}(u){\cal{Y}}_{-}(uq)-\frac{(q-q^{-1})}{\bar{\rho}}{\cal{Z}}_{-}(u){\cal{Z}}_{+}(uq)-{\cal{Z}}_{-}(u)-{\cal{Z}}_{+}(uq)\
.$

Inserting (2.36) into the l.h.s. of (2.38), the quantum determinant reduces
to:

(2.42) $\displaystyle\text{\rm
tr}_{12}\big{(}P^{-}_{12}(K(u)\otimes{\mathbb{I}})\
R^{(0)}({\mathbb{I}}\otimes K(uq))\big{)}=\frac{1}{2(q-q^{-1})}\left({\cal
C}(u)+\sigma({\cal C}(u))-\frac{2\bar{\rho}}{(q-q^{-1})}\right)\ .$

Using the exchange relations between the generating functions (2.33)-(2.34)
extracted from (2.37), one shows $\sigma({\cal C}(u))={\cal C}(u)$. Thus, the
condition (2.38) is equivalent to:

(2.43) $\displaystyle{\cal C}(u)=0\ .$

Extracting the set of constraints on the coefficients of the generating
function ${\cal C}(u)$, one gets (2.19). ∎

Note that eqs. (2.37), (2.38), are left invariant under the transformation
$(u,v)\mapsto(\lambda u,\lambda v)$ for $\lambda$ invertible and
$[\lambda,U_{q}^{T,+}]=0$. This property will be used in further analysis.

##  3\. Relating Terwilliger’s and Drinfeld’s second realizations

It is natural to ask for the precise relationship between the equitable and
Drinfeld’s generators. As shown in this section, the Freidel-Maillet type
presentation of Theorem 2.10 combined with the framework of FRT algebras
[FRT89, RS90, DF93] gives a suitable framework for answering this question. In
addition, it provides a tool for constructing left or right coaction maps that
ensure a comodule algebra structure for $U_{q}^{T,+}$.

Below, as a preliminary the FRT presentation for $U_{q}^{Dr}$ is first
recalled, and Drinfeld type ‘alternating’ subalgebras
$\\{U_{q}^{Dr,a,\pm}\\}$, their extensions $\\{{U^{\prime}_{q}}^{Dr,a,\pm}\\}$
for $a=\triangleright,\triangleleft$, are introduced. Then, a K-operator
satisfying a Freidel-Maillet type equation is constructed, and used to derive
an injective homomorphism $\nu:\
U_{q}^{T,+}\rightarrow{U^{\prime}_{q}}^{Dr,\triangleright,+}$. Using the
comodule algebra structure of the Freidel-Maillet type presentation, a left
coaction map $\delta:\
U_{q}^{T,+}\rightarrow{U^{\prime}_{q}}^{Dr,\triangleright,+}\otimes
U_{q}^{T,+}$ is also derived. For the specialization $\bar{\delta}:\
U_{q}^{T,+}\rightarrow{U^{\prime}_{q}}^{Dr,\triangleright,+}/_{C=1}\otimes
U_{q}^{T,+}$ the image of the generating functions for the equitable
generators (2.33), (2.34) is given.

###  3.1. FRT presentation

For the quantum affine Lie algebra $U_{q}(\widehat{gl_{2}})$, a FRT
presentation is known [RS90, DF93]. Define the $R$-matrix:

(3.1) $\tilde{R}(z)=\begin{pmatrix}1&0&0&0\\\
0&\frac{z-1}{zq-q^{-1}}&\frac{z(q-q^{-1})}{zq-q^{-1}}&0\\\
0&\frac{(q-q^{-1})}{zq-q^{-1}}&\frac{z-1}{zq-q^{-1}}&0\\\
0&0&0&1\end{pmatrix}\ $

where $z$ is an indeterminate. It is known that $\tilde{R}(z)$ satisfies the
quantum Yang-Baxter equation

(3.2)
$\displaystyle\tilde{R}_{12}(z_{1}/z_{2})\tilde{R}_{13}(z_{1})\tilde{R}_{23}(z_{2})=\tilde{R}_{23}(z_{2})\tilde{R}_{13}(z_{1})\tilde{R}_{12}(z_{1}/z_{2})\
.$

In terms of ${\tilde{R}}(z)$, the permutation operator reads $P=\tilde{R}(1)$.
Note that $\tilde{R}_{12}(z)=\tilde{R}_{21}^{t_{1}t_{2}}(z)$.

######  Theorem 3.1.

[RS90, DF93] $U_{q}(\widehat{gl_{2}})$ admits a FRT presentation given by a
unital associative algebra with generators
$\\{{{\textsf{x}}}_{k}^{\pm},{\textsf{k}}^{+}_{j,-\ell},{\textsf{k}}^{-}_{j,\ell},q^{\pm
c/2}|k\in{\mathbb{Z}},\ell\in{\mathbb{N}},j=1,2\\}$. The generators $q^{\pm
c/2}$ are central and mutally inverse. Define:

(3.3) $\displaystyle
L^{\pm}(z)=\begin{pmatrix}{\textsf{k}}_{1}^{\pm}(z)&{\textsf{k}}_{1}^{\pm}(z){\textsf{f}}^{\pm}(z)\\\
{\textsf{e}}^{\pm}(z){\textsf{k}}_{1}^{\pm}(z)&{\textsf{k}}_{2}^{\pm}(z)+{\textsf{e}}^{\pm}(z){\textsf{k}}_{1}^{\pm}(z){\textsf{f}}^{\pm}(z)\end{pmatrix}\
$

in terms of the generating functions in the indeterminate $z$:

(3.4) $\displaystyle{{\textsf{e}}}^{+}(z)$ $\displaystyle=$
$\displaystyle(q-q^{-1})\sum_{k=0}^{\infty}q^{k(c/2-1)}{{\textsf{x}}}_{-k}^{-}z^{k}\
,\quad{{\textsf{e}}}^{-}(z)=-(q-q^{-1})\sum_{k=1}^{\infty}q^{k(c/2+1)}{{\textsf{x}}}^{-}_{k}z^{-k}\
,$ (3.5) $\displaystyle{{\textsf{f}}}^{+}(z)$ $\displaystyle=$
$\displaystyle(q-q^{-1})\sum_{k=1}^{\infty}q^{-k(c/2+1)}{{\textsf{x}}}^{+}_{-k}z^{k}\
,\quad{{\textsf{f}}}^{-}(z)=-(q-q^{-1})\sum_{k=0}^{\infty}q^{-k(c/2-1)}{{\textsf{x}}}^{+}_{k}z^{-k}\
,$ (3.6) $\displaystyle\quad{{\textsf{k}}}_{j}^{+}(z)$ $\displaystyle=$
$\displaystyle\sum_{k=0}^{\infty}{{{\textsf{k}}}}^{+}_{j,-k}z^{k}\ \
,\quad\qquad\qquad{{\textsf{k}}}_{j}^{-}(z)=\sum_{k=0}^{\infty}{{{\textsf{k}}}}^{-}_{j,k}z^{-k}\
,\quad j=1,2\ .$

The defining relations are the following:

(3.7) $\displaystyle{\textsf{k}}^{+}_{i,0}{\textsf{k}}^{-}_{i,0}$
$\displaystyle=$ $\displaystyle{\textsf{k}}^{-}_{i,0}{\textsf{k}}^{+}_{i,0}=1\
,$ (3.8) $\displaystyle\tilde{R}(z/w)\ (L^{\pm}(z)\otimes{\mathbb{I}})\
({\mathbb{I}}\otimes L^{\pm}(w))$ $\displaystyle=$
$\displaystyle({\mathbb{I}}\otimes L^{\pm}(w))\
(L^{\pm}(z)\otimes{\mathbb{I}})\ \tilde{R}(z/w)\ ,$ (3.9)
$\displaystyle\tilde{R}(q^{c}z/w)\ (L^{+}(z)\otimes{\mathbb{I}})\
({\mathbb{I}}\otimes L^{-}(w))$ $\displaystyle=$
$\displaystyle({\mathbb{I}}\otimes L^{-}(w))\ (L^{+}(z)\otimes{\mathbb{I}})\
\tilde{R}(q^{-c}z/w)\ .$

For (3.8), the expansion direction of $\tilde{R}(z/w)$ can be chosen in $z/w$
or $w/z$, but for (3.8) the expansion direction is only in $z/w$. The Hopf
algebra structure is characterized as follows. The coproduct222The index $[j]$
characterizes the ‘quantum space’ $V_{[j]}$ on which the entries of
$L^{\pm}(z)$ act. With respect to the ordering $V_{[1]}\otimes V_{[2]}$, one
has:  (3.10)
$\displaystyle((T)_{[\textsf{1}]}(T^{\prime})_{[\textsf{2}]})_{ij}=\sum_{k=1}^{2}(T)_{ik}\otimes(T^{\prime})_{kj}\
.$ $\Delta$, antipode ${\cal S}$ and counit ${\cal E}$ are such that:

(3.11) $\displaystyle\Delta(L^{\pm}(z))=(L^{\pm}(zq^{\pm(1\otimes
c/2)}))_{[\textsf{1}]}(L^{\pm}(zq^{\mp(c/2\otimes 1)}))_{[\textsf{2}]}\ ,$
(3.12) $\displaystyle{\cal S}(L^{\pm}(z))=L^{\pm}(z)^{-1}\ ,\quad{\cal
E}(L^{\pm}(z))={\mathbb{I}}\ .$

The complete isomorphism between the FRT presentation of Theorem 3.1 and
Drinfeld second presentation of $U_{q}(\widehat{gl_{2}})$ is given in [GJ02,
Section 4] (see also [FMu02]). Following [GJ02, Section 4], introduce the
generating functions

(3.13)
$\displaystyle{\textsf{k}}_{i}^{\pm}(z)={\textsf{k}}_{i,0}^{\pm}\exp\left(\pm(q-q^{-1})\sum_{n=1}^{\infty}a_{i,\mp
n}z^{\pm n}\right)\ $

in terms of the new generators $a_{i,\mp n}$. In terms of Drinfeld generators
${\textsf{h}}_{m}$, the new generators $a_{1,m},a_{2,m}$ decompose as:

(3.14) $\displaystyle
a_{1,m}=\frac{1}{q^{m}+q^{-m}}({\textsf{h}}_{m}+\gamma_{m})\ ,\qquad
a_{2,m}=-\frac{1}{q^{m}+q^{-m}}(q^{2m}{\textsf{h}}_{m}-\gamma_{m})\ \ .$

where $\gamma_{m}$ are central elements of $U_{q}(\widehat{gl_{2}})$. For our
purpose, introduce the surjective map
$\gamma^{\prime}_{D}:U_{q}(\widehat{gl_{2}})\rightarrow U_{q}^{Dr}$ that is
defined as follows. Let $\gamma^{\prime}_{m}$ be Laurent polynomials in
$C^{1/2}$, that will be specified later on. We define:

(3.15) $\displaystyle\gamma^{\prime}_{D}(q^{c/2})\mapsto C^{1/2}\ ,$ (3.16)
$\displaystyle\gamma^{\prime}_{D}({\textsf{x}}_{k}^{\pm})\mapsto{\textsf{x}}_{k}^{\pm}\
,$ (3.17)
$\displaystyle\gamma^{\prime}_{D}(a_{1,m})\mapsto\frac{1}{q^{m}+q^{-m}}({\textsf{h}}_{m}+\gamma^{\prime}_{m})\
,\qquad\gamma_{D}(a_{2,m})\mapsto-\frac{1}{q^{m}+q^{-m}}(q^{2m}{\textsf{h}}_{m}-\gamma^{\prime}_{m})\
,$ (3.18)
$\displaystyle\gamma^{\prime}_{D}({\textsf{k}}_{2,0}^{\mp}({\textsf{k}}_{1,0}^{\mp})^{-1})\mapsto{\textsf{K}}^{\pm
1}\
,\qquad\quad\gamma_{D}({\textsf{k}}_{1,0}^{\pm}{\textsf{k}}_{2,0}^{\pm})\mapsto
1\ .$

Note that the map $\gamma^{\prime}_{D}$ slightly differs from the map chosen
in [B20, eq. (5.62)-(5.65)].

###  3.2. Alternating subalgebras of $U_{q}^{Dr}$

Certain ‘alternating’ subalgebras of $U_{q}(\widehat{sl_{2}})$ have been
introduced in [B20], that are now reviewed for further analysis.

######  Definition 3.2.

$\displaystyle U_{q}^{Dr,\triangleright,\pm}$ $\displaystyle=$
$\displaystyle\\{C^{\mp
k/2}{\textsf{K}}^{-1}{\textsf{x}}_{k}^{\pm},C^{\pm(k+1)/2}{\textsf{x}}^{\mp}_{k+1},{\textsf{h}}_{k+1}|k\in{\mathbb{N}}\\}\
,$ $\displaystyle U_{q}^{Dr,\triangleleft,\pm}$ $\displaystyle=$
$\displaystyle\\{C^{\mp
k/2}{\textsf{x}}_{-k}^{\pm},C^{\pm(k+1)/2}{\textsf{x}}^{\mp}_{-k-1}{\textsf{K}},{\textsf{h}}_{-k-1}|k\in{\mathbb{N}}\\}\
.$

We call $U_{q}^{Dr,\triangleright,\pm}$ and $U_{q}^{Dr,\triangleleft,\pm}$ the
right and left alternating subalgebras of $U_{q}^{Dr}$. The subalgebra
generated by $\\{{\textsf{K}}^{\pm 1},C^{\pm 1/2}\\}$ is denoted
$U_{q}^{Dr,\diamond}$.

The defining relations of the alternating subalgebras are identified using
(A.4)-(A.9). Consider for instance $U_{q}^{Dr,\triangleright,+}$. If we denote
$A_{k}^{+}=C^{-k/2}{\textsf{K}}^{-1}{\textsf{x}}_{k}^{+}$,
$A_{\ell}^{-}=C^{\ell/2}{\textsf{x}}_{\ell}^{-}$ and
$B_{k}={\textsf{K}}^{-1}\psi_{k}$, using the relations in Appendix A one gets
the defining relations:

(3.19) $\displaystyle\big{[}{\textsf{h}}_{k},{\textsf{h}}_{\ell}\big{]}=0\
,\quad\big{[}{\textsf{h}}_{k},B_{\ell}\big{]}=0\ ,$ (3.20)
$\displaystyle\big{[}{\textsf{h}}_{k},A_{\ell}^{\pm}\big{]}=\pm\frac{\big{[}2k\big{]}_{q}}{k}A_{k+\ell}^{\pm}\
,$ (3.21) $\displaystyle A^{\pm}_{k+1}A^{\pm}_{\ell}-q^{\pm
2}A^{\pm}_{\ell}A^{\pm}_{k+1}=q^{\pm
2}A^{\pm}_{k}A^{\pm}_{\ell+1}-A^{\pm}_{\ell+1}A^{\pm}_{k}\ ,$ (3.22)
$\displaystyle\big{[}A^{+}_{k},A^{-}_{\ell}\big{]}_{q^{-1}}=\frac{q^{-1}B_{k+\ell}}{q-q^{-1}}\
.$

The defining relations of the other alternating subalgebras can be similarly
written.

For $U_{q}(\widehat{sl_{2}})$, it is known that given a certain ordering the
elements
$\\{{{\textsf{x}}}_{k}^{\pm},{\textsf{h}}_{\ell},{\textsf{K}}^{\pm},C^{\pm
1/2}\\}$ generate a PBW basis. See [Be94, Proposition 6.1] with [BCP98, Lemma
1.5]. For the alternating subalgebras, PBW bases follow naturally. If one
considers the subalgebra $U_{q}^{Dr,\triangleright,+}$, let us choose the
ordering:

(3.23) $\displaystyle
C^{1/2}{\textsf{x}}_{1}^{-}<C{\textsf{x}}_{2}^{-}<\cdots<{\textsf{h}}_{1}<{\textsf{h}}_{2}<\cdots<C^{-1/2}{\textsf{K}}^{-1}{\textsf{x}}_{1}^{+}<{\textsf{K}}^{-1}{\textsf{x}}_{0}^{+}\
,$

whereas for the subalgebra $U_{q}^{Dr,\triangleleft,-}$ we choose the
ordering:

(3.24)
$\displaystyle{\textsf{x}}_{0}^{-}<C^{1/2}{\textsf{x}}_{-1}^{-}<\cdots<{\textsf{h}}_{-1}<{\textsf{h}}_{-2}<\cdots<C^{-1}{\textsf{x}}_{-2}^{+}{\textsf{K}}<C^{-1/2}{\textsf{x}}_{-1}^{+}{\textsf{K}}\
.$

It follows:

######  Proposition 3.3.

The vector space $U_{q}^{Dr,\triangleright,+}$ (resp.
$U_{q}^{Dr,\triangleleft,-}$) has a linear basis consisting of the products
$x_{1}x_{2}\cdots x_{n}$ $(n\in{\mathbb{N}})$ with $x_{i}\in
U_{q}^{Dr,\triangleright,+}$ (resp. $x_{i}\in U_{q}^{Dr,\triangleleft,-}$)
such that $x_{1}\leq x_{2}\leq\cdots\leq x_{n}$.

Using the automorphism (A.12), PBW bases for $U_{q}^{Dr,\triangleright,-}$ and
$U_{q}^{Dr,\triangleleft,+}$ are similarly obtained.

Extensions of the alternating subalgebras are now introduced, that will be
useful in the analysis below.

######  Definition 3.4.

${U^{\prime}_{q}}^{Dr,\triangleright,\pm}$ (resp.
${U^{\prime}_{q}}^{Dr,\triangleleft,\pm}$) denote the subalgebras of
$U_{q}^{Dr}$ generated by $U_{q}^{Dr,\triangleright,\pm}$ (resp.
$U_{q}^{Dr,\triangleleft,\pm}$) and $\\{{\textsf{K}}^{\pm 1},C^{\pm 1/2}\\}$.

If one considers for instance ${U^{\prime}_{q}}^{Dr,\triangleright,+}$, in
addition to the relations (3.19)-(3.22) one has:

(3.25) $\displaystyle\big{[}{\textsf{h}}_{k},{\textsf{K}}^{\pm 1}\big{]}=0\
,\quad\big{[}B_{k},{\textsf{K}}^{\pm 1}\big{]}=0\ ,\quad C^{1/2}\ \
\mbox{central}\ ,$ (3.26)
$\displaystyle{\textsf{K}}A_{k}^{\pm}{\textsf{K}}^{-1}=q^{\pm 2}A_{k}^{\pm}\
.$

###  3.3. The homomorphism
$\nu:U_{q}^{T,+}\rightarrow{U^{\prime}_{q}}^{Dr,\triangleright,+}$

Consider the following Freidel-Maillet type equation (for a non-symmetric
R-matrix)

(3.27) $\displaystyle\tilde{R}_{12}(z/w)\ (\tilde{K}(z)\otimes{\mathbb{I}})\
R^{(0)}\ ({\mathbb{I}}\otimes\tilde{K}(w))\ =\
({\mathbb{I}}\otimes\tilde{K}(w))\ R^{(0)}\ (\tilde{K}(z)\otimes{\mathbb{I}})\
\tilde{R}_{21}(z/w)\ .$

Assume there exists a matrix $\tilde{K}^{0}(z)$ with scalar entries and two
quantum Lax operators $L(z),L^{0}$, such that the following relations hold
(recall that $\tilde{R}_{21}(z)=P\tilde{R}_{12}(z)P$):

(3.28) $\displaystyle\tilde{R}_{12}(z/w)\ \tilde{K}^{0}_{1}(z)\ R^{(0)}\
\tilde{K}^{0}_{2}(w)\ $ $\displaystyle=$ $\displaystyle\ \tilde{K}^{0}_{2}(w)\
R^{(0)}\ \tilde{K}^{0}_{1}(z)\ \tilde{R}_{21}(z/w)\ ,$ (3.29)
$\displaystyle\tilde{R}_{12}(z/w)L_{1}(z)L_{2}(w)$ $\displaystyle=$
$\displaystyle L_{2}(w)L_{1}(z)\tilde{R}_{12}(z/w)\ ,$ (3.30)
$\displaystyle\tilde{R}_{21}(z/w)(L^{0})_{1}(L^{0})_{2}$ $\displaystyle=$
$\displaystyle(L^{0})_{2}(L^{0})_{1}\tilde{R}_{21}(z/w)\ ,$ (3.31)
$\displaystyle(L^{0})_{1}R^{(0)}L_{2}(w)$ $\displaystyle=$ $\displaystyle
L_{2}(w)R^{(0)}(L^{0})_{1}\ ,$ (3.32) $\displaystyle
L_{1}(z)R^{(0)}(L^{0})_{2}$ $\displaystyle=$
$\displaystyle(L^{0})_{2}R^{(0)}L_{1}(z)\ .$

Adapting [Sk88, Proposition 2], using the above relations one finds that :

(3.33) $\displaystyle\tilde{K}(z)\mapsto L(z\lambda)\tilde{K}^{0}(z)L^{0}$

satisfies (3.27) provided $\lambda$ is invertible and
$[\lambda,U_{q}(\widehat{gl_{2}})]=0$. For instance, define:

(3.34)
$\displaystyle\tilde{K}^{0}(z)=\begin{pmatrix}\bar{\epsilon}_{+}&\frac{{\bar{k}}_{+}(q+q^{-1})}{(q-q^{-1})}\\\
\frac{{\bar{k}}_{-}(q+q^{-1})}{(q-q^{-1})}&\frac{\bar{\epsilon}_{-}}{z}\end{pmatrix}\
,$

where ${\bar{k}}_{\pm}\in{\mathbb{C}}(q)$ and
$\big{[}\bar{\epsilon}_{\pm},U_{q}(\widehat{gl_{2}})\big{]}=0$. It satisfies
(3.28). It follows:

######  Lemma 3.5.

The K-operator

(3.35)
$\displaystyle\tilde{K}(z)\mapsto\tilde{K}^{-}(z)=L^{-}(z\lambda)\tilde{K}^{0}(z)L^{-,0}\
$

satisfies (3.27) for any invertible $\lambda$ such that
$[\lambda,U_{q}(\widehat{gl_{2}})]=0$.

###### Proof.

By previous comment, it is sufficient to check that (3.29)-(3.32) hold. For
the choices

(3.36) $\displaystyle L(z)\mapsto L^{-}(z)\quad\mbox{and}\quad L^{0}\mapsto
L^{-,0}=diag(({\textsf{k}}_{2,0}^{-})^{-1},({\textsf{k}}^{-}_{1,0})^{-1})\ ,$

eq. (3.29) holds by definition and it is checked that eqs. (3.30)-(3.32) hold.
∎

The R-matrices $R(u)$ (symmetric) and $\tilde{R}(z)$ (non-symmetric) given by
(2.29) and (3.1), respectively, are related through the similarity
transformations:

$\displaystyle\qquad\left(\frac{u}{v}q-\frac{v}{u}q^{-1}\right)^{-1}R_{12}(u/v)$
$\displaystyle=$ $\displaystyle\cal M(u)_{1}\cal
M(v)_{2}\tilde{R}_{12}(u^{2}/v^{2})\cal M(v)_{2}^{-1}\cal M(u)_{1}^{-1}\ ,$
$\displaystyle=$ $\displaystyle{\cal M}(u)_{1}^{-1}{\cal
M}(v)_{2}^{-1}\tilde{R}_{21}(u^{2}/v^{2}){\cal M}(v)_{2}{\cal
M}(u)_{1}\quad\mbox{with}\quad{\cal M}(u)=\begin{pmatrix}u^{-1/2}&0\\\
0&u^{1/2}\end{pmatrix}\ .$

Using this transformation, one relates (3.27) to (2.37): there exists an
injective homomorphism from the Freidel-Maillet algebra (2.37) to the Yang-
Baxter algebra (3.7)-(3.9) given by:

(3.37) $\displaystyle K(u)\mapsto\cal M(u){\tilde{K}}^{-}(qu^{2}){\cal M}(u)\
.$

The explicit expression for (3.37) is a generalization of the K-operator in
[B20, Lemma 5.15]. Here the difference relies on the additional elements
$\bar{\epsilon}_{\pm}\neq 0$ in (3.34).

The map (3.37) allows to establish the precise relation between the equitable
generators
$\\{{y}^{+}_{-k},{y}^{+}_{k+1},{z}^{+}_{k+1},{\tilde{z}}^{+}_{k+1}\\}$ and the
generators of alternating subalgebras. In the expressions below, for
normalization convenience we set:

(3.38) $\displaystyle{\bar{k}}_{+}=q^{-1}(q-q^{-1})\
,\quad{\bar{k}}_{-}=q-q^{-1}\ ,\quad\bar{\epsilon}_{+}=q+q^{-1}\
,\quad\bar{\epsilon}_{-}=q(q+q^{-1})C^{-1}\ ,\quad\lambda=C^{3/2}\ .$

######  Proposition 3.6.

There exists an injective homomorphism $\nu:\
U_{q}^{T,+}\rightarrow{U^{\prime}_{q}}^{Dr,\triangleright,+}$ such that:

(3.39) $\displaystyle{{\cal{Y}}}_{+}(u)\mapsto
g(u)\left(-{\bar{k}}_{-}(q^{2}+1)(qu^{2})^{-1}\sum_{k=0}^{\infty}q^{k}C^{-k/2}{\textsf{K}}^{-1}{{\textsf{x}}}_{k}^{+}(qu^{2}\lambda)^{-k}+\bar{\epsilon}_{+}(qu^{2})^{-1}{\textsf{K}}^{-1}\right)\
,$ (3.40)
$\displaystyle{{\cal{Y}}}_{-}(u)\mapsto\left(-{\bar{k}}_{+}(q^{-2}+1)\sum_{k=0}^{\infty}q^{k+1}C^{(k+1)/2}{{\textsf{x}}}_{k+1}^{-}(qu^{2}\lambda)^{-k-1}\right.$
$\displaystyle\qquad\qquad\quad\left.+\
\bar{\epsilon}_{-}q^{-1}(qu^{2})^{-1}\left(\psi(u^{2}\lambda)+(q-q^{-1})^{2}\sum_{k,\ell=0}^{\infty}q^{k-\ell}C^{(k-\ell+1)/2}{{\textsf{x}}}_{k+1}^{-}{{\textsf{x}}}_{\ell}^{+}(qu^{2}\lambda)^{-k-\ell-1}\right)\right)g(u)\
,$ (3.41)
$\displaystyle{{\cal{Z}}}_{+}(u)\mapsto\left(\frac{\bar{\rho}}{q-q^{-1}}-\bar{\epsilon}_{-}{\bar{k}}_{-}q^{-1}(q^{2}-q^{-2})(qu^{2})^{-1}\sum_{k=0}^{\infty}q^{-k}C^{-k/2}{{\textsf{x}}}_{k}^{+}(qu^{2}\lambda)^{-k}\right)g(u)-\frac{\bar{\rho}}{q-q^{-1}}\
,$ (3.42) $\displaystyle{{\cal{Z}}}_{-}(u)\mapsto
g(u)\left(\frac{\bar{\rho}}{q-q^{-1}}{\textsf{K}}^{-1}\psi(u^{2}\lambda)-\
\bar{\epsilon}_{+}{\bar{k}}_{+}(q^{2}-q^{-2})\sum_{k=0}^{\infty}q^{-k+1}C^{(k+1)/2}{\textsf{x}}_{k+1}^{-}{\textsf{K}}^{-1}(qu^{2}\lambda)^{-k-1}\right.$
$\displaystyle\qquad\qquad\qquad\qquad\left.+\
\bar{\rho}(q-q^{-1})\sum_{k,\ell=0}^{\infty}q^{-k+\ell}C^{(k-\ell+1)/2}{\textsf{K}}^{-1}{{\textsf{x}}}_{k+1}^{-}{{\textsf{x}}}_{\ell}^{+}(qu^{2}\lambda)^{-k-\ell-1}\right)-\frac{\bar{\rho}}{q-q^{-1}}\
,$

where

(3.43) $\displaystyle
g(u)=\exp\left(-(q-q^{-1})\sum_{n=1}^{\infty}\frac{({\textsf{h}}_{n}+\gamma^{\prime}_{n})}{q^{n}+q^{-n}}(qu^{2}\lambda)^{-n}\right)\
\quad\mbox{with}\quad\gamma^{\prime}_{n}=-\frac{(q-q^{-1})^{2n-1}}{n}\left(\frac{\bar{\epsilon}_{+}\bar{\epsilon}_{-}\lambda}{\bar{\rho}q}\right)^{n}\
.$

###### Proof.

The first part of the proof concerns the derivation of the expressions on the
r.h.s of (3.39)-(3.42). Recall Lemma 3.5. Then, one expands explicitly (3.35)
using (3.3). Consider for instance the entry $({\tilde{K}}^{-}(z))_{11}$,
where some commutation relations given in [B20, eqs. (5.55)-(5.56)] are used:

$\displaystyle(\tilde{K}^{-}(z))_{11}$ $\displaystyle=$
$\displaystyle\frac{{\bar{k}}_{-}(q+q^{-1})}{q-q^{-1}}{\textsf{k}}_{1}^{-}(z\lambda)\underbrace{{\textsf{f}}^{-}(z\lambda)({\textsf{k}}^{-}_{2,0})^{-1}}_{=q({\textsf{k}}^{-}_{2,0})^{-1}{\textsf{f}}^{-}(z\lambda)}+\bar{\epsilon}_{+}{\textsf{k}}_{1}^{-}(z\lambda)({\textsf{k}}^{-}_{2,0})^{-1}$
$\displaystyle=$
$\displaystyle\frac{{\bar{k}}_{-}(q+q^{-1})}{q-q^{-1}}\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\underbrace{{\textsf{k}}_{1}^{-}(z\lambda)({\textsf{k}}^{-}_{2,0})^{-1}}_{={\textsf{K}}^{-1}\exp\left(-(q-q^{-1})\sum_{n=1}^{\infty}a_{1,n}(z\lambda)^{-n}\right)}\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!{\textsf{f}}^{-}(z\lambda)+\bar{\epsilon}_{+}\underbrace{{\textsf{k}}_{1}^{-}(z\lambda)({\textsf{k}}^{-}_{2,0})^{-1}}\qquad\mbox{by}\quad(\ref{kpmz})\
.$

Inserting (3.5), one gets:

$\displaystyle(\tilde{K}^{-}(z))_{11}$ $\displaystyle=$
$\displaystyle-{\bar{k}}_{-}(q+q^{-1})\exp\left(-(q-q^{-1})\sum_{n=1}^{\infty}a_{1,n}(z\lambda)^{-n}\right)\sum_{k=0}^{\infty}q^{k}q^{-ck/2}{\textsf{K}}^{-1}{{\textsf{x}}}_{k}^{+}(z\lambda)^{-k}$
$\displaystyle\ +\
\bar{\epsilon}_{+}{\textsf{K}}^{-1}\exp\left(-(q-q^{-1})\sum_{n=1}^{\infty}a_{1,n}(z\lambda)^{-n}\right)\
.$

Applying $\gamma^{\prime}_{D}$ according to (3.15)-(3.18), one finds
$\gamma^{\prime}_{D}\left(\tilde{K}^{-}(z)_{11}\right)$ is a power series in
the elements of ${U^{\prime}_{q}}^{Dr,\triangleright,+}$. Proceeding similarly
for the other entries,
$\gamma^{\prime}_{D}\left(K^{-}(z)_{ij}\right)\in{U^{\prime}_{q}}^{Dr,\triangleright,+}\otimes{\mathbb{C}}[[z]]$.
Also, the entries are reordered using the defining relations for Drinfeld’s
currents [DF93]. In particular, one introduces (A.10) and uses:

(3.45)
$\displaystyle{\textsf{x}}_{\ell+1}^{-}g(u)=q^{-2\ell}g(u){\textsf{x}}_{\ell+1}^{-}\
,\qquad{\textsf{x}}_{\ell}^{+}g(u)=q^{2\ell}g(u){\textsf{x}}_{\ell}^{+}\ .$

Then, using (3.37) one compares (2.36) to $\cal
M(u)\gamma^{\prime}_{D}\left({\tilde{K}}^{-}(qu^{2})\right){\cal M}(u)$. This
gives (3.39)-(3.42).

The second part of the proof concerns the identification of the elements
$\gamma^{\prime}_{n}$ such that (2.38) holds, i.e. (2.43). In the r.h.s. of
(2.41), insert the explicit expressions previously obtained to get the image
of ${\cal C}(u)$ in ${U^{\prime}_{q}}^{Dr,\triangleright,+}$. By [B20,
Corollary 3.4, Remark 3.5] (recall the substitutions (2.39), (2.40)),
$\nu({\cal C}(u))$ is central: it can be reduced to a function of $C^{1/2}$.
To determine this function, it is sufficient to extract all terms of
$\nu({\cal C}(u))$ that belong to the center. According to the ordering (3.23)
and the reduction rules (A.4)-(A.9), one identifies the subset of terms in the
images of $\\{{{\cal{Y}}}_{\pm}(u),{{\cal{Z}}}_{\pm}(u)\\}$ that are relevant.
One finds:

$\displaystyle{{\cal{Y}}}_{+}(u)\mapsto\bar{\epsilon}_{+}{\textsf{K}}^{-1}(qu^{2})^{-1}c(u)+\cdots\
,\qquad\quad{{\cal{Y}}}_{-}(uq)\mapsto\bar{\epsilon}_{-}q^{-3}{\textsf{K}}(qu^{2})^{-1}c(uq)+\cdots\
,$
$\displaystyle{{\cal{Z}}}_{+}(uq)\mapsto\frac{\bar{\rho}}{(q-q^{-1})}(c(uq)-1)+\cdots\
,\qquad{{\cal{Z}}}_{-}(u)\mapsto\frac{\bar{\rho}}{(q-q^{-1})}(c(u)-1)+\cdots\
,$

where

(3.46) $\displaystyle
c(u)=\exp\left(-(q-q^{-1})\sum_{n=1}^{\infty}\frac{\gamma^{\prime}_{n}}{q^{n}+q^{-n}}(qu^{2}\lambda)^{-n}\right)\
,$

and the ‘dots’ correspond to terms that will not contribute. After
simplifications, one gets the factorized expression:

(3.47) $\displaystyle\nu({\cal
C}(u))-\frac{\bar{\rho}}{(q-q^{-1})}=\left((q-q^{-1})q^{-2}\bar{\epsilon}_{+}\bar{\epsilon}_{-}(qu^{2})^{-1}-\frac{\bar{\rho}}{(q-q^{-1})}\right)c(u)c(uq)\
.$

The condition $\nu({\cal C}(u))=0$ leads to

(3.48)
$\displaystyle\exp\left(-(q-q^{-1})\sum_{n=1}^{\infty}(q\lambda)^{-n}\gamma^{\prime}_{n}(qu^{2})^{-n}\right)=1-\frac{(q-q^{-1})^{2}}{\bar{\rho}q^{2}}\bar{\epsilon}_{+}\bar{\epsilon}_{-}(qu^{2})^{-1}\
.$

Taking the logarithm on both sides, the corresponding formal power series are
identified. It yields to (3.43). ∎

Recall (2.33), (2.34). Identifying the leading terms of the power series, from
the proposition above with (3.38) one gets for instance:

######  Example 3.7.

The image in ${U^{\prime}_{q}}^{Dr,\triangleright,+}$ of $U_{q}^{T,+}$ is such
that:

$\displaystyle y_{0}^{+}$ $\displaystyle\mapsto$ $\displaystyle
C^{-1}{\textsf{K}}-q^{-1}(q-q^{-1})C^{-1}{\textsf{x}}_{1}^{-}\ ,$
$\displaystyle y_{1}^{+}$ $\displaystyle\mapsto$
$\displaystyle{\textsf{K}}^{-1}-q(q-q^{-1}){\textsf{K}}^{-1}{\textsf{x}}_{0}^{+}\
,$ $\displaystyle{\tilde{z}}_{1}^{+}$ $\displaystyle\mapsto$
$\displaystyle-(q-q^{-1})^{2}\left(q^{-1}C^{-3/2}{\textsf{h}}_{1}+(q+q^{-1})C^{-1}{\textsf{x}}_{0}^{+}\right)+(q-q^{-1})C^{-1}\
,$ $\displaystyle z_{1}^{+}$ $\displaystyle\mapsto$
$\displaystyle(q-q^{-1})^{2}\left(qC^{-3/2}{\textsf{h}}_{1}-(q+q^{-1})C^{-1}{\textsf{x}}_{1}^{-}{\textsf{K}}^{-1}+q(q^{2}-q^{-2})C^{-1}{\textsf{x}}_{1}^{-}{\textsf{K}}^{-1}{\textsf{x}}_{0}^{+}\right)+(q-q^{-1})C^{-1}\
.$

Using (A.13), it is checked that the images of $\\{y_{i}^{+}\\}_{i=0,1}$ match
with (2.6).

######  Remark 3.8.

Alternative expressions for (3.39)-(3.42) can be written using the
commutations relations (3.45) and $\big{[}\psi(z),{\textsf{K}}^{\pm
1}\big{]}=\big{[}\psi(z),g(u)\big{]}=0$.

######  Remark 3.9.

The image of the equitable generators in terms of Lusztig’s root vectors and
$\\{K_{0},K_{1}\\}$ is obtained as follows. According to the definitions of
the root vectors $\\{E_{n\delta+\alpha_{i}},E_{n\delta}|i=0,1\\}\in
U_{q}^{DJ,+}$ (or $\\{F_{n\delta+\alpha_{i}},F_{n\delta}|i=0,1\\}\in
U_{q}^{DJ,-}$) given in [Be94, BCP98], one uses the correspondence:

(3.49) $\displaystyle{\textsf{x}}^{+}_{k}$ $\displaystyle=$ $\displaystyle
E_{k\delta+\alpha_{1}}\
,\qquad{\textsf{x}}^{-}_{k+1}=-C^{-k-1}{\textsf{K}}E_{k\delta+\alpha_{0}}\
,\qquad\qquad{\textsf{h}}_{k+1}=C^{-(k+1)/2}E_{(k+1)\delta}\ ,$ (3.50)
$\displaystyle{\textsf{x}}^{-}_{-k}$ $\displaystyle=$ $\displaystyle
F_{k\delta+\alpha_{1}}\
,\qquad{\textsf{x}}^{+}_{-k-1}=-F_{k\delta+\alpha_{0}}{\textsf{K}}^{-1}C^{k+1}\
,\qquad{\textsf{h}}_{-k-1}=C^{(k+1)/2}F_{(k+1)\delta}$

for $k\in{\mathbb{N}}$ and ${\textsf{K}}=K_{1}$, $C{\textsf{K}}^{-1}=K_{0}$ in
Proposition 3.6, where $\bar{k}_{\pm},\bar{\epsilon}_{\pm},\lambda$, are
chosen such that (2.6) is recovered at the leading order of the power series.

###  3.4. The homomorphism
$\delta:U_{q}^{T,+}\rightarrow{U^{\prime}_{q}}^{Dr,\triangleright,+}\otimes
U_{q}^{T,+}$

For $U_{q}^{T,+}$, a comodule algebra structure can be exhibited as follows.
Starting from any K-operator satisfying (3.27) and following standard
arguments [Sk88], left or right coactions can be constructed using the FRT
presentation. Consider the K-operator in the r.h.s. of (3.35). A new
K-operator can be constructed using a dressing procedure [Sk88], which leads
naturally to a left or right coaction map. For instance333With respect to the
ordering $V_{[1]}\otimes V_{[2]}$:  (3.51)
$\displaystyle((T)_{[\textsf{1}]}(T^{\prime})_{[\textsf{2}]}(T^{\prime\prime})_{[\textsf{1}]})_{ij}=\sum_{k,\ell=1}^{2}(T)_{ik}(T^{\prime\prime})_{\ell
j}\otimes(T^{\prime})_{k\ell}\ .$

######  Proposition 3.10.

$U_{q}^{T,+}$ is a left comodule algebra over
${U^{\prime}_{q}}^{Dr,\triangleright,+}$ with coaction map
$\delta:U_{q}^{T,+}\rightarrow{U^{\prime}_{q}}^{Dr,\triangleright,+}\otimes
U_{q}^{T,+}$ such that

(3.52)
$\displaystyle\delta(\tilde{K}^{-}(z))=(\gamma^{\prime}_{D}\otimes\gamma^{\prime}_{D})\left((L^{-}(zq^{(1\otimes
c/2)}))_{[\textsf{1}]}(\tilde{K}^{-}(zq^{(c/2\otimes
1)}))_{[\textsf{2}]}(L^{-,0})_{[\textsf{1}]}\right)\ .$

###### Proof.

By construction, the r.h.s. satisfies (3.27) for the non-symmetric R-matrix
(3.1). For $U_{q}^{T,+}$ to be a comodule algebra, we need to check:

(3.53) $\displaystyle(\Delta\otimes id)\circ\delta$ $\displaystyle=$
$\displaystyle(id\otimes\delta)\circ\delta\ ,$ (3.54) $\displaystyle({\cal
E}\otimes id)\circ\delta$ $\displaystyle\cong$ $\displaystyle id\ .$

Firstly, consider (3.53). Apply the l.h.s. of (3.53) to $\tilde{K}^{-}(z)$ in
(3.35) and use the Lax operator coproduct rule (3.11). Compare the result with
the r.h.s. of (3.53) applied on $\tilde{K}^{-}(z)$. Both expressions coincide.
Secondly, consider (3.54). Apply the l.h.s of (3.54) to $\tilde{K}^{-}(z)$ in
(3.35) and use the counit rule (3.12). Thus, we conclude that $U_{q}^{T,+}$ is
a left comodule algebra. ∎

If needed, the image of the equitable generators by $\delta$ can be extracted
in a straightforward manner. As an example, for simplicity let us consider a
specialization of (3.52), namely the left coaction map
$\bar{\delta}:U_{q}^{T,+}\rightarrow{U^{\prime}_{q}}^{Dr,\triangleright,+}/_{C=1}\otimes
U_{q}^{T,+}$. For the symmetric R-matrix (2.29) using (3.3) and (3.52) at
$c=0$ (i.e. $C=1$) it yields to:

(3.55) $\displaystyle\bar{\delta}(K(u))=(\gamma^{\prime}_{D}\otimes
1)\left(\left(\cal M(u)L^{-}(qu^{2})\cal
M(u)^{-1}\right)_{[\textsf{1}]}(K(u))_{[\textsf{2}]}(L^{-,0})_{[\textsf{1}]}\right)/_{C_{[\textsf{1}]}=1}\
.$

Now, recall the generating functions (2.33), (2.34).

######  Lemma 3.11.

There exists a coaction map $\bar{\delta}:\
U_{q}^{T,+}\rightarrow{U^{\prime}_{q}}^{Dr,\triangleright,+}/_{C=1}\otimes
U_{q}^{T,+}$ such that:

$\displaystyle\bar{\delta}({\cal{Y}}_{+}(u))$ $\displaystyle\mapsto$
$\displaystyle(qu^{2})^{-1}q\gamma^{\prime}_{D}\left({\textsf{k}}^{-}_{1}(qu^{2})({\textsf{k}}^{-}_{2,0})^{-1}{\textsf{f}}^{-}(qu^{2})\right)/_{C=1}\otimes\left(\frac{1}{{\bar{k}}_{+}(q+q^{-1})}{\cal{Z}}_{-}(u)+\frac{{\bar{k}}_{-}(q+q^{-1})}{(q-q^{-1})}\right)$
$\displaystyle+\
\gamma^{\prime}_{D}\left({\textsf{k}}^{-}_{1}(qu^{2})({\textsf{k}}^{-}_{2,0})^{-1}\right)/_{C=1}\otimes{\cal{Y}}_{+}(u)\
,$ $\displaystyle\qquad\bar{\delta}({\cal{Y}}_{-}(u))$ $\displaystyle\mapsto$
$\displaystyle
q^{-1}\gamma^{\prime}_{D}\left({\textsf{e}}^{-}(qu^{2}){\textsf{k}}^{-}_{1}(qu^{2})({\textsf{k}}^{-}_{1,0})^{-1}\right)/_{C=1}\otimes\left(\frac{1}{{\bar{k}}_{-}(q+q^{-1})}{\cal{Z}}_{+}(u)+\frac{{\bar{k}}_{+}(q+q^{-1})}{(q-q^{-1})}\right)$
$\displaystyle+\
\gamma^{\prime}_{D}\left({\textsf{k}}^{-}_{2}(qu^{2})({\textsf{k}}^{-}_{1,0})^{-1}+\
q^{-1}{\textsf{e}}^{-}(qu^{2}){\textsf{k}}_{1}^{-}(qu^{2})({\textsf{k}}^{-}_{1,0})^{-1}{\textsf{f}}^{-}(qu^{2})\right)/_{C=1}\otimes{\cal{Y}}_{-}(u)\
,$ $\displaystyle\bar{\delta}({\cal{Z}}_{+}(u))$ $\displaystyle\mapsto$
$\displaystyle\gamma^{\prime}_{D}\left({\textsf{k}}^{-}_{1}(qu^{2})({\textsf{k}}^{-}_{1,0})^{-1}\right)/_{C=1}\otimes{\cal{Z}}_{+}(u)+\frac{\bar{\rho}}{q-q^{-1}}\left(\gamma^{\prime}_{D}\left({\textsf{k}}^{-}_{1}(qu^{2})({\textsf{k}}^{-}_{1,0})^{-1}\right)-1\right)/_{C=1}\otimes
1$ $\displaystyle+\
{\bar{k}}_{-}(q+q^{-1})\gamma^{\prime}_{D}\left({\textsf{k}}^{-}_{1}(qu^{2})({\textsf{k}}^{-}_{1,0})^{-1}{\textsf{f}}^{-}(qu^{2})\right)/_{C=1}\otimes{\cal{Y}}_{-}(u)\
,$ $\displaystyle\bar{\delta}({\cal{Z}}_{-}(u))$ $\displaystyle\mapsto$
$\displaystyle\gamma^{\prime}_{D}\left({\textsf{k}}^{-}_{2}(qu^{2})({\textsf{k}}^{-}_{2,0})^{-1}+q{\textsf{e}}^{-}(qu^{2})k_{1}^{-}(qu^{2})({\textsf{k}}^{-}_{2,0})^{-1}{\textsf{f}}^{-}(qu^{2})\right)/_{C=1}\otimes{\cal{Z}}_{-}(u)$
$\displaystyle+\frac{\bar{\rho}}{q-q^{-1}}\gamma^{\prime}_{D}\left({\textsf{k}}^{-}_{2}(qu^{2})({\textsf{k}}^{-}_{2,0})^{-1}+q{\textsf{e}}^{-}(qu^{2})k_{1}^{-}(qu^{2})({\textsf{k}}^{-}_{2,0})^{-1}{\textsf{f}}^{-}(qu^{2})-1\right)/_{C=1}\otimes
1$ $\displaystyle+\
{\bar{k}}_{+}qu^{2}(q+q^{-1})\gamma^{\prime}_{D}\left({\textsf{e}}^{-}(qu^{2}){\textsf{k}}^{-}_{1}(qu^{2})({\textsf{k}}^{-}_{2,0})^{-1}\right)/_{C=1}\otimes{\cal{Y}}_{+}(u)\
.$

###### Proof.

Compute (3.55) using (3.3), (3.3) and (2.36) . Compare the entries of the
resulting matrix to $\bar{\delta}(K(u))$ with (2.36). Applying (3.15)-(3.18)
to (3.4), (3.5) and (3.13), one finds
$\gamma^{\prime}_{D}\left(\bar{\delta}(K(u))_{ij}\right)\in{U^{\prime}_{q}}^{Dr,\triangleright,+}/_{C=1}\otimes
U_{q}^{T,+}\otimes{\mathbb{C}}[[u^{2}]]$. ∎

Other examples of left and right coaction maps can be derived along the same
lines. Now, expanding the power series on both sides of the above equations
using (2.33), (2.34), (3.4)-(3.6) with (3.13), one gets the image by
$\bar{\delta}$ of the generators of $U_{q}^{T,+}$.

######  Example 3.12.

$\displaystyle\bar{\delta}(y_{1}^{+})$ $\displaystyle=$
$\displaystyle-q(q-q^{-1}){\textsf{K}}^{-1}{\textsf{x}}_{0}^{+}\otimes
1+{\textsf{K}}^{-1}\otimes y_{1}^{+}\ ,$
$\displaystyle\bar{\delta}(y_{0}^{+})$ $\displaystyle=$
$\displaystyle-q^{-1}(q-q^{-1}){\textsf{x}}_{1}^{-}\otimes
1+{\textsf{K}}\otimes y_{0}^{+}\ .$

Using (A.13) at $C=1$, for (2.6) one finds $\bar{\delta}$ coincides with
$\Delta$, see (A.3).

Note that tensor product representations for $U_{q}^{T,+}$ can be obtained
from [B20, Section 4], adapting the definitions of the generators and
conventions. See [B20, Proposition 4.5].

##  4\. The central extension of $U_{q}^{T,+}$

In this section, following [T19b] the central extension of ${U}_{q}^{T,+}$ and
its center are considered. Below, they are denoted respectively ${\cal
U}_{q}^{T,+}$ and ${\textsf{C}}^{+}$. Specializing the results of [B20], a
Freidel-Maillet type presentation for ${\cal U}_{q}^{T,+}$ is given. Then,
following [B20] the alternating subalgebras
$U_{q}(\widehat{gl_{2}})^{\triangleright,+}$,
${U^{\prime}_{q}}(\widehat{gl_{2}})^{\triangleright,+}$ and center
${\textsf{C}}^{\triangleright}$ are introduced. By analogy with the analysis
in previous section, the Freidel-Maillet type presentation is used to compute
the images of the generators of ${\cal U}_{q}^{T,+}$ and ${\textsf{C}}^{+}$ in
${U^{\prime}_{q}}(\widehat{gl_{2}})^{\triangleright,+}$ and
${\textsf{C}}^{\triangleright}$, respectively.

The following definitions are straightforward adaptations of [T19b].

######  Definition 4.1.

${\cal U}_{q}^{T,+}$ is the unital associative $\mathbb{C}(q)$-algebra with
equitable generators
$\\{Y_{-k}^{+},Y_{k+1}^{+},Z_{k+1}^{+},{\tilde{Z}}_{k+1}^{+}|k\in{\mathbb{N}}\\}$
subject to the relations (2.8)-(2.18) with the substitutions:

(4.1) $\displaystyle y_{-k}^{+}\rightarrow Y_{-k}^{+}\ ,\quad
y_{k+1}^{+}\rightarrow Y_{k+1}^{+}\ ,\quad z_{k+1}^{+}\rightarrow Z_{k+1}^{+}\
,\quad{\tilde{z}}_{k+1}^{+}\rightarrow{\tilde{Z}}_{k+1}^{+}\ .$

######  Definition 4.2.

The center ${\textsf{C}}^{+}$ is the subalgebra of ${\cal U}_{q}^{T,+}$
generated by the elements:

(4.2)
$\displaystyle{\textsf{C}}_{n+1}=(q^{2}-q^{-2})\sum_{k=0}^{n}q^{-2n+2k-1}Y_{k+1}^{+}Y_{-n+k}^{+}-\frac{(q-q^{-1})}{\bar{\rho}}\sum_{k=0}^{n+1}q^{2k-2n-2}Z_{k}^{+}\tilde{Z}_{n+1-k}^{+}\
,\quad n\in{\mathbb{N}}\ $

with $Z^{+}_{0}=\tilde{Z}^{+}_{0}=\bar{\rho}/(q-q^{-1})$.

Note that the automorphisms $\sigma,S$, given by (2.21), (2.22), naturally
extend from $U_{q}^{T,+}$ to ${\cal U}_{q}^{T,+}$; See [T19b, Section 8] for
details. Importantly, using the defining relations of ${\cal U}_{q}^{T,+}$ one
shows that the central elements ${\textsf{C}}_{n+1}$ are fixed under the
action of $\sigma,S$ [T19b, Proposition 8.3].

A Freidel-Maillet type presentation for ${\cal U}_{q}^{T,+}$ follows from
[B20, Theorem 3.1], adapting the notations. We refer the reader to this work
for the proof of the Theorem below. Introduce the generating functions:

(4.3)
$\displaystyle{{\textsf{Y}}}_{+}(u)=\sum_{k\in{\mathbb{N}}}{Y}^{+}_{k+1}U^{-k-1}\
,\quad{{\textsf{Y}}}_{-}(u)=\sum_{k\in{\mathbb{N}}}{Y}^{+}_{-k}U^{-k-1}\ ,$
(4.4)
$\displaystyle\quad{{\textsf{Z}}}_{+}(u)=\sum_{k\in{\mathbb{N}}}{\tilde{Z}}^{+}_{k+1}U^{-k-1}\
,\quad{{\textsf{Z}}}_{-}(u)=\sum_{k\in{\mathbb{N}}}{Z}^{+}_{k+1}U^{-k-1}\ .$

######  Theorem 4.3.

${\cal U}_{q}^{T,+}$ has a presentation of Freidel-Maillet type. Let
${\textsf{K}}(u)$ be a square matrix such that

(4.5)
$\displaystyle{\textsf{K}}(u)=\begin{pmatrix}uq{\textsf{Y}}_{+}(u)&\frac{1}{{\bar{k}}_{-}(q+q^{-1})}{\textsf{Z}}_{+}(u)+\frac{{\bar{k}}_{+}(q+q^{-1})}{(q-q^{-1})}\\\
\frac{1}{{\bar{k}}_{+}(q+q^{-1})}{\textsf{Z}}_{-}(u)+\frac{{\bar{k}}_{-}(q+q^{-1})}{(q-q^{-1})}&uq{\textsf{Y}}_{-}(u)\end{pmatrix}$

with (4.3)-(4.4). The defining relations are given by:

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

In the Freidel-Maillet framework, a generating function for central elements
of ${\textsf{C}}^{+}$ is derived from the quantum determinant of the
K-operator (4.5). Introduce the generating function with coefficients (4.2)

(4.7)
$\displaystyle{\textsf{C}}(u)=\sum_{k=0}^{\infty}{\textsf{C}}_{k+1}U^{-k-1}\
.$

By previous comments, note that $\sigma({\textsf{C}}(u))={\textsf{C}}(u)$. The
following proposition is an alternative to [T19b, Section 13], adapted to the
equitable case.

######  Proposition 4.4.

The quantum determinant

(4.8) $\displaystyle\Gamma(u)=\text{\rm
tr}_{12}\big{(}P^{-}_{12}({\textsf{K}}(u)\otimes I\\!\\!I)\
R^{(0)}(I\\!\\!I\otimes{\textsf{K}}(uq))\big{)}\ \ $

generates ${\textsf{C}}^{+}$.

###### Proof.

Firstly, one shows that $\Gamma(u)$ is central i.e.
$\big{[}\Gamma(u),({\textsf{K}}(u))_{ij}\big{]}=0$. We refer the reader to
[B20, Proposition 3.3] for details. Secondly, inserting (4.5) into the r.h.s.
of (4.8), one gets

(4.9)
$\displaystyle\Gamma(u)=\frac{1}{(q-q^{-1})}\left({\textsf{C}}(u)-\frac{\bar{\rho}}{(q-q^{-1})}\right)\
$

where

(4.10) $\displaystyle{\textsf{C}}(u)$ $\displaystyle=$
$\displaystyle(q-q^{-1})u^{2}q^{2}{\textsf{Y}}_{+}(u){\textsf{Y}}_{-}(uq)-\frac{(q-q^{-1})}{\bar{\rho}}{\textsf{Z}}_{-}(u){\textsf{Z}}_{+}(uq)-{\textsf{Z}}_{-}(u)-{\textsf{Z}}_{+}(uq)\
.$

∎

Also, the analogs of [T19b, Lemma 3.3, Lemma 2.8] take the following form.
Recall Theorems 2.10, 4.3.

######  Lemma 4.5.

There exists a surjective homomorphism ${\cal U}_{q}^{T,+}\rightarrow
U_{q}^{T,+}$ that sends

(4.11) $\displaystyle{\textsf{K}}(u)\mapsto K(u)\
,\qquad\Gamma(u)\mapsto-\frac{\bar{\rho}}{(q-q^{-1})^{2}}\ .$

An embedding of ${\cal U}_{q}^{T,+}$ into a subalgebra of
$U_{q}(\widehat{gl_{2}})$ can be obtained using the the FRT presentation of
Theorem 3.1. To prepare the analysis below, some results from [B20] are
needed. We refer to [B20, Section 5] for details and references. The
definition below is a variation of [B20, Definition 5.12, eqs. (5.52)-(5.53)].

######  Definition 4.6.

$\displaystyle U_{q}(\widehat{gl_{2}})^{\triangleright,\pm}$ $\displaystyle=$
$\displaystyle\\{C^{\mp
k/2}{\textsf{K}}^{-1}{\textsf{x}}_{k}^{\pm},C^{\pm(k+1)/2}{\textsf{x}}^{\mp}_{k+1},a_{1,k+1},a_{2,k+1}|k\in{\mathbb{N}}\\}\
,$ $\displaystyle U_{q}(\widehat{gl_{2}})^{\triangleleft,\pm}$
$\displaystyle=$ $\displaystyle\\{C^{\mp
k/2}{\textsf{x}}_{-k}^{\pm},C^{\pm(k+1)/2}{\textsf{x}}^{\mp}_{-k-1}{\textsf{K}},a_{1,-k-1},a_{2,-k-1}|k\in{\mathbb{N}}\\}\
.$

We call $U_{q}(\widehat{gl_{2}})^{\triangleright,\pm}$ and
$U_{q}(\widehat{gl_{2}})^{\triangleleft,\pm}$ the right and left alternating
subalgebras of $U_{q}(\widehat{gl_{2}})$. The subalgebra generated by
$\\{{\textsf{K}}^{\pm 1},C^{\pm 1/2}\\}$ is denoted
$U_{q}(\widehat{gl_{2}})^{\diamond}$.

Importantly, it is known that the elements

(4.12) $\displaystyle\gamma_{m}=q^{m}a_{1,m}+q^{-m}a_{2,m}\quad\mbox{for}\quad
m\in{\mathbb{Z}}^{*}\ $

generate the center ${\cal C}$ of $U_{q}(\widehat{gl_{2}})$ (see e.g.
[FMu02]). It follows:

######  Definition 4.7.

The center $\cal C^{\triangleright}$ (resp. $\cal C^{\triangleleft}$) of
$U_{q}(\widehat{gl_{2}})^{\triangleright,\pm}$ (resp.
$U_{q}(\widehat{gl_{2}})^{\triangleleft,\pm}$) is generated by $\gamma_{m}$
(resp. $\gamma_{-m}$) with $m\in{\mathbb{N}}^{*}$.

Extensions of the alternating subalgebras of $U_{q}(\widehat{gl_{2}})$ are now
introduced.

######  Definition 4.8.

${U^{\prime}_{q}}(\widehat{gl_{2}})^{\triangleright,\pm}$ (resp.
${U^{\prime}_{q}}(\widehat{gl_{2}})^{\triangleleft,\pm}$) denote the
subalgebras of $U_{q}(\widehat{gl_{2}})$ generated by
${U}_{q}(\widehat{gl_{2}})^{\triangleright,\pm}$ (resp.
${U}_{q}(\widehat{gl_{2}})^{\triangleleft,\pm}$) and $\\{{\textsf{K}}^{\pm
1},C^{\pm 1/2}\\}$.

For the quantum algebra $U_{q}(\widehat{gl_{2}})$, it is known that
$U_{q}(\widehat{gl_{2}})\cong U_{q}^{Dr}\otimes{\cal C}$. Thus, for the
alternating subalgebras analog properties hold. Recall Definition 3.4 and
(3.14). One has:

$\displaystyle{U^{\prime}_{q}}(\widehat{gl_{2}})^{\triangleright,\pm}\cong{U^{\prime}_{q}}^{Dr,\triangleright,\pm}\otimes\cal
C^{\triangleright}\
,\qquad{U^{\prime}_{q}}(\widehat{gl_{2}})^{\triangleleft,\pm}\cong{U^{\prime}_{q}}^{Dr,\triangleleft,\pm}\otimes\cal
C^{\triangleleft}\ .$

The embedding of the Freidel-Maillet algebra (4.6) into the FRT presentation
of $U_{q}(\widehat{gl_{2}})$ is now studied. Generalizing the results of
previous section, the map (3.37) allows to establish the precise relation
between the equitable generators
$\\{{Y}^{+}_{-k},{Y}^{+}_{k+1},{Z}^{+}_{k+1},{\tilde{Z}}^{+}_{k+1}\\}$ and the
generators of alternating subalgebras of $U_{q}(\widehat{gl_{2}})$. Recall
(3.46) and Proposition 3.6. Introduce the generating function in the central
elements:

(4.13)
$\displaystyle{\textsf{c}}(u)=\exp\left(-(q-q^{-1})\sum_{n=1}^{\infty}\frac{\gamma_{n}}{q^{n}+q^{-n}}(qu^{2}\lambda)^{-n}\right)\
\in{\cal C}^{\triangleright}\otimes{\mathbb{C}}[[u^{2}]].$

######  Proposition 4.9.

There exists an injective homomorphism $\mu:\ {\cal
U}_{q}^{T,+}\rightarrow{U^{\prime}_{q}}(\widehat{gl_{2}})^{\triangleright,+}$
such that

(4.14) $\displaystyle{\textsf{Y}}_{\pm}(u)$ $\displaystyle\mapsto$
$\displaystyle\nu({\cal{Y}}_{\pm}(u))c(u)^{-1}{\textsf{c}}(u)\ ,$ (4.15)
$\displaystyle{\textsf{Z}}_{\pm}(u)$ $\displaystyle\mapsto$
$\displaystyle\left(\nu({\cal{Z}}_{\pm}(u))+\frac{\bar{\rho}}{(q-q^{-1})}\right)c(u)^{-1}{\textsf{c}}(u)-\frac{\bar{\rho}}{(q-q^{-1})}\
.$

###### Proof.

One expands explicitly (3.35) using (3.3). For the entry
$({\tilde{K}}^{-}(z))_{11}$, we previously obtained (3.3). Inserting (3.14)
and using (3.17), it factorizes as:

(4.16) $\displaystyle(\tilde{K}^{-}(z))_{11}$ $\displaystyle=$
$\displaystyle\exp\left(-(q-q^{-1})\sum_{n=1}^{\infty}\frac{(\gamma_{n}-\gamma^{\prime}_{n})}{q^{n}+q^{-n}}(z\lambda)^{-n}\right)\gamma_{D}^{\prime}((\tilde{K}^{-}(z))_{11})\
.$

Actually, other entries
$(\tilde{K}^{-}(z))_{ij}\in{U^{\prime}_{q}}(\widehat{gl_{2}})^{\triangleright,\pm}\otimes{\mathbb{C}}[[z]]$
and similarly factorize in terms of
$\gamma_{D}^{\prime}((\tilde{K}^{-}(z))_{ij})$. Thus, a solution of (4.6) is
given by the K-operator:

(4.17) $\displaystyle\cal M(u)\left({\tilde{K}}^{-}(qu^{2})\right){\cal
M}(u)={\textsf{c}}(u)c(u)^{-1}\cal
M(u)\gamma^{\prime}_{D}\left({\tilde{K}}^{-}(qu^{2})\right){\cal M}(u)$

with (3.46), (4.13). Then, one compares (2.36) to (4.17) which gives
(4.14)-(4.15). ∎

In terms of central elements of $U_{q}(\widehat{gl_{2}})$, the quantum
determinant takes a rather simple form.

######  Corollary 4.10.

(4.18)
$\displaystyle\Gamma(u)\stackrel{{\scriptstyle\mu}}{{\mapsto}}\left(\frac{\bar{\epsilon}_{+}\bar{\epsilon}_{-}(q-q^{-1})^{2}-\bar{k}_{+}\bar{k}_{-}(q+q^{-1})^{2}}{q^{3}u^{2}(q-q^{-1})^{2}}\right)\exp\left(-(q-q^{-1})\sum_{n=1}^{\infty}\gamma_{n}(q^{2}u^{2}\lambda)^{-n}\right)\
$

with (3.38).

###### Proof.

Applying $\mu$ to (4.10) and using (4.14), (4.15), one gets:

(4.19)
$\displaystyle\mu({\textsf{C}}(u))-\frac{\bar{\rho}}{(q-q^{-1})}={\textsf{c}}(u){\textsf{c}}(uq)c(u)^{-1}c(uq)^{-1}\left(\nu({\cal
C}(u))-\frac{\bar{\rho}}{(q-q^{-1})}\right)\ .$

Using (2.35), (3.47), (4.9), (4.13), the r.h.s. of (4.18) follows. ∎

######  Remark 4.11.

An alternative expression for $\mu(\Gamma(u))$ is derived as follows.
Inserting (4.12) into (4.13) and using the second eq. of (3.14), (A.5) and
(A.10), one gets:

(4.20)
$\displaystyle\Gamma(u)\stackrel{{\scriptstyle\mu}}{{\mapsto}}\left(\frac{\bar{\epsilon}_{+}\bar{\epsilon}_{-}(q-q^{-1})^{2}-\bar{k}_{+}\bar{k}_{-}(q+q^{-1})^{2}}{q^{3}u^{2}(q-q^{-1})^{2}}\right){\textsf{g}}(u)\psi(q^{2}u^{2}\lambda){\textsf{g}}(uq)\
$

with

(4.21)
$\displaystyle{\textsf{g}}(u)=\exp\left(-(q-q^{-1})\sum_{n=1}^{\infty}a_{1,n}(qu^{2}\lambda)^{-n}\right)\
.$

##  5\. Concluding remarks

The results of this paper together with [B20] can be summarized as follows. In
[B20], it was shown that the alternating presentation for an algebra
$U_{q}^{+}$ and its central extension ${\cal U}_{q}^{+}$ introduced and
studied in [T18, T19a, T19b] admits a presentation of Freidel-Maillet type.
For ${\cal U}_{q}^{+}$, this presentation consists of a K-operator satisfying
(4.6), which entries are generating functions in the alternating generators of
${\cal U}_{q}^{+}$ [T19a, T19b]; See [B20, Theorem 3.1]. To get the analog
presentation for $U_{q}^{+}$, a condition for the quantum determinant of the
K-operator is asserted. It reads (2.38). Now, by Lemmas 2.5, 2.6, two
different embeddings of the alternating presentation of $U_{q}^{+}$ into
$U_{q}(\widehat{sl_{2}})$ can be considered :

(i) The Drinfeld-Jimbo (or Chevalley) type: the alternating presentation for
$U_{q}^{DJ,+}$ \- the standard positive part of $U_{q}(\widehat{sl_{2}})$;

(ii) The equitable (or Ito-Terwilliger) type: the alternating presentation for
$U_{q}^{IT,+}$ \- the non-standard positive part of $U_{q}(\widehat{sl_{2}})$.

In this paper, the alternating presentation for the equitable type is denoted
$U_{q}^{T,+}$, and its central extension ${\cal U}_{q}^{T,+}$. Its generators
are called the equitable generators to avoid any confusion with the Drinfeld-
Jimbo type.

The alternating presentations (i),(ii), are studied in details in [B20, Ter21]
and the present paper. The approach followed in [B20] and here is based on
Freidel-Maillet type presentations. In this framework, to the K-operator of
$U_{q}^{+}$ (see [B20, eq. (3.8)] one associates a K-operator of Drinfeld-
Jimbo type for (i), or of equitable type (2.36) for (ii). For each type,
embeddings into Yang-Baxter subalgebras of $U_{q}^{RS}$ [RS90, DF93] and
Drinfeld second realization $U_{q}^{Dr}$ [D88] are studied in details. They
are characterized explicitly using L-operators and Drinfeld generators as
follows:

(i’) The K-operator of Drinfeld-Jimbo type is the image of (3.37) by
$\gamma^{\prime}_{D}$ for
$\bar{\epsilon}_{\pm}=0,\bar{k}_{+}=q^{2},\bar{k}_{-}=-q^{-1}$ and
$\gamma^{\prime}_{n}=0$, $\forall n$. As a corollary, the image of the
alternating generators in $U_{q}^{Dr}$ is obtained in terms of Drinfeld
generators/root vectors from (3.39)-(3.42); See [B20, Propositions 5.27];

(ii’) The K-operator of equitable type is the image of (3.37) by
$\gamma^{\prime}_{D}$ for (3.38), (3.43). As a corollary, the image of the
equitable generators in terms of Drinfeld generators is obtained. It follows
from (3.39)-(3.42), thus generalizing (2.6) of [IT03]. See Remark 3.9 for the
corresponding expressions in terms of root vectors.

Analogous results hold for their central extensions related with subalgebras
of $U_{q}(\widehat{gl_{2}})$, see [B20, Section 5.2]) and Section 4 of this
paper. Furthermore, for the central extension of any type (i) and (ii) the
image of the quantum determinant (4.18) coincides, up to an overall factor,
with a generating function for ‘half’ of central elements (4.12) of
$U_{q}(\widehat{gl_{2}})$.

The results of [B20] and the present paper show that Freidel-Maillet type
algebras provide a unified framework for $U_{q}^{DJ,+}$ and $U_{q}^{IT,+}$.
Actually, this unified framework can be extended to $U_{q}(\widehat{sl_{2}})$
(see [B20, Section 6] for the Drinfeld-Jimbo type), thus providing an
alternative to the FRT presentation for $U_{q}(\widehat{sl_{2}})$ [DF93]. For
the Drinfeld-Jimbo type, the existence of a Freidel-Maillet type presentation
can be understood from the FRT presentation using a Drinfeld twist. The most
interesting case is the Freidel-Maillet presentation for
$U_{q}(\widehat{sl_{2}})$ of equitable type. Details will be considered
elsewhere. As a preliminary, for the case of $U_{q}(sl_{2})$ the Freidel-
Maillet type presentation unifying the Drinfeld-Jimbo and equitable
presentations is introduced and studied in [B21].

Let us also mention that FRT presentations for higher rank affine Lie algebras
have been recently achieved, see [JLM19, JLM20, LP21]. By analogy, Freidel-
Maillet and alternating presentations of Drinfeld-Jimbo or equitable type for
higher rank cases are expected.

From the perspective of physics, it seems natural to study further Freidel-
Maillet type algebras of Drinfeld-Jimbo or equitable type. Indeed, it is known
that FRT presentations of quantum algebras provide a powerful framework for
the explicit construction and analysis of quantum integrable models such as
spin chains, using the Bethe ansatz or q-vertex operators’s techniques for
instance. By analogy, it would be natural to investigate the class of quantum
integrable models generated from Freidel-Maillet type presentations.

Acknowledgments: I thank Paul Terwilliger for many discussions, kind
explanations of his work and important comments on the manuscript. Also, I
thank him for sharing the unpublished results of [Ter21] which motivated the
analysis of Section 4. P.B. is supported by C.N.R.S.

##  Appendix A Drinfeld-Jimbo and Drinfeld (second realization) presentation
of $U_{q}(\widehat{sl_{2}})$

For the quantum affine Kac-Moody algebra $U_{q}(\widehat{sl_{2}})$, two
standard presentations are recalled. The Drinfeld-Jimbo presentation
$U_{q}^{DJ}$ and the Drinfeld (second) presentation $U_{q}^{Dr}$, see e.g.
[CP94, p.392], [Da14].

###  A.1. Drinfeld-Jimbo presentation $U_{q}^{DJ}$

Define the extended Cartan matrix $\\{a_{ij}\\}$ ($a_{ii}=2$, $a_{ij}=-2$ for
$i\neq j$). The quantum affine algebra $U_{q}(\widehat{sl_{2}})$ over
${\mathbb{C}}(q)$ is generated by $\\{E_{j},F_{j},K_{j}^{\pm 1}\\}$,
$j\in\\{0,1\\}$ which satisfy the defining relations

$\displaystyle K_{i}K_{j}=K_{j}K_{i}\ ,\quad
K_{i}K_{i}^{-1}=K_{i}^{-1}K_{i}=1\ ,\quad
K_{i}E_{j}K_{i}^{-1}=q^{a_{ij}}E_{j}\ ,\quad
K_{i}F_{j}K_{i}^{-1}=q^{-a_{ij}}F_{j}\
,\quad[E_{i},F_{j}]=\delta_{ij}\frac{K_{i}-K_{i}^{-1}}{q-q^{-1}}\ $

together with the $q-$Serre relations ($i\neq j$)

(A.1)
$\displaystyle\big{[}E_{i},\big{[}E_{i},\big{[}E_{i},E_{j}\big{]}_{q}\big{]}_{q^{-1}}\big{]}$
$\displaystyle=$ $\displaystyle 0\ ,$ (A.2)
$\displaystyle\big{[}F_{i},\big{[}F_{i},\big{[}F_{i},F_{j}\big{]}_{q}\big{]}_{q^{-1}}\big{]}$
$\displaystyle=$ $\displaystyle 0\ .$

The product $C=K_{0}K_{1}$ is the central element of the algebra.

The Hopf algebra structure is ensured by the existence of a comultiplication
$\Delta$ , antipode ${\cal S}$ and a counit ${\cal E}$ with

(A.3) $\displaystyle\Delta(E_{i})$ $\displaystyle=$ $\displaystyle 1\otimes
E_{i}+E_{i}\otimes K_{i}\ ,$ $\displaystyle\Delta(F_{i})$ $\displaystyle=$
$\displaystyle F_{i}\otimes 1+K_{i}^{-1}\otimes F_{i}\ ,$
$\displaystyle\Delta(K_{i})$ $\displaystyle=$ $\displaystyle K_{i}\otimes
K_{i}\ ,$ $\displaystyle{\cal S}(E_{i})=-E_{i}K_{i}^{-1}\ ,\quad{\cal
S}(F_{i})=-K_{i}F_{i}\ ,\quad{\cal S}(K_{i})=K_{i}^{-1}\qquad{\cal S}({1})=1\
$

and

$\displaystyle{\cal E}(E_{i})={\cal E}(F_{i})=0\ ,\quad{\cal E}(K_{i})=1\
,\qquad{\cal E}(1)=1\ .$

###  A.2. Drinfeld’s second realization $U_{q}^{Dr}$

A second presentation for the quantum affine algebra
$U_{q}(\widehat{sl_{2}})$, known as the Drinfeld’s second realization, is now
recalled. In [D88], it is shown that $U_{q}(\widehat{sl_{2}})$ is isomorphic
to the associative algebra over ${\mathbb{C}}(q)$ with generators
$\\{{{\textsf{x}}}_{k}^{\pm},{\textsf{h}}_{\ell},{\textsf{K}}^{\pm
1}|k\in{\mathbb{Z}},\ell\in{\mathbb{Z}}\backslash\\{0\\}\\}$, central elements
$C^{\pm 1/2}$ and the following relations (see e.g. [CP94, Theorem 12.2.1]):

(A.4) $\displaystyle C^{1/2}C^{-1/2}=1\
,\quad{\textsf{K}}{\textsf{K}}^{-1}={\textsf{K}}^{-1}{\textsf{K}}=1\ ,$ (A.5)
$\displaystyle\big{[}{\textsf{h}}_{k},{\textsf{h}}_{\ell}\big{]}=\delta_{k+\ell,0}\frac{1}{k}\big{[}2k\big{]}_{q}\frac{C^{k}-C^{-k}}{q-q^{-1}}\
,$ (A.6)
$\displaystyle\big{[}{\textsf{h}}_{k},{\textsf{x}}^{\pm}_{\ell}\big{]}=\pm\frac{1}{k}\big{[}2k\big{]}_{q}C^{\mp|k|/2}{\textsf{x}}^{\pm}_{k+\ell}\
,$ (A.7)
$\displaystyle{\textsf{K}}{\textsf{x}}^{\pm}_{k}{\textsf{K}}^{-1}=q^{\pm
2}{\textsf{x}}^{\pm}_{k}\ ,$ (A.8)
$\displaystyle{\textsf{x}}^{\pm}_{k+1}{\textsf{x}}^{\pm}_{\ell}-q^{\pm
2}{\textsf{x}}^{\pm}_{\ell}{\textsf{x}}^{\pm}_{k+1}=q^{\pm
2}{\textsf{x}}^{\pm}_{k}{\textsf{x}}^{\pm}_{\ell+1}-{\textsf{x}}^{\pm}_{\ell+1}{\textsf{x}}^{\pm}_{k}\
,$ (A.9)
$\displaystyle\big{[}{\textsf{x}}^{+}_{k},{\textsf{x}}^{-}_{\ell}\big{]}=\frac{(C^{(k-\ell)/2}\psi_{k+\ell}-C^{-(k-\ell)/2}\phi_{k+\ell})}{q-q^{-1}}\
,$

where the $\psi_{k}$ and $\phi_{k}$ are defined by the following equalities of
formal power series in the indeterminate $z$:

(A.10) $\displaystyle\psi(z)$ $\displaystyle=$
$\displaystyle\sum_{k=0}^{\infty}\psi_{k}z^{-k}={\textsf{K}}\exp\left((q-q^{-1})\sum_{k=1}^{\infty}{\textsf{h}}_{k}z^{-k}\right)\
,$ (A.11) $\displaystyle\phi(z)$ $\displaystyle=$
$\displaystyle\sum_{k=0}^{\infty}\phi_{-k}z={\textsf{K}}^{-1}\exp\left(-(q-q^{-1})\sum_{k=1}^{\infty}{\textsf{h}}_{-k}z\right)\
.$

Note that there exists an automorphism such that:

(A.12) $\displaystyle\theta:$
$\displaystyle{\textsf{x}}^{\pm}_{k}\mapsto{\textsf{x}}^{\mp}_{k}\
,\quad{\textsf{h}}_{k}\mapsto-{\textsf{h}}_{k}\
,\quad{\textsf{K}}\mapsto{\textsf{K}}\ ,\quad C\mapsto C^{-1},\ \quad q\mapsto
q^{-1}\ .$

An isomorphism $U_{q}^{DJ}\rightarrow U_{q}^{Dr}$ is given by (see e.g [CP94,
p. 393]:

(A.13) $\displaystyle\qquad K_{0}\mapsto C{\textsf{K}}^{-1}\ ,\quad
K_{1}\mapsto{\textsf{K}}\ ,\quad E_{1}\mapsto{\textsf{x}}_{0}^{+}\ ,\quad
E_{0}\mapsto{\textsf{x}}_{1}^{-}{\textsf{K}}^{-1}\ ,\quad
F_{1}\mapsto{\textsf{x}}_{0}^{-}\ ,\quad
F_{0}\mapsto{\textsf{K}}{\textsf{x}}_{-1}^{+}\ .$

Note that it is still an open problem to find the complete Hopf algebra
isomorphism between $U_{q}^{DJ}$ and $U_{q}^{Dr}$. Only partial information is
known, see e.g. [CP91, Section 4.4].

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