# A new current algebra and the reflection equation

P. Baseilhac  Laboratoire de Mathématiques et Physique Théorique CNRS/UMR
6083, Fédération Denis Poisson, Université de Tours, Parc de Grammont, 37200
Tours, FRANCE<EMAIL_ADDRESS>and  K. Shigechi  Institute for
Theoretical Physics, Valckenierstraat 65, 1018 XE Amsterdam, THE NETHERLANDS
<EMAIL_ADDRESS>
###### Abstract.

We establish an explicit algebra isomorphism between the quantum reflection
algebra for the $U_{q}(\widehat{sl_{2}})$ $R$-matrix and a new type of current
algebra. These two algebras are shown to be two realizations of a special case
of tridiagonal algebras ($q-$Onsager).

MSC: 81R50; 81R10; 81U15.

Keywords: Current algebra; Reflection equation; $q-$Onsager algebra; Quantum
integrable models

##  1\. Introduction

Discovered in the context of the quantum inverse scattering method for solving
quantum integrable systems, quantum groups appeared in the literature through
different ways (see [Cha] for references). On one hand, starting from the
fundamental independent discovery of Drinfeld [Dr1] and Jimbo [Jim] the
quantum affine algebras $U_{q}({\widehat{g}})$ were initially formulated using
a $q-$deformed version of the commutation relations between the elements of
the Chevalley presentation of ${\widehat{g}}$. Later on [Dr2], Drinfeld
proposed a new realization of $U_{q}({\widehat{g}})$ in terms of elements
$\\{x_{i,k}^{\pm},\varphi_{i,m},\psi_{i,n}|i=1,...,l;k\in{\mathbb{Z}},m\in-{\mathbb{Z}}_{+},n\in{\mathbb{Z}}_{+}\\}$
with $l=rank(g)$ generated through operator-valued functions
$x^{\pm}_{i}(u),\varphi_{i}(u),\psi_{i}(u)$ of the formal variable $u$, the
so-called currents. In some sense, the Drinfeld’s realization is a quantum
analogue of the loop realization of affine Lie algebras. Although Drinfeld
stated the isomorphism between the two realizations, the proof only appeared
later on [Be, Jin]. In particular, in [Be] (see also [Dam]) Lusztig’s theory
of braid group action [L] on the quantum enveloping algebras was used from
which an explicit homomorphism from Drinfeld’s new realization [Dr2] to the
initial one [Dr1, Jim] was obtained. On the other hand, an alternative
realization of quantum affine algebras $U_{q}({\widehat{g}})$ by means of
solutions of the quantum Yang-Baxter equation [KRS, KS, F1] \- called the
$R-$matrix - and the “RLL” algebraic relations of the quantum inverse
scattering method was proposed by Reshetikhin and Semenov-Tian-Shansky in
[RS], extending the previous results of Faddeev-Reshetikhin-Takhtajan [FRT1]
for finite dimensional simple Lie algebra $g$.

In view of these realizations, in [DiF] Ding and Frenkel exhibited an explicit
isomorphism between the “RLL” formulation and Drinfeld’s second realization.
Namely, $L-$operators were shown to admit a unique (Gauss) decomposition in
terms of Drinfeld’s currents $x^{\pm}_{i}(u),\varphi_{i}(u),\psi_{i}(u)$. So,
all these different realizations may be summarized by the following picture
which provides an unifying algebraic scheme for quantum affine algebras:

“RLL” algebra [FRT1] Yang-Baxter equation [DiF] Current algebra [Dr2]
Drinfeld’s presentation $\\{x_{i,k}^{\pm},\varphi_{i,m},\psi_{i,n}\\}$
$U_{q}({\widehat{g}})$ [RS],[DiF] [Be] [Jin] Drinfeld-Jimbo [Dr1], [Jim]

Beyond the interest of the algebraic structures involved, the explicit
relation between the two different realizations (“RLL” and Drinfeld’s one) of
$U_{q}({\widehat{g}})$ has found many interesting applications in the study of
quantum integrable systems and representation theory.

For quantum integrable systems with boundaries, Cherednik [Cher] and later on
[Sk] introduced another example of quadratic algebra associated with the so-
called reflection equations. In this case, given an $R-$matrix associated with
$U_{q}(\widehat{g})$ one is looking for a $K-$operator (sometimes called a
Sklyanin’s operator) satisfying “RKRK” algebraic relations. Motivated by the
study of related integrable systems, several examples of $K-$operators acting
on finite dimensional representations have been constructed. However, a
formulation of $K-$operators in terms of current algebras i.e. a “boundary” -
in reference to boundary integrable models - version of Ding-Frenkel [DiF]
analysis has never been explicitly presented, nor a “boundary” analogue of
Drinfeld’s presentation even in the simplest case $U_{q}(\widehat{sl_{2}})$.

In this paper, we argue that the $q-$Onsager algebra ${\mathbb{T}}$ (a type of
tridiagonal algebra) which independently appeared in the context of orthogonal
polynomial association schemes [Ter2] and hidden symmetries of boundary
integrable models [Bas] admits analogously two alternative realizations. One
realization is given in terms of a $K-$operator satisfying “RKRK” defining
relations for the $U_{q}(\widehat{sl_{2}})$ $R$-matrix, and the other
realization in terms of a new type of current algebra associated with the
generating set $\\{{\cal W}_{-k},{\cal W}_{k+1},{\cal G}_{k+1},{\tilde{\cal
G}}_{k+1}|k\in{\mathbb{Z}}_{+}\\}$ introduced in [BasK]. A new algebraic
scheme follows, which extends to the family of reflection equation algebras
the standard scheme relating the Faddeev-Reshetikhin-Takhtajan, Jimbo and
Drinfeld (first and second) realizations of quantum affine algebras (see above
picture). Although it is not considered here, the extension of our work to
other classical Lie algebra - technically more complicated - is an interesting
and open problem.

The paper is organized as follows. In Section 2, a new current algebra -
denoted $O_{q}(\widehat{sl_{2}})$ below - with generators $\cal
W_{\pm}(u),\cal G_{\pm}(u)$ and formal variable $u$ is introduced. It is shown
to be isomorphic to the “RKRK” algebra. A coaction map, the analogue of the
coproduct for Hopf’s algebras, is also explicitly derived. In Section 3, the
new currents are found to be generating functions in the symmetric variable
$U=(qu^{2}+q^{-1}u^{-2})/(q+q^{-1})$ which coefficients coincide with the
elements of the infinite dimensional algebra - denoted ${\cal A}_{q}$ below -
introduced in [BasK]. In the last section, based on the commuting properties
of the $K-$operator with the two generators of the $q-$Onsager algebra we
establish the isomorphism between ${\mathbb{T}}$ and the “RKRK” algebra. A new
algebraic scheme unifying these realizations is then proposed.

######  Notation .

In this paper, ${\mathbb{R}}$, ${\mathbb{C}}$, ${\mathbb{Z}}$ denote the field
of real, complex numbers and integers, respectively. We denote
${\mathbb{R}}^{*}={\mathbb{R}}\backslash\\{0\\}$,
${\mathbb{C}}^{*}={\mathbb{C}}\backslash\\{0\\}$,
${\mathbb{Z}}^{*}={\mathbb{Z}}\backslash\\{0\\}$ and ${\mathbb{Z}}_{+}$ for
nonnegative integers. We introduce the $q-$commutator
$\big{[}X,Y\big{]}_{q}=qXY-q^{-1}YX$ where $q$ is the deformation parameter,
assumed not to be a root of unity.

##  2\. A new current algebra

Let ${\cal V}$ be a finite dimensional space. Let the operator-valued function
$R:{\mathbb{C}}^{*}\mapsto\mathrm{End}({\cal V}\otimes{\cal V})$ be the
intertwining operator (quantum $R-$matrix) between the tensor product of two
fundamental representations ${\cal V}={\mathbb{C}}^{2}$ associated with the
algebra $U_{q}(\widehat{sl_{2}})$. The element $R(u)$ depends on the
deformation parameter $q$ and is defined by [Baxter]

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

where $u$ is called the spectral parameter. Then $R(u)$ satisfies the quantum
Yang-Baxter equation in the space ${\cal V}_{1}\otimes{\cal V}_{2}\otimes{\cal
V}_{3}$. Using the standard notation $R_{ij}(u)\in\mathrm{End}({\cal
V}_{i}\otimes{\cal V}_{j})$, it reads

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

Let us now consider an extension related with the reflection equation or
boundary quantum Yang-Baxter equation - which was first introduced in the
context of boundary quantum inverse scattering theory (see [Cher],[Sk] for
details) -. For simplicity and without loosing generality we consider the
simplest case, i.e. the $U_{q}(\widehat{sl_{2}})$ $R-$matrix defined above.

######  Definition 2.1 (“RKRK” Reflection equation algebra).

Define $R(u)$ to be (2.5). $B_{q}(\widehat{sl_{2}})$ is an associative algebra
with unit $1$ and generators $K_{11}(u)\equiv A(u)$, $K_{12}(u)\equiv B(u)$,
$K_{21}(u)\equiv C(u)$, $K_{22}(u)\equiv D(u)$ considered as the elements of
the $2\times 2$ square matrix $K(u)$ which obeys the defining relations
$\forall u,v$

(2.7) $\displaystyle R_{12}(u/v)\ (K(u)\otimes I\\!\\!I)\ R_{12}(uv)\
(I\\!\\!I\otimes K(v))\ =\ (I\\!\\!I\otimes K(v))\ R_{12}(uv)\ (K(u)\otimes
I\\!\\!I)\ R_{12}(u/v)\ .$

It is known that given a solution $K(u)$ of the reflection equation (2.7), one
can construct by induction other solutions using suitable combinations of Lax
operators $L(u)$. This is sometimes named as the “dressing” procedure. In
particular, for the simplest case one has:

######  Proposition 2.1 (see [Sk]).

Given $R(u)$ defined by (2.5), let $L(u)$ be a solution of the quantum Yang-
Baxter algebra with defining relations $\forall u,v$

(2.8) $\displaystyle R(u/v)(L(u)\otimes I\\!\\!I)(I\\!\\!I\otimes L(v))\ =\
(I\\!\\!I\otimes L(v))(L(u)\otimes I\\!\\!I)R(u/v)\ .$

Let $K(u)$ be a solution of (2.7). Then, the matrix $L(u)K(u)L^{-1}(u^{-1})$
is a solution of the reflection equation (2.7).

For instance, using the generating set $\\{S_{\pm},s_{3}\\}$ of the quantum
algebra $U_{q}(sl_{2})$ with defining relations $[s_{3},S_{\pm}]=\pm S_{\pm}$
and $[S_{+},S_{-}]=(q^{2s_{3}}-q^{-2s_{3}})/(q-q^{-1})$ , it is known that the
Lax operator

(2.11) $\displaystyle{L}(u)=\left(\begin{array}[]{cc}uq^{{1\over
2}}q^{s_{3}}-u^{-1}q^{-{1\over 2}}q^{-s_{3}}&(q-q^{-1})S_{-}\\\
(q-q^{-1})S_{+}&uq^{{1\over 2}}q^{-s_{3}}-u^{-1}q^{-{1\over 2}}q^{s_{3}}\\\
\end{array}\right)\ $

satisfies the quantum Yang-Baxter algebra (2.8). In quantum integrable lattice
models with boundaries, the “dressing” procedure is often used. Starting from
an elementary solution with $c-$number entries (associated with one boundary
of the system) and dressing the $K-$operator with a product of $N$
$L-$operators acting on different quantum spaces, one reconstructs a whole
spin chain with $N$ sites including inhomogeneities, if necessary [Sk].

In order to exhibit the new current algebra starting from the “RKRK”
reflection equation algebra, based on previous works on boundary quantum
integrable systems on the lattice [Bas, BasK] it seems rather natural to write
the elements $A(u)$, $B(u)$, $C(u)$, $D(u)$ in terms of new currents as
follows. It may be important to stress that Proposition 2.1 plays an essential
role (see [Bas, BasK]) in suggesting such a decomposition.

######  Lemma 2.1.

Suppose $q\neq 1$, $u\neq q^{-1}$ and $k_{\pm}\in{\mathbb{C}}^{*}$. Any
solution of the reflection equation algebra $B_{q}(\widehat{sl_{2}})$ admits
the following decomposition in terms of new elements $\cal W_{\pm}(u)$, $\cal
G_{\pm}(u)$:

(2.12) $\displaystyle A(u)=uq\cal W_{+}(u)-u^{-1}q^{-1}\cal W_{-}(u)\ ,$
(2.13) $\displaystyle D(u)=uq\cal W_{-}(u)-u^{-1}q^{-1}\cal W_{+}(u)\ ,$
(2.14) $\displaystyle B(u)=\frac{1}{k_{-}(q+q^{-1})}\cal
G_{+}(u)+\frac{k_{+}(q+q^{-1})}{(q-q^{-1})}\ ,$ (2.15) $\displaystyle
C(u)=\frac{1}{k_{+}(q+q^{-1})}\cal
G_{-}(u)+\frac{k_{-}(q+q^{-1})}{(q-q^{-1})}\ .$

Given the elements $A(u),B(u),C(u)$ of this form, this decomposition is
unique.

###### Proof.

The reflection equation being satisfied for arbitrary $u,v\in{\mathbb{C}}^{*}$
and generic $q$, in view of (2.5) the elements $A(u)$, $B(u)$, $C(u)$, $D(u)$
are a priori formal power series in $u$. With no restrictions, let us choose
$A(u)$, $B(u)$, $C(u)$ to be (2.12), (2.14), (2.15), respectively. We have to
show that $D(u)$ is uniquely defined by (2.13). To prove it, assume the set
$\\{A,B,C,D\\}$ given by (2.12)-(2.15) satisfies the reflection equation
algebra with (2.5). In terms of these elements, explicitly (2.7) reads

$\displaystyle(i)$ $\displaystyle
a_{-}c_{+}\left(BC^{\prime}-B^{\prime}C\right)+a_{-}a_{+}[A,A^{\prime}]=0\ ,$
$\displaystyle(i^{\prime})$ $\displaystyle
a_{-}c_{+}\left(CB^{\prime}-C^{\prime}B\right)+a_{-}a_{+}[D,D^{\prime}]=0\ ,$
$\displaystyle(ii)$ $\displaystyle
b_{-}b_{+}[A,D^{\prime}]+c_{-}c_{+}[D,D^{\prime}]+\
c_{-}a_{+}\big{(}CB^{\prime}-C^{\prime}B\big{)}=0\ ,$
$\displaystyle(ii^{\prime})$ $\displaystyle
b_{-}b_{+}[D,A^{\prime}]+c_{-}c_{+}[A,A^{\prime}]+\
c_{-}a_{+}\big{(}BC^{\prime}-B^{\prime}C\big{)}=0\ ,$ $\displaystyle(iii)$
$\displaystyle
c_{-}b_{+}\big{(}DA^{\prime}-D^{\prime}A\big{)}+b_{-}c_{+}\big{(}AA^{\prime}-D^{\prime}D\big{)}+\
b_{-}a_{+}[B,C^{\prime}]=0\ ,$ $\displaystyle(iii^{\prime})$ $\displaystyle
c_{-}b_{+}\big{(}AD^{\prime}-A^{\prime}D\big{)}+b_{-}c_{+}\big{(}DD^{\prime}-A^{\prime}A\big{)}+\
b_{-}a_{+}[C,B^{\prime}]=0\ ,$ $\displaystyle(iv)$ $\displaystyle
b_{-}b_{+}AC^{\prime}+c_{-}c_{+}DC^{\prime}+\ c_{-}a_{+}CA^{\prime}-\
a_{-}a_{+}C^{\prime}A-\ a_{-}c_{+}D^{\prime}C=0\ ,$ $\displaystyle(v)$
$\displaystyle b_{-}b_{+}B^{\prime}A+\ c_{-}c_{+}B^{\prime}D+\
c_{-}a_{+}A^{\prime}B-\ a_{-}a_{+}AB^{\prime}-\ a_{-}c_{+}BD^{\prime}=0\ ,$
$\displaystyle(vi)$ $\displaystyle b_{-}b_{+}C^{\prime}D+\
c_{-}c_{+}C^{\prime}A+\ c_{-}a_{+}D^{\prime}C-\ a_{-}a_{+}DC^{\prime}-\
a_{-}c_{+}CA^{\prime}=0\ ,$ $\displaystyle(vii)$ $\displaystyle
b_{-}b_{+}DB^{\prime}+\ c_{-}c_{+}AB^{\prime}+\ c_{-}a_{+}BD^{\prime}-\
a_{-}a_{+}B^{\prime}D-\ a_{-}c_{+}A^{\prime}B=0\ ,$ $\displaystyle(viii)$
$\displaystyle b_{-}a_{+}BD^{\prime}+\ c_{-}b_{+}DB^{\prime}+\
b_{-}c_{+}AB^{\prime}-\ a_{-}b_{+}D^{\prime}B=0\ ,$ $\displaystyle(ix)$
$\displaystyle b_{-}a_{+}A^{\prime}B+\ c_{-}b_{+}B^{\prime}A+\
b_{-}c_{+}B^{\prime}D-\ a_{-}b_{+}BA^{\prime}=0\ ,$ $\displaystyle(x)$
$\displaystyle b_{-}a_{+}D^{\prime}C+\ c_{-}b_{+}C^{\prime}D+\
b_{-}c_{+}C^{\prime}A-\ a_{-}b_{+}CD^{\prime}=0\ ,$ $\displaystyle(xi)$
$\displaystyle b_{-}a_{+}CA^{\prime}+\ c_{-}b_{+}AC^{\prime}+\
b_{-}c_{+}DC^{\prime}-\ a_{-}b_{+}A^{\prime}C=0\ ,$ $\displaystyle(xii)$
$\displaystyle a_{-}b_{+}[B,B^{\prime}]=0\ ,$ $\displaystyle(xiii)$
$\displaystyle a_{-}b_{+}[C,C^{\prime}]=0\ ,$

where $a(u)=uq-u^{-1}q^{-1}$, $b(u)=u-u^{-1}$, $c_{\pm}=q-q^{-1}$ and we used
the shorthand notations $a_{-}=a(u/v)$, $a_{+}=a(uv)$ and similarly for $b$.
Also $A=A(u)$, $A^{\prime}=A(v)$ and similarly for $B,C$ and $D$. Now,
consider another set, say $\\{A,B,C,{\overline{D}}\\}$,
${\overline{D}(u)}=D(u)+f(u)$ where $f(u)$ is an unknown function of $u$ and
the elements of the reflection equation algebra. If
$\\{A,B,C,{\overline{D}}\\}$ is also a solution of the reflection equation
algebra, then $f(u)\equiv f(A,B,C,D;u)$ \- the equations $(i)-(xiii)$ being
the complete set of defining relations. Replacing ${\overline{D}(u)}$ in
$(iv)-(xi)$, we obtain $B(u)f(v)=f(u)B(v)=C(u)f(v)=f(u)C(v)=0$ $\forall u,v$.
On the other hand, from $(i)-(iii^{\prime})$ one gets
$\big{[}A(u),f(v)\big{]}=0$. Acting with the l.h.s of $(ix)$ on $f(w)$ and
using previous equations it follows $\big{[}D(u),f(w)\big{]}=0$ $\forall u,w$.
All these equations imply that $f(u)\equiv 0$ $\forall u$. ∎

The next step is to prove the equivalence between the (sixteen in total)
independent equations coming from the reflection equation algebra (2.7) with
(2.5) and a closed system of commutation relations among the currents. The
relations below are among the main results of the paper.

######  Definition 2.2 (Current algebra).

$O_{q}(\widehat{sl_{2}})$ is an associative algebra with unit $1$, current
generators $\cal W_{\pm}(u)$, $\cal G_{\pm}(u)$ and parameter
$\rho\in{\mathbb{C}}^{*}$. Define the formal variables
$U=(qu^{2}+q^{-1}u^{-2})/(q+q^{-1})$ and $V=(qv^{2}+q^{-1}v^{-2})/(q+q^{-1})$
$\forall u,v$. The defining relations are:

(2.16) $\displaystyle\big{[}{\cal W}_{\pm}(u),{\cal W}_{\pm}(v)\big{]}=0\
,\qquad\qquad\qquad\qquad\qquad\qquad\qquad$ (2.17) $\displaystyle\big{[}{\cal
W}_{+}(u),{\cal W}_{-}(v)\big{]}+\big{[}{\cal W}_{-}(u),{\cal
W}_{+}(v)\big{]}=0\ ,\qquad\qquad\qquad\qquad\qquad\qquad\qquad$ (2.18)
$\displaystyle(U-V)\big{[}{\cal W}_{\pm}(u),{\cal
W}_{\mp}(v)\big{]}=\frac{(q-q^{-1})}{\rho(q+q^{-1})}\left({\cal
G}_{\pm}(u){\cal G}_{\mp}(v)-{\cal G}_{\pm}(v){\cal
G}_{\mp}(u)\right)\qquad\qquad\qquad$
$\displaystyle\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad+\frac{1}{(q+q^{-1})}\big{(}{\cal
G}_{\pm}(u)-{\cal G}_{\mp}(u)+{\cal G}_{\mp}(v)-{\cal G}_{\pm}(v)\big{)}\ ,$
(2.19) $\displaystyle{\cal W}_{\pm}(u){\cal W}_{\pm}(v)-{\cal W}_{\mp}(u){\cal
W}_{\mp}(v)+\frac{1}{\rho(q^{2}-q^{-2})}\big{[}{\cal G}_{\pm}(u),{\cal
G}_{\mp}(v)\big{]}\qquad\qquad\qquad\qquad\qquad$
$\displaystyle\qquad\qquad\qquad\qquad\qquad\qquad+\
\frac{1-UV}{U-V}\big{(}{\cal W}_{\pm}(u){\cal W}_{\mp}(v)-{\cal
W}_{\pm}(v){\cal W}_{\mp}(u)\big{)}=0\ ,$ (2.20) $\displaystyle U\big{[}{\cal
G}_{\mp}(v),{\cal W}_{\pm}(u)\big{]}_{q}-V\big{[}{\cal G}_{\mp}(u),{\cal
W}_{\pm}(v)\big{]}_{q}-(q-q^{-1})\big{(}{\cal W}_{\mp}(u){\cal
G}_{\mp}(v)-{\cal W}_{\mp}(v){\cal G}_{\mp}(u)\big{)}$
$\displaystyle\qquad\qquad\qquad\qquad\qquad\qquad\qquad\quad+\
\rho\big{(}U{\cal W}_{\pm}(u)-V{\cal W}_{\pm}(v)-{\cal W}_{\mp}(u)+{\cal
W}_{\mp}(v)\big{)}=0\ ,$ (2.21) $\displaystyle U\big{[}{\cal W}_{\mp}(u),{\cal
G}_{\mp}(v)\big{]}_{q}-V\big{[}{\cal W}_{\mp}(v),{\cal
G}_{\mp}(u)\big{]}_{q}-(q-q^{-1})\big{(}{\cal W}_{\pm}(u){\cal
G}_{\mp}(v)-{\cal W}_{\pm}(v){\cal G}_{\mp}(u)\big{)}$
$\displaystyle\qquad\qquad\qquad\qquad\qquad\qquad\qquad\quad+\
\rho\big{(}U{\cal W}_{\mp}(u)-V{\cal W}_{\mp}(v)-{\cal W}_{\pm}(u)+{\cal
W}_{\pm}(v)\big{)}=0\ ,$ (2.22) $\displaystyle\big{[}{\cal
G}_{\epsilon}(u),{\cal W}_{\pm}(v)\big{]}+\big{[}{\cal W}_{\pm}(u),{\cal
G}_{\epsilon}(v)\big{]}=0\ ,\quad\forall\epsilon=\pm\
,\qquad\qquad\qquad\qquad\qquad\qquad\qquad$ (2.23) $\displaystyle\big{[}{\cal
G}_{\pm}(u),{\cal G}_{\pm}(v)\big{]}=0\
,\qquad\qquad\qquad\qquad\qquad\qquad\qquad$ (2.24) $\displaystyle\big{[}{\cal
G}_{+}(u),{\cal G}_{-}(v)\big{]}+\big{[}{\cal G}_{-}(u),{\cal
G}_{+}(v)\big{]}=0\ .\qquad\qquad\qquad\qquad\qquad\qquad\qquad\ $

######  Remark 1.

There exists an automorphism $\Omega$ defined by:

(2.25) $\displaystyle\Omega(\cal W_{\pm}(u))=\cal W_{\mp}(u)\
,\qquad\Omega(\cal G_{\pm}(u))=\cal G_{\mp}(u)\ .$

Contrary to all known examples of Drinfeld currents associated with quantum
affine Lie algebras or superalgebras, it is important to notice that the
variables $u,v$ only arise through the symmetric $(qx^{2}\leftrightarrow
q^{-1}x^{-2},\ \forall x\in u,v$) combinations $U,V$, respectively. In view of
the connections with algebraic structures that appear in boundary quantum
integrable models [Bas, Bas2], such a fact is not surprising although not
obvious from (2.7). We now turn to the derivation of all equations above.

######  Theorem 1.

The map $\Phi:B_{q}(\widehat{sl_{2}})\mapsto O_{q}(\widehat{sl_{2}})$ defined
by (2.12-2.15) is an algebra isomorphism.

###### Proof.

First, according to Lemma 2.1 we have to show that the map $\Phi$ defined by
(2.12-2.15) is an algebra homomorphism from $B_{q}(\widehat{sl_{2}})$ to
$O_{q}(\widehat{sl_{2}})$. Set $\rho\equiv k_{+}k_{-}(q+q^{-1})^{2}$ and
define

$\displaystyle\qquad\qquad X_{1}\equiv\big{[}{\cal W}_{+}(u),{\cal
W}_{+}(v)\big{]}\ ,\qquad X_{2}\equiv\big{[}{\cal W}_{-}(u),{\cal
W}_{-}(v)\big{]}\ ,$ $\displaystyle\qquad\qquad X_{3}\equiv\big{[}{\cal
W}_{+}(u),{\cal W}_{-}(v)\big{]}+\big{[}{\cal W}_{-}(u),{\cal
W}_{+}(v)\big{]}\ ,$ $\displaystyle\qquad\qquad X_{4}\equiv\big{[}{\cal
G}_{+}(u),{\cal G}_{-}(v)\big{]}+\big{[}{\cal G}_{-}(u),{\cal
G}_{+}(v)\big{]}\ ,$ $\displaystyle\qquad\qquad
X_{5}\equiv(q+q^{-1})(U-V)\big{[}{\cal W}_{+}(u),{\cal
W}_{-}(v)\big{]}-\frac{(q-q^{-1})}{\rho}\left({\cal G}_{+}(u){\cal
G}_{-}(v)-{\cal G}_{+}(v){\cal G}_{-}(u)\right)\qquad\qquad\qquad$
$\displaystyle\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad-\big{(}{\cal
G}_{+}(u)-{\cal G}_{-}(u)+{\cal G}_{-}(v)-{\cal G}_{+}(v)\big{)}\ ,$

where the variables $U\equiv(qu^{2}+q^{-1}u^{-2})/(q+q^{-1})$ and similarly
for $V$ are introduced. In terms of the combinations $X_{i}$, it is
straightforward to show that the equations
$(i),(i^{\prime}),(ii),(ii^{\prime})$ above can be simply written,
respectively, as

$\displaystyle(i)$ $\displaystyle\Leftrightarrow\quad
a(uv)uvq^{2}X_{1}+a(uv)u^{-1}v^{-1}q^{-2}X_{2}-a(uv)u^{-1}vX_{3}-X_{5}=0\ ,$
$\displaystyle(i^{\prime})$ $\displaystyle\Leftrightarrow\quad
a(uv)uvq^{2}X_{2}+a(uv)u^{-1}v^{-1}q^{-2}X_{1}-a(uv)uv^{-1}X_{3}+\frac{q-q^{-1}}{\rho}X_{4}+X_{5}=0\
,$ $\displaystyle(ii)$
$\displaystyle\Leftrightarrow\quad\big{(}b(u/v)b(uv)uv^{-1}-(q-q^{-1})^{2}u^{-1}v^{-1}q^{-2}\big{)}X_{1}$
$\displaystyle\qquad+\big{(}b(u/v)b(uv)u^{-1}v-(q-q^{-1})^{2}uvq^{2}\big{)}X_{2}$
$\displaystyle\qquad-\big{(}b(u/v)b(uv)u^{-1}v^{-1}q^{-2}-(q-q^{-1})^{2}uv^{-1}\big{)}X_{3}$
$\displaystyle\qquad-a(uv)\frac{(q-q^{-1})}{\rho}X_{4}-a(uv)X_{5}=0\ ,$
$\displaystyle(ii^{\prime})$
$\displaystyle\Leftrightarrow\quad\big{(}b(u/v)b(uv)u^{-1}v-(q-q^{-1})^{2}uvq^{2}\big{)}X_{1}$
$\displaystyle\qquad+\big{(}b(u/v)b(uv)uv^{-1}-(q-q^{-1})^{2}u^{-1}v^{-1}q^{-2}\big{)}X_{2}$
$\displaystyle\qquad-\big{(}b(u/v)b(uv)uvq^{2}-(q-q^{-1})^{2}u^{-1}v\big{)}X_{3}$
$\displaystyle\qquad-a(uv)X_{5}=0\ .$

Simplifying these expressions, in particular it follows

$\displaystyle a(uv)(i)-(ii^{\prime})$ $\displaystyle\Leftrightarrow\quad
v^{2}q^{2}X_{1}+v^{-2}q^{-2}X_{2}-X_{3}=0\ ,$ $\displaystyle
a(uv)(i^{\prime})-(ii)$ $\displaystyle\Leftrightarrow\quad
v^{2}q^{2}X_{2}+v^{-2}q^{-2}X_{1}-X_{3}=0\ .$

Considering both equations for $v$ arbitrary, it implies $X_{1}=X_{2}$. Then
it is important to notice that the combinations $X_{i}|_{u\leftrightarrow
v}=-X_{i}$ for $i=1,2,3$. As now $X_{3}=(v^{2}q^{2}+v^{-2}q^{-2})X_{1}$ and
$u$ is arbitrary, it immediately follows $X_{3}\equiv X_{1}\equiv X_{2}\equiv
0$. Plugged into $(ii)$, $(ii^{\prime})$ we obtain $X_{4}\equiv X_{5}\equiv
0$. In terms of the currents, these equalities lead to the commutation
relations (2.16), (2.17), (2.18), (2.24).

As a consequence of these relations, after some straightforward calculations
one finds that the equations $(iii),(iii^{\prime})$ drastically simplify into
the relations (2.19).

Let us now consider the equations $(iv),(vi),(x),(xi)$ above. Proceeding
similarly, let us introduce

$\displaystyle Y_{1}\equiv(q+q^{-1})\big{(}U\big{[}C(v),{\cal
W}_{+}(u)\big{]}_{q}-V\big{[}C(u),{\cal
W}_{+}(v)\big{]}_{q}+(q-q^{-1})\big{(}{\cal W}_{-}(v)C(u)-{\cal
W}_{-}(u)C(v)\big{)}\big{)}\ ,$ $\displaystyle
Y_{2}\equiv(q+q^{-1})\big{(}U\big{[}{\cal
W}_{-}(u),C(v)\big{]}_{q}-V\big{[}{\cal
W}_{-}(v),C(u)\big{]}_{q}+(q-q^{-1})\big{(}{\cal W}_{+}(v)C(u)-{\cal
W}_{+}(u)C(v)\big{)}\big{)}\ ,$ $\displaystyle Y_{3}\equiv\big{[}C(u),{\cal
W}_{+}(v)\big{]}+\big{[}{\cal W}_{+}(u),C(v)\big{]}\ ,$ $\displaystyle
Y_{4}\equiv\big{[}C(u),{\cal W}_{-}(v)\big{]}+\big{[}{\cal
W}_{-}(u),C(v)\big{]}\ .$

In terms of these combinations, the equations $(iv),(vi),(x),(xi)$ become,
respectively,

$\displaystyle(iv)$ $\displaystyle\Leftrightarrow\quad
u\big{(}qY_{1}+q(v^{2}+v^{-2})Y_{3}+(q-q^{-1})Y_{4})\big{)}$
$\displaystyle\qquad\quad+\
u^{-1}\big{(}q^{-1}Y_{2}-q^{-1}(v^{2}+v^{-2})Y_{4}+(q-q^{-1})Y_{3})\big{)}=0\
,$ $\displaystyle(vi)$ $\displaystyle\Leftrightarrow\quad
u\big{(}qY_{2}-q(v^{2}+v^{-2})Y_{4}+q^{2}(q-q^{-1})Y_{3})\big{)}$
$\displaystyle\qquad\quad+\
u^{-1}\big{(}q^{-1}Y_{1}+q^{-1}(v^{2}+v^{-2})Y_{3}+q^{-2}(q-q^{-1})Y_{4})\big{)}=0\
,$ $\displaystyle(x)$ $\displaystyle\Leftrightarrow\quad
v\big{(}Y_{2}-q(q+q^{-1})UY_{4}+(q^{2}-q^{-2})Y_{3})\big{)}$
$\displaystyle\qquad\quad+\
v^{-1}\big{(}Y_{1}+q^{-1}(q+q^{-1})UY_{3}+(q^{2}-q^{-2})Y_{4})\big{)}=0\ ,$
$\displaystyle(xi)$ $\displaystyle\Leftrightarrow\quad
v\big{(}Y_{1}+q(q+q^{-1})UY_{3})\big{)}$ $\displaystyle\qquad\quad+\
v^{-1}\big{(}Y_{2}-q^{-1}(q+q^{-1})UY_{4})\big{)}=0\ .$

The variables $u,v$ and deformation parameter $q$ being arbitrary,
compatibility of these equations implies $Y_{1}\equiv Y_{2}\equiv Y_{3}\equiv
Y_{4}\equiv 0$. Replacing the explicit expression of $C(u)$ into $Y_{i}$, one
ends up with the commutation relations (2.20), (2.21), (2.22) for the current
${\cal G}_{-}(u)$. Similar analysis for the remaining equations
$(v),(vii),(viii),(ix)$ imply (2.20), (2.21), (2.22) for ${\cal G}_{+}(u)$.
Finally, from $(xii),(xiii)$ we immediately obtain (2.23).

Surjectivity of the map being shown, the injectivity of the homomorphism
follows from the fact that $\Phi$ is invertible for $u$ generic. This
completes the proof. ∎

Quantum affine algebras are known to be Hopf algebras, thanks to the existence
of a coproduct, counit and antipode actions. Although the explicit Hopf
algebra isomorphism between Drinfeld’s new realization (currents) and
Drinfeld-Jimbo construction is still an open problem, several results are
already known (see for instance [DiF]). For the new current algebra
(2.16)-(2.24), it is also important to exhibit analogous properties. Actually,
solely using the results of [Sk] a coaction map [Cha] can be easily
identified.

######  Proposition 2.2.

For any $k_{\pm},w\in{\mathbb{C}}^{*}$, there exists an algebra homomorphism
$\delta_{w}:O_{q}(\widehat{sl_{2}})\mapsto U_{q}(sl_{2})\times
O_{q}(\widehat{sl_{2}})$ such that

$\displaystyle\delta_{w}(\cal W_{\pm}(u))\\!\\!$ $\displaystyle=$
$\displaystyle\\!\\!\\!\left((q-q^{-1})^{2}S_{\pm}S_{\mp}-q(q^{\pm
2s_{3}}-q^{\mp 2s_{3}})\right)\otimes{\cal
W}_{\mp}(u)-(w^{2}+w^{-2})I\\!\\!I\otimes{\cal W}_{\pm}(u)$
$\displaystyle\\!\\!\\!\\!+\frac{(q-q^{-1})}{k_{+}k_{-}(q+q^{-1})}\left(k_{+}w^{\pm
1}q^{\pm 1/2}S_{+}q^{\pm s_{3}}\otimes{\cal G}_{+}(u)+k_{-}w^{\mp 1}q^{\mp
1/2}S_{-}q^{\pm s_{3}}\otimes{\cal G}_{-}(u)\right)$
$\displaystyle\\!\\!\\!\\!+(q+q^{-1})\left((k_{+}w^{\pm 1}q^{\pm
1/2}S_{+}q^{\pm s_{3}}+k_{-}w^{\mp 1}q^{\mp 1/2}S_{-}q^{\pm s_{3}})\otimes
I\\!\\!I+q^{\pm 2s_{3}}\otimes U{\cal W}_{\pm}(u)\right),$
$\displaystyle\delta_{w}(\cal G_{\pm}(u))$ $\displaystyle=$
$\displaystyle\\!\\!\\!\frac{k_{\mp}}{k_{\pm}}(q-q^{-1})^{2}S_{\mp}^{2}\otimes{\cal
G}_{\mp}(u)-(w^{2}q^{\pm 2s_{3}}+w^{-2}q^{\mp 2s_{3}})\otimes{\cal
G}_{\pm}(u)+I\\!\\!I\otimes(q+q^{-1})U{\cal G}_{\pm}(u)$ $\displaystyle\ \ +\
(q+q^{-1})^{2}(q-q^{-1})\left(k_{\mp}w^{\pm 1}q^{\mp
1/2}S_{\mp}q^{s_{3}}\otimes(U{\cal W}_{+}(u)-{\cal W}_{-}(u))\right.$
$\displaystyle\qquad\qquad\qquad\qquad\qquad\quad+\left.k_{\mp}w^{\mp 1}q^{\pm
1/2}S_{\mp}q^{-s_{3}}\otimes(U{\cal W}_{-}(u)-{\cal W}_{+}(u))\right)$
$\displaystyle\ \ +\
\frac{k_{+}k_{-}(q+q^{-1})^{2}}{(q-q^{-1})}\left((q+q^{-1})U+\frac{k_{\mp}}{k_{\pm}}(q-q^{-1})^{2}S_{\mp}^{2}-(w^{2}q^{\pm
2s_{3}}+w^{-2}q^{\mp 2s_{3}}+1)\right)\otimes I\\!\\!I\ .$

###### Proof.

According to [Sk] (see Proposition 2.1) and the Lax operator (2.11),
$L(uw)K(u)L(uw^{-1})$ is a solution $\forall w$ of (2.7). Expanding this
expression using (2.12)-(2.15), the new entries of $L(uw)K(u)L(uw^{-1})$ are
found to take the form (2.12)-(2.15) replacing $\cal
W_{\pm}(u)\rightarrow\delta_{w}(\cal W_{\pm}(u))$, $\cal
G_{\pm}(u)\rightarrow\delta_{w}(\cal G_{\pm}(u))$. For more details, we refer
the reader to [BasK] where similar calculations have been performed. ∎

##  3\. Another presentation

In [BasK], an infinite dimensional algebra denoted below ${\cal A}_{q}$ was
proposed in order to solve boundary integrable systems with hidden symmetries
related with a coideal subalgebra of $U_{q}(\widehat{sl_{2}})$. However, its
defining relations were essentially conjectured based on the commutation
relations and properties of certain operators acting on irreducible finite
dimensional tensor product of evaluation representations. The aim of this
Section is to construct an analogue of Drinfeld’s presentation for the current
algebra (2.16)-(2.24). As a consequence, it provides a rigorous derivation of
the relations conjectured in [BasK].

######  Definition 3.1 ([BasK]).

${\cal A}_{q}$ is an associative algebra with parameter
$\rho\in{\mathbb{C}}^{*}$, unit $1$ and generators $\\{{\cal W}_{-k},{\cal
W}_{k+1},{\cal G}_{k+1},{\tilde{\cal G}}_{k+1}|k\in{\mathbb{Z}}_{+}\\}$
satisfying the following relations:

(3.1) $\displaystyle\big{[}{\cal W}_{0},{\cal W}_{k+1}\big{]}=\big{[}{\cal
W}_{-k},{\cal W}_{1}\big{]}=\frac{1}{(q+q^{-1})}\big{(}{\tilde{\cal
G}_{k+1}}-{{\cal G}_{k+1}}\big{)}\ ,$ (3.2) $\displaystyle\big{[}{\cal
W}_{0},{\cal G}_{k+1}\big{]}_{q}=\big{[}{\tilde{\cal G}}_{k+1},{\cal
W}_{0}\big{]}_{q}=\rho{\cal W}_{-k-1}-\rho{\cal W}_{k+1}\ ,$ (3.3)
$\displaystyle\big{[}{\cal G}_{k+1},{\cal W}_{1}\big{]}_{q}=\big{[}{\cal
W}_{1},{\tilde{\cal G}}_{k+1}\big{]}_{q}=\rho{\cal W}_{k+2}-\rho{\cal W}_{-k}\
,$ (3.4) $\displaystyle\big{[}{\cal W}_{-k},{\cal W}_{-l}\big{]}=0\
,\quad\big{[}{\cal W}_{k+1},{\cal W}_{l+1}\big{]}=0\ ,\quad$ (3.5)
$\displaystyle\big{[}{\cal W}_{-k},{\cal W}_{l+1}\big{]}+\big{[}{{\cal
W}}_{k+1},{\cal W}_{-l}\big{]}=0\ ,$ (3.6) $\displaystyle\big{[}{\cal
W}_{-k},{\cal G}_{l+1}\big{]}+\big{[}{{\cal G}}_{k+1},{\cal W}_{-l}\big{]}=0\
,$ (3.7) $\displaystyle\big{[}{\cal W}_{-k},{\tilde{\cal
G}}_{l+1}\big{]}+\big{[}{\tilde{\cal G}}_{k+1},{\cal W}_{-l}\big{]}=0\ ,$
(3.8) $\displaystyle\big{[}{\cal W}_{k+1},{\cal G}_{l+1}\big{]}+\big{[}{{\cal
G}}_{k+1},{\cal W}_{l+1}\big{]}=0\ ,$ (3.9) $\displaystyle\big{[}{\cal
W}_{k+1},{\tilde{\cal G}}_{l+1}\big{]}+\big{[}{\tilde{\cal G}}_{k+1},{\cal
W}_{l+1}\big{]}=0\ ,$ (3.10) $\displaystyle\big{[}{\cal G}_{k+1},{\cal
G}_{l+1}\big{]}=0\ ,\quad\big{[}{\tilde{\cal G}}_{k+1},\tilde{{\cal
G}}_{l+1}\big{]}=0\ ,$ (3.11) $\displaystyle\big{[}{\tilde{\cal
G}}_{k+1},{\cal G}_{l+1}\big{]}+\big{[}{{\cal G}}_{k+1},\tilde{{\cal
G}}_{l+1}\big{]}=0\ .$

A natural ordering of ${\cal A}_{q}$ arises from the study of the commutation
relations above. Indeed, starting from monomials of lowest $k=0,1,...$ and
using (3.1) possible definitions of ${\cal G}_{1},{\tilde{\cal G}}_{1}$ are
such that $\mathrm{d}[{\cal G}_{1}]=\mathrm{d}[{\tilde{\cal G}}_{1}]\leq 2$,
where $\mathrm{d}$ denotes the degree of the monomials in the elements ${\cal
W}_{0},{\cal W}_{1}$. By induction, from (3.2), (3.3) with (3.1) one
immediately gets:

######  Corollary 3.1.

The elements of ${\cal A}_{q}$ are monomials in ${\cal W}_{0},{\cal W}_{1}$ of
degree:

(3.12) $\displaystyle\qquad\qquad\mathrm{d}[{\cal W}_{-k}]=\mathrm{d}[{{\cal
W}}_{k+1}]\leq 2k+1\qquad\mbox{and}\qquad\mathrm{d}[{\cal
G}_{k+1}]=\mathrm{d}[{\tilde{\cal G}}_{k+1}]\leq 2k+2\ ,\qquad
k\in{\mathbb{Z}}_{+}.$

Note that writing explicitly all higher elements of ${\cal A}_{q}$ in terms of
${\cal W}_{0},{\cal W}_{1}$ is essentially related with the construction of a
Poincare-Birkoff-Witt basis for the algebra considered in the next Section, a
problem that will be considered elsewhere.

######  Remark 2.

According to the ordering (3.12), the elements ${\cal G}_{1},{\tilde{\cal
G}}_{1}\in{\cal A}_{q}$ are uniquely determined:

(3.13) $\displaystyle{\cal G}_{1}=\big{[}{\cal W}_{1},{\cal
W}_{0}\big{]}_{q}+\alpha\qquad\mbox{and}\qquad{\tilde{\cal
G}}_{1}=\big{[}{\cal W}_{0},{\cal W}_{1}\big{]}_{q}+\alpha\qquad\forall\alpha\
\in{\mathbb{C}}\ .$

For the derivation of the second theorem, several other equalities will be
required which can all be deduced from the relations above and (3.13). Indeed,
let us show the following.

######  Proposition 3.1.

If (3.1)-(3.11) are satisfied, then the following relations hold:

(3.14) $\displaystyle\qquad\qquad\quad\big{[}{\cal W}_{-k-1},{\cal
W}_{l+1}\big{]}-\big{[}{\cal W}_{-k},{\cal
W}_{l+2}\big{]}=\frac{q-q^{-1}}{\rho(q+q^{-1})}\big{(}{\cal
G}_{k+1}\tilde{{\cal G}}_{l+1}-{\cal G}_{l+1}\tilde{{\cal G}}_{k+1}\big{)}\ ,$
(3.15) $\displaystyle\qquad\qquad\quad-{\cal W}_{-k}{\cal W}_{0}+{\cal
W}_{k+1}{\cal W}_{1}-{\cal W}_{-k-1}{\cal W}_{1}+{\cal W}_{0}{\cal
W}_{k+2}-\frac{1}{\rho(q^{2}-q^{-2})}\big{[}{\cal G}_{k+1},{\tilde{\cal
G}}_{1}\big{]}=0\ ,$ (3.16) $\displaystyle\qquad\qquad\quad{\cal
W}_{-k-1}{\cal W}_{-l}-{\cal W}_{k+2}{\cal W}_{l+1}-{\cal W}_{-k}{\cal
W}_{-l-1}+{\cal W}_{k+1}{\cal W}_{l+2}$ $\displaystyle\qquad\qquad\quad+{\cal
W}_{-k}{\cal W}_{l+1}-{\cal W}_{-l}{\cal W}_{k+1}-{\cal W}_{-k-1}{\cal
W}_{l+2}+{\cal W}_{-l-1}{\cal W}_{k+2}$
$\displaystyle\qquad\qquad\qquad\qquad\quad+\frac{1}{\rho(q^{2}-q^{-2})}\big{(}\big{[}{\cal
G}_{k+2},{\tilde{\cal G}}_{l+1}\big{]}-\big{[}{\cal G}_{k+1},{\tilde{\cal
G}}_{l+2}\big{]}\big{)}=0\ ,$ (3.17)
$\displaystyle\qquad\qquad\quad\big{[}{\cal G}_{l+1},{\cal
W}_{k+2}\big{]}_{q}-\big{[}{\cal G}_{k+1},{\cal
W}_{l+2}\big{]}_{q}-(q-q^{-1})\big{(}{\cal W}_{-k}{\cal G}_{l+1}-{\cal
W}_{-l}{\cal G}_{k+1}\big{)}=0\ ,$ (3.18)
$\displaystyle\qquad\qquad\quad\big{[}{\cal W}_{-k-1},{\cal
G}_{l+1}\big{]}_{q}-\big{[}{\cal W}_{-l-1},{\cal
G}_{k+1}\big{]}_{q}-(q-q^{-1})\big{(}{\cal W}_{k+1}{\cal G}_{l+1}-{\cal
W}_{l+1}{\cal G}_{k+1}\big{)}=0\ ,$ (3.19)
$\displaystyle\qquad\qquad\quad\big{[}{\tilde{\cal G}}_{l+1},{\cal
W}_{-k-1}\big{]}_{q}-\big{[}{\tilde{\cal G}_{k+1}},{\cal
W}_{-l-1}\big{]}_{q}-(q-q^{-1})\big{(}{\cal W}_{k+1}{\tilde{\cal
G}}_{l+1}-{\cal W}_{l+1}{\tilde{\cal G}_{k+1}}\big{)}=0\ ,$ (3.20)
$\displaystyle\qquad\qquad\quad\big{[}{\cal W}_{k+2},{\tilde{\cal
G}}_{l+1}\big{]}_{q}-\big{[}{\cal W}_{l+2},{\tilde{\cal
G}}_{k+1}\big{]}_{q}-(q-q^{-1})\big{(}{\cal W}_{-k}{\tilde{\cal
G}}_{l+1}-{\cal W}_{-l}{\tilde{\cal G}}_{k+1}\big{)}=0\ .$

###### Proof.

To show (3.14), let us consider the first commutator. Expand it using (3.2).
Combining ${\cal W}_{0}$ and ${\cal W}_{l+1}$ using (3.1), one finds:

$\displaystyle\big{[}{\cal W}_{-k-1},{\cal W}_{l+1}\big{]}$ $\displaystyle=$
$\displaystyle\frac{q}{\rho(q+q^{-1})}\big{(}\tilde{{\cal G}}_{l+1}{\cal
G}_{k+1}-\tilde{{\cal G}}_{k+1}{\cal G}_{l+1}\big{)}$
$\displaystyle+\frac{q^{-1}}{\rho(q+q^{-1})}\big{(}{\cal G}_{l+1}\tilde{{\cal
G}}_{k+1}-{\cal G}_{k+1}\tilde{{\cal G}}_{l+1}\big{)}+\big{[}{\cal
W}_{-l-1},{\cal W}_{k+1}\big{]}\ .$

Then, using (3.5) and (3.11) one obtains (3.14).

Consider now (3.15). Introduce (3.13) in the last commutator, and expand using
(3.2) and (3.3). Collecting terms and simplifying, one obtains (3.15).
Equation (3.16), although technically slightly more complicated, is derived
along the same line.

To show (3.17)-(3.20), the same procedure will be used so we only explain
(3.17). Consider the two commutators and expand using (3.3). Then, using (3.8)
and (3.11), one verifies that (3.17) is indeed satisfied. ∎

By analogy with Drinfeld’s construction, we are now looking for an infinite
dimensional set of elements of an algebra in terms of which the currents $\cal
W_{\pm}(u)$, $\cal G_{\pm}(u)$ can be expanded. According to the structure of
the equations (2.16)-(2.24) defining the current algebra - in particular the
dependence in the formal variable $U,V$ \- we obtain the second main result of
the paper.

######  Theorem 2.

Define the formal variable $U=(qu^{2}+q^{-1}u^{-2})/(q+q^{-1})$. Let
$\Psi:O_{q}(\widehat{sl_{2}})\mapsto{\cal A}_{q}$ be the map defined by

(3.21) $\displaystyle{\cal W}_{+}(u)=\sum_{k\in{\mathbb{Z}}_{+}}{\cal
W}_{-k}U^{-k-1}\ ,\quad{\cal W}_{-}(u)=\sum_{k\in{\mathbb{Z}}_{+}}{\cal
W}_{k+1}U^{-k-1}\ ,$ (3.22) $\displaystyle\quad{\cal
G}_{+}(u)=\sum_{k\in{\mathbb{Z}}_{+}}{\cal G}_{k+1}U^{-k-1}\ ,\quad{\cal
G}_{-}(u)=\sum_{k\in{\mathbb{Z}}_{+}}\tilde{{\cal G}}_{k+1}U^{-k-1}\ .$

Then, $\Psi$ is an algebra isomorphism between $O_{q}(\widehat{sl_{2}})$ and
${\cal A}_{q}$.

###### Proof.

Plugging (3.21), (3.22) into (2.16)-(2.24), expanding and identifying terms of
same order in $U^{-k}V^{-l}$ one finds all defining relations (3.1)-(3.11),
together with the set of higher relations (3.14)-(3.20). From Proposition 3.1,
it follows that the sixteen independent algebraic relations (3.1)-(3.11) are
sufficient i.e. the map is surjective. The currents being analytic in the
variable $U\in{\mathbb{C}}$, according to Cauchy’s theorem any element of
${\cal A}_{q}$ is uniquely determined from the currents using contour
integrals. The injectivity of the map follows, which completes the proof. ∎

It is important to stress that in [BasK], commutation relations among the so-
called transfer matrix were used to derive some of the relations (3.1)-(3.11).
However, the derivation described above uses solely the reflection equation
algebra. Consequently, this theorem establishes a rigorous proof of the
relations conjectured in [BasK]. In addition, for the case of the reflection
equation algebra with the $U_{q}(\widehat{sl_{2}})$ $R$-matrix it shows that
the presentation $\\{{\cal W}_{-k},{\cal W}_{k+1},{\cal G}_{k+1},{\tilde{\cal
G}}_{k+1}|k\in{\mathbb{Z}}_{+}\\}$ is the “boundary” analogue of Drinfeld’s
one.

##  4\. intertwiner of the $q-$Onsager (tridiagonal) algebra
and the reflection equation

The purpose of this Section is to exhibit an intertwiner $K(u)$ of the
$q-$Onsager algebra, to show its uniqueness and that it coincides exactly with
the solution $K(u)$ of the reflection equation (2.7). The final aim is
actually to establish the isomorphism between the new current algebra and the
$q-$Onsager algebra. Although the reader may be familiar with the ideas of
[Jim], it will be useful to first recall some well-known results. Indeed, the
procedure we follow to construct the intertwiner is analogous to the one
described in [Jim]. In the context of quantum integrable systems, note that
for finite dimensional representations intertwiners have already been obtained
along the same line in [MN, DeMS, Nep, DeG, DeM].

a. The $R-$matrix as an intertwiner of $U_{q}(\widehat{sl_{2}})$ [Jim].
In [Jim], Jimbo pointed out that intertwiners $R$ of quantum loop algebras
lead to trigonometric solutions of the quantum Yang-Baxter equation (2.8). Any
tensor product of two evaluation representations with generic evaluation
parameters $u$ and $v$ being indecomposable, by Schur’s lemma the solution $R$
is unique up to an overall scalar factor. In particular, considering the
quantum affine algebra $U_{q}(\widehat{sl_{2}})$ the construction of the
solution $R(u)$ given by (2.5) goes as follow.

First, we need to recall the realization of the quantum affine algebra
$U_{q}(\widehat{sl_{2}})$ in the Chevalley presenation
$\\{H_{j},E_{j},F_{j}\\}$, $j\in\\{0,1\\}$ (see e.g [Cha]):

######  Definition 4.1.

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

$\displaystyle[H_{i},H_{j}]=0\ ,\quad[H_{i},E_{j}]=a_{ij}E_{j}\
,\quad[H_{i},F_{j}]=-a_{ij}F_{j}\
,\quad[E_{i},F_{j}]=\delta_{ij}\frac{q^{H_{i}}-q^{-H_{i}}}{q-q^{-1}}\ $

together with the $q-$Serre relations

(4.1) $\displaystyle[E_{i},[E_{i},[E_{i},E_{j}]_{q}]_{q^{-1}}]=0\
,\quad\mbox{and}\quad[F_{i},[F_{i},[F_{i},F_{j}]_{q}]_{q^{-1}}]=0\ .$

The sum ${\it K}=H_{0}+H_{1}$ is the central element of the algebra. The Hopf
algebra structure is ensured by the existence of a comultiplication
$\Delta:U_{q}(\widehat{sl_{2}})\mapsto U_{q}(\widehat{sl_{2}})\otimes
U_{q}(\widehat{sl_{2}})$, antipode ${\cal S}:U_{q}(\widehat{sl_{2}})\mapsto
U_{q}(\widehat{sl_{2}})$ and a counit ${\cal
E}:U_{q}(\widehat{sl_{2}})\mapsto{\mathbb{C}}$ with

$\displaystyle\Delta(E_{i})$ $\displaystyle=$ $\displaystyle E_{i}\otimes
q^{-H_{i}/2}+q^{H_{i}/2}\otimes E_{i}\ ,$ $\displaystyle\Delta(F_{i})$
$\displaystyle=$ $\displaystyle F_{i}\otimes q^{-H_{i}/2}+q^{H_{i}/2}\otimes
F_{i}\ ,$ (4.2) $\displaystyle\Delta(H_{i})$ $\displaystyle=$ $\displaystyle
H_{i}\otimes I\\!\\!I+I\\!\\!I\otimes H_{i}\ ,$ $\displaystyle{\cal
S}(E_{i})=-E_{i}q^{-H_{i}}\ ,\quad{\cal S}(F_{i})=-q^{H_{i}}F_{i}\ ,\quad{\cal
S}(H_{i})=-H_{i}\qquad{\cal S}({I\\!\\!I})=1\ $

and

$\displaystyle{\cal E}(E_{i})={\cal E}(F_{i})={\cal E}(H_{i})=0\ ,\qquad{\cal
E}({I\\!\\!I})=1\ .$

Note that the opposite coproduct $\Delta^{\prime}$ can be similarly defined
with $\Delta^{\prime}\equiv\sigma\circ\Delta$ where the permutation map
$\sigma(x\otimes y)=y\otimes x$ for all $x,y\in U_{q}(\widehat{sl_{2}})$ is
used.

Then, by definition the intertwiner $R(u/v):{\cal V}_{u}\otimes{\cal
V}_{v}\mapsto{\cal V}_{v}\otimes{\cal V}_{u}$ between two fundamental
$U_{q}(\widehat{sl_{2}})-$evaluation representations obeys

(4.3) $\displaystyle
R(u/v)(\pi_{u}\times\pi_{v})\big{[}\Delta(x)\big{]}=(\pi_{u}\times\pi_{v})\big{[}\Delta^{\prime}(x)\big{]}R(u/v)\qquad\forall
x\in U_{q}(\widehat{sl_{2}})\ ,$

where the (evaluation) endomorphism
$\pi_{u}:U_{q}(\widehat{sl_{2}})\mapsto\mathrm{End}({\cal V}_{u})$ is chosen
such that $({\cal V}\equiv{\mathbb{C}}^{2})$

$\displaystyle\pi_{u}[E_{1}]=uq^{1/2}\sigma_{+}\ ,\qquad\ \ \ \ \
\pi_{u}[E_{0}]=uq^{1/2}\sigma_{-}\ ,$
$\displaystyle\pi_{u}[F_{1}]=u^{-1}q^{-1/2}\sigma_{-}\
,\qquad\pi_{u}[F_{0}]=u^{-1}q^{-1/2}\sigma_{+}\ ,$ (4.4)
$\displaystyle\pi_{u}[q^{H_{1}}]=q^{\sigma_{3}}\ ,\qquad\qquad\quad\
\pi_{u}[q^{H_{0}}]=q^{-\sigma_{3}}\ $

in terms of the Pauli matrices $\sigma_{\pm},\sigma_{3}$:

(4.11) $\displaystyle\sigma_{+}=\left(\begin{array}[]{cc}0&1\\\
0&0\end{array}\right)\ ,\qquad\sigma_{-}=\left(\begin{array}[]{cc}0&0\\\
1&0\end{array}\right)\ ,\qquad\sigma_{3}=\left(\begin{array}[]{cc}1&0\\\
0&-1\end{array}\right)\ .$

As one can easily check, the matrix $R(u)$ given by (2.5) indeed satisfies the
required conditions (4.3). The tensor product ${\cal V}_{u}\otimes{\cal
V}_{v}$ being indecomposable with respect to $U_{q}(\widehat{sl_{2}})$
evaluation representations for generic evaluation parameters $u,v$, the
intertwiner $R(u)$ is unique (up to an overall scalar factor). As a
consequence, it automatically satisfies the Yang-Baxter equation (2.8) which
may be depicted by the following commutative diagram setting $w=1$:

(4.12) $\begin{CD}{\cal V}_{u}\otimes{\cal V}_{v}\otimes{\cal
V}_{w}@>{R(u/v)\,\otimes\,\text{id}}>{}>{\cal V}_{v}\otimes{\cal
V}_{u}\otimes{\cal V}_{w}@>{id\,\otimes\,R(u/w)}>{}>{\cal V}_{v}\otimes{\cal
V}_{w}\otimes{\cal V}_{u}\\\ @V{}V{\text{id}\otimes
R(v/w)}V@V{R(v/w)\otimes\text{id}}V{}V\\\ {\cal V}_{u}\otimes{\cal
V}_{w}\otimes{\cal V}_{v}@>{R(u/w)\,\otimes\,\text{id}}>{}>{\cal
V}_{w}\otimes{\cal V}_{u}\otimes{\cal V}_{v}@>{id\,\otimes\,R(u/v)}>{}>{\cal
V}_{w}\otimes{\cal V}_{v}\otimes{\cal V}_{u}\end{CD}$

b. The $K-$matrix as an intertwiner of ${\mathbb{T}}$.
Tridiagonal algebras have been introduced and studied in [Ter1, ITTer, Ter2],
where they first appeared in the context of $P-$ and $Q-$polynomial
association schemes. A tridiagonal algebra is an associative algebra with unit
which consists of two generators A and ${\textsf{A}}^{*}$ called the standard
generators. In general, the defining relations depend on five scalars
$\rho,\rho^{*},\gamma,\gamma^{*}$ and $\beta$. In the following, we will focus
on the reduced parameter sequence $\gamma=0,\gamma^{*}=0$,
$\beta=q^{2}+q^{-2}$ and $\rho=\rho^{*}$ which exhibits all interesting
properties that can be extended to more general parameter sequences. We call
the corresponding algebra the $q-$Onsager algebra denoted ${\mathbb{T}}$, in
view of its closed relationship with the Onsager algebra [Ons] and the Dolan-
Grady relations [DoG]. In particular, the isomorphism between the Onsager and
Dolan-Grady algebraic structures has been studied in [Pe, AMPT, Dav] and shown
explicitly in [DaRo].

######  Definition 4.2 (see also [Ter2]).

The $q-$Onsager algebra $\mathbb{T}$ is the associative algebra with unit and
standard generators $\textsf{A},\textsf{A}^{*}$ subject to the following
relations

(4.13)
$\displaystyle[\textsf{A},[\textsf{A},[\textsf{A},\textsf{A}^{*}]_{q}]_{q^{-1}}]=\rho[\textsf{A},\textsf{A}^{*}]\
,\qquad[\textsf{A}^{*},[\textsf{A}^{*},[\textsf{A}^{*},\textsf{A}]_{q}]_{q^{-1}}]=\rho[\textsf{A}^{*},\textsf{A}]\
.$

######  Remark 3.

For $\rho=0$ the relations (4.13) reduce to the $q-$Serre relations of
$U_{q}(\widehat{sl_{2}})$. For $q=1$, $\rho=16$ they coincide with the Dolan-
Grady relations [DoG].

By analogy with the construction described above for the $R-$matrix and along
the lines described in [DeM, DeG], an intertwiner for ${\mathbb{T}}$ can be
easily constructed. Before, we need to introduce the concept of comodule
algebra using the analogue of the Hopf’s algebra coproduct action called the
coaction map.

######  Definition 4.3 ([Cha]).

Given a Hopf algebra ${\cal H}$ with comultiplication $\Delta$ and counit
${\cal E}$, ${\cal I}$ is called a left ${\cal H}-$comodule if there exists a
left coaction map $\delta:\ \ {\cal I}\rightarrow{\cal H}\otimes{\cal I}$ such
that

(4.14) $\displaystyle(\Delta\times id)\circ\delta=(id\times\delta)\circ\delta\
,\qquad({\cal E}\times id)\circ\delta\cong id\ .$

Right ${\cal H}-$comodules are defined similarly.

######  Proposition 4.1 (see also [Bas]).

Let $k_{\pm}\in{\mathbb{C}}^{*}$ and set $\rho\equiv
k_{+}k_{-}(q+q^{-1})^{2}$. The q-Onsager algebra ${\mathbb{T}}$ is a left
$U_{q}(\widehat{sl_{2}})-$comodule algebra with coaction map
$\delta:{\mathbb{T}}\rightarrow U_{q}(\widehat{sl_{2}})\otimes{\mathbb{T}}$
such that

$\displaystyle\delta({\textsf{A}})$ $\displaystyle=$
$\displaystyle(k_{+}E_{1}q^{H_{1}/2}+k_{-}F_{1}q^{H_{1}/2})\otimes
1+q^{H_{1}}\otimes{\textsf{A}}\ ,$ (4.15)
$\displaystyle\delta({\textsf{A}}^{*})$ $\displaystyle=$
$\displaystyle(k_{-}E_{0}q^{H_{0}/2}+k_{+}F_{0}q^{H_{0}/2})\otimes
1+q^{H_{0}}\otimes{\textsf{A}}^{*}\ .$

###### Proof.

The verification of the comodule algebra axioms (4.14) is immediate using
(4.2). One also has to check that $\delta$ is an algebra homomorphism i.e
$\delta({\textsf{A}}),\delta({\textsf{A}}^{*})$ satisfy (4.13). This
calculation is rather long but straightforward, so we omit the details (see
also [Bas2, Bas3]). ∎

Having identified such a coaction map, we are now in position to consider an
intertwiner relating representations of ${\mathbb{T}}$, a key ingredient in
relating the $q-$Onsager algebra and the reflection equation algebra.

######  Proposition 4.2.

Let $\pi_{u}:U_{q}(\widehat{sl_{2}})\mapsto\mathrm{End}({\cal V}_{u})$ be the
evaluation endomorphism for ${\cal V}\equiv{\mathbb{C}}^{2}$. Let $W$ denote a
vector space over ${\mathbb{C}}$ on which the elements of ${\mathbb{T}}$ act.
There exists an intertwinner

$\displaystyle K(u):{\cal V}_{u}\otimes W\mapsto{\cal V}_{u^{-1}}\otimes W$

such that

(4.16) $\displaystyle K(u)(\pi_{u}\times
id)\big{[}\delta(a)\big{]}=(\pi_{u^{-1}}\times id)\big{[}\delta(a)\big{]}K(u)\
,\qquad\forall a\in{\mathbb{T}}\ .$

It is unique (up to an overall scalar factor), and it satisfies the reflection
equation (2.7).

###### Proof.

First, let us identify one solution of (4.16). By definition, ${\cal V}_{u}$
is a two-dimensional vector space. Then $K(u)$ is a $2\times 2$ matrix, which
entries are formal power series in the variable $u$ in view of (4.4) and
(4.15). Define

(4.19) $\displaystyle K(u)=\left(\begin{array}[]{cc}A(u)&B(u)\\\
C(u)&D(u)\end{array}\right)\ .$

Replacing $K(u)$ in (4.16), we find that the entries must satisfy the
following system of equations

$\displaystyle\big{[}{\textsf{A}},A(u)\big{]}=q^{-1}u^{-1}\big{(}k_{-}B(u)-k_{+}C(u)\big{)}\
,$ (4.20)
$\displaystyle\big{[}{\textsf{A}},D(u)\big{]}=-qu\big{(}k_{-}B(u)-k_{+}C(u)\big{)}\
,$
$\displaystyle\big{[}{\textsf{A}},B(u)\big{]}_{q}=k_{+}\big{(}uA(u)-u^{-1}D(u)\big{)}\
,$
$\displaystyle\big{[}{\textsf{A}},C(u)\big{]}_{q^{-1}}=-k_{-}\big{(}uA(u)-u^{-1}D(u)\big{)}\
$

and similar relations for ${\textsf{A}}^{*}$, provided one substitutes
$q\rightarrow q^{-1},u\rightarrow u^{-1}$ in (4.20). Then, using (3.13) in
(3.2) for $k=0$ it is easy to notice that the defining relations (4.13) are
nothing but (3.4) for $k=0,l=1$, provided we consider the following
homomorphism

(4.21) $\displaystyle{\textsf{A}}\mapsto{\cal W}_{0}\
,\qquad{\textsf{A}}^{*}\mapsto{\cal W}_{1}\ .$

Now, identify the entries of $K(u)$ with (2.12)-(2.15). Expanding and using
the defining relations (3.1)-(3.3) of the algebra ${\cal A}_{q}$, it is easy
to check (4.20) as well as all other relations for ${\textsf{A}}^{*}$. So, at
least one solution $K(u)$ exists and it is written in terms of elements of
${\cal A}_{q}$. For generic $u$, the tensor product $\mathrm{End}({\cal
V}_{u})\otimes W$ is not decomposable with respect to ${\mathbb{T}}$
representations. By Schur’s lemma, this means that given $W$, the solution to
the intertwining relation (4.16) is unique (up to an overall scalar factor).
It remains to show that $K(u)$ satisfying (4.16) is automatically a solution
of the reflection equation algebra (2.7). To this end, let us recall that
$K(u):{\cal V}_{u}\otimes W\mapsto{\cal V}_{u^{-1}}\otimes W$ and
$R(u/v):{\cal V}_{u}\otimes{\cal V}_{v}\mapsto{\cal V}_{v}\otimes{\cal
V}_{u}$. Then, the proof that this solution $K(u)$ satisfies the reflection
equation (2.7) follows from the commutativity of the following diagram (up to
an overall scalar factor):

$\begin{CD}{\cal V}_{u}\otimes{\cal V}_{v}\otimes
W@>{id\,\otimes\,K(v)}>{}>{\cal V}_{u}\otimes{\cal V}_{v^{-1}}\otimes
W@>{R(uv)\,\otimes\,id}>{}>{\cal V}_{v^{-1}}\otimes{\cal V}_{u}\otimes W\\\
@V{}V{R(u/v)\otimes id}V@V{id\otimes K(u)}V{}V\\\ {\cal V}_{v}\otimes{\cal
V}_{u}\otimes W\qquad{\cal V}_{v^{-1}}\otimes{\cal V}_{u^{-1}}\otimes W\\\
@V{}V{id\otimes K(u)}V@V{R(u/v)\otimes id}V{}V\\\ {\cal V}_{v}\otimes{\cal
V}_{u^{-1}}\otimes W@>{R(uv)\,\otimes\,\text{id}}>{}>{\cal
V}_{u^{-1}}\otimes{\cal V}_{v}\otimes W@>{id\,\otimes\,K(v)}>{}>{\cal
V}_{u^{-1}}\otimes{\cal V}_{v^{-1}}\otimes W\end{CD}$

∎

Combining previous results, we obtain the third main result of the paper:

######  Theorem 3.

The $q-$Onsager algebra ${\mathbb{T}}$ and the current algebra
$O_{q}(\widehat{sl_{2}})$ are isomorphic.

###### Proof.

According to Proposition 4.2, $K(u)$ with (2.12)-(2.15) is the unique
intertwiner of ${\mathbb{T}}$ satisfying (4.16). Also, it satisfies the
reflection equation algebra (2.7). So, $K(u)$ establishes the isomorphism
between ${\mathbb{T}}$ and the reflection equation algebra (2.7) for the
$U_{q}(\widehat{sl_{2}})$ $R-$matrix. Theorem 1 then establishes the
isomorphism between the reflection equation algebra (2.7) and
$O_{q}(\widehat{sl_{2}})$, which supports the claim. ∎

Although the isomorphism between ${\mathbb{T}}$ and
$O_{q}({\widehat{sl_{2}}})\cong{\cal A}_{q}$ is now established, an
interesting problem remains to construct an explicit homomorphism from ${\cal
A}_{q}$ to ${\mathbb{T}}$, i.e. to write all higher elements of ${\cal A}_{q}$
solely in terms of ${\cal W}_{0},{\cal W}_{1}$. This problem will be
considered elsewhere.

To conclude, the $q-$Onsager algebra ${\mathbb{T}}$ admits two different
realizations: one [see Proposition 4.2] in terms of the reflection equation
algebra for the $U_{q}({\widehat{sl_{2}}})$ $R-$matrix and another one in
terms [see Theorems 1, 2, 3] of the current algebra
$O_{q}({\widehat{sl_{2}}})\cong{\cal A}_{q}$. Previous results are resumed by
the picture below.

“RKRK” algebra [Cher, Sk] Reflection equation for $U_{q}(\widehat{sl_{2}})$
Theorems 1,2 Current algebra (Def. 2.2) Presentation $\\{{\cal W}_{-k},{\cal
W}_{k+1},{\cal G}_{k+1},{\tilde{\cal G}}_{k+1}\\}$ [BasK]
$O_{q}({\widehat{sl_{2}}})$ Proposition 4.2 Theorem 3 $q-$Onsager algebra
${\mathbb{T}}$ [Ter2] Figure 1.  An algebraic scheme for
$O_{q}({\widehat{sl_{2}}})$

Acknowledgements: Part of this work has been supported by the ANR Research
project “Boundary integrable models: algebraic structures and correlation
functions”, contract number JC05-52749. P.B thanks S. Pakuliak for detailed
explanations and notes about Drinfeld’s construction at the early stage of
this work, as well as P. Terwilliger for reading the manuscript and helpful
comments.

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# Spin-orbit torque emerging from orbital textures in centrosymmetric
materials

Luis M. Canonico  Catalan Institute of Nanoscience and Nanotechnology (ICN2),
CSIC and BIST, Campus UAB, Bellaterra, 08193 Barcelona, Spain  Jose H. Garcia
<EMAIL_ADDRESS>Catalan Institute of Nanoscience and Nanotechnology
(ICN2), CSIC and BIST, Campus UAB, Bellaterra, 08193 Barcelona, Spain
Corresponding author. Email<EMAIL_ADDRESS>Stephan Roche  Catalan
Institute of Nanoscience and Nanotechnology (ICN2), CSIC and BIST, Campus UAB,
Bellaterra, 08193 Barcelona, Spain  ICREA, Institució Catalana de Recerca i
Estudis Avançats, 08070 Barcelona, Spain

###### Abstract

We unveil a hitherto concealed spin-orbit torque mechanism driven by orbital
degrees of freedom in centrosymmetric two-dimensional transition metal
dichalcogenides (focusing on PtSe2). Using first-principles simulations,
tight-binding models and large-scale quantum transport calculations, we show
that such a mechanism fundamentally stems from a spatial localization of
orbital textures at opposite sides of the material, which imprints their
symmetries onto spin-orbit coupling effects, further producing efficient and
tunable spin-orbit torque. Our study suggests that orbital-spin entanglement
at play in centrosymmetric materials can be harnessed as a resource for
outperforming conventional spin-orbit torques generated by the Rashba-type
effects.

††preprint: APS/123-QED

Spin-orbit torque (SOT) is a central mechanism mediated by charge-to-spin
conversion that enables efficient manipulation of magnetization in spintronic
devices Gambardella and Miron (2011); Manchon  _et al._ (2019); Shao  _et al._
(2021). SOT is usually generated by nonequilibrium manifestations of the spin-
orbit coupling (SOC) such the spin-Hall and Rashba-Edelstein effects which
accumulate intrinsic angular momentum source that exerts some torque on nearby
magnets. These mechanisms are well described in _noncentrosymmetric_ systems
Bihlmayer  _et al._ (2022) and are fundamentally related to spin-momentum
locking in reciprocal space. Other mechanisms have been discussed for low-
dimensional systems, including the contribution of the Berry curvature
Kurebayashi  _et al._ (2014) or skew scattering Sousa  _et al._ (2020)
effects, but they require inversion symmetry breaking, so to date, all known
SOT mechanisms exclude the large family of _centrosymmetric_ materials.

Recently, the possibility of unconventional spatially-dependent spin-momentum
locking properties has been discussed for a particular family of
_centrosymmetric_ materials, namely the 1T-phase transition metal
dichalcogenides (TMDs). In these materials, the emergence of hidden spin
textures is dictated by the presence of dipolar fields within the crystal
structure Zhang  _et al._ (2014); Yuan  _et al._ (2019), as confirmed by ARPES
experiments Yao  _et al._ (2017); Clark  _et al._ (2022). On the other hand,
the electrical manipulation of the orbital degrees of freedom can give rise to
the orbital-Hall effect, where a longitudinal current drives a sizable
transverse flow of orbital angular momentum even in systems with negligible
SOC Bernevig  _et al._ (2005); Go  _et al._ (2018); Jo  _et al._ (2018);
Bhowal and Satpathy (2020); Bhowal and Vignale (2021); Canonico  _et al._
(2020); Cysne  _et al._ (2021); Pezo  _et al._ (2022); Choi  _et al._ (2023).
Such orbital effects have been shown to also signal the presence of high-order
topological phases Costa  _et al._ (2023), pointing to novel paradigms in
“orbitronics". Importantly, the role of the orbital degrees of freedom and
intertwined spin response was recognized in a more general description of the
Rashba SOC Park  _et al._ (2011, 2012); Bihlmayer  _et al._ (2022); Kim and
Otani (2022) and the driving or enhancement of SOT Go  _et al._ (2020); Ding
_et al._ (2020); Lee  _et al._ (2021). However, a fundamental orbital-based
description of complex spin phenomena such as the hidden spin textures and
their ultimate nature in generating novel SOT effects, remains largely
unexplored.

In this letter, we predict a novel SOT mechanism in _centrosymmetric_
materials (such as monolayer PtSe2) whose origin stems from intrinsic orbital
textures of the electronic states entangled with their spin features. The
hybridization between orbitals with different symmetries is the mechanism
behind the emergence of this unconventional SOT, which can be further tailored
using external electric fields. The theoretical results are obtained using
realistic tight-binding models elaborated with first-principles simulations
and implemented in large-scale (real-space) quantum transport simulations.
These discovered new SOT components in _centrosymmetric_ structures enlarge
the portfolio of available materials for designing ultracompact 2D materials-
based spintronic architectures in the context of non-volatile memory
technologies Yang  _et al._ (2022).

Figure 1:  (a) Bandstructure of free-standing monolayer PtSe2 computed using
fully relativistic band structure DFT (red dotted lines) and PAO Hamiltonian
(blue solid lines). Inset: Side view of PtSe2 monolayer (left) where its
height is chosen to match the $y$-axis of the dipolar electric field as a
function of the $z-$direction computed from DFT (right). (b) Orbital texture
of PtSe2 projected in the top Se atoms, computed in the energy window shown in
(a) (grey area), where $L_{x}(\bm{k})$ and $L_{y}(\bm{k})$ are represented as
the $x$ and $y$ components of each arrow and $L_{z}(\bm{k})$ as the background
color. (c) Spin texture of PtSe2 represented following the convention in (b)
but replacing the orbital angular momentum with the spin operator.

_Methods_. We performed Density Functional Theory (DFT) calculations using the
plane-wave-based code Quantum Espresso Giannozzi  _et al._ (2017). The
correlation and exchange functionals are treated within the generalized
gradient approximation (GGA) Perdew  _et al._ (1996). We described the ionic
cores using fully relativistic projected augmented wave potentials (PAW)
Kresse and Joubert (1999) and set an energy cutoff of $120$ Ry and $800$ Ry
for computing the wave functions and charge densities, respectively. For the
self-consistent run, we used a $k$-mesh of $24\times 24\times 1$ points, and
for subsequent calculations, we set a $32\times 32\times 1$ mesh. To avoid
spurious interactions due to periodic boundary conditions, we added a vacuum
spacing of $17.57$ $\AA$ and set the forces and the self-consistent cutoffs to
$5.142$ meV/$\AA$ and $4.03\times 10^{-9}$ Ry, respectively. We obtained an
energy gap of 1.17 eV and a lattice constant of 3.75 $\AA$ in good agreement
with previous first-principles calculations. We then constructed an effective
tight-binding Hamiltonians from our DFT calculations using the pseudo-atomic
orbital projection (PAO) method Agapito  _et al._ (2013, 2015, 2016a, 2016b)
implemented in PAOFLOW Buongiorno Nardelli  _et al._ (2018); Cerasoli  _et
al._ (2021). This approach consists in projecting the Kohn-Sham energy states
into the compact subspace spanned by the pseudo-atomic orbitals built in the
PAW potentials. We used PAW potentials for the Pt and Se constructed with an
$spd$ and $sp$ basis, respectively. The computation of the spin-orbit torque
is achieved using
$\bm{T}=\sum_{i}\hat{\bm{m}}\Delta_{xc}\times\langle\bm{S}^{i}\rangle$, where
$\hat{\bm{m}}$ is the magnetization direction, $\Delta_{xc}$ is the exchange
coupling strength and $\langle\bm{S}^{i}\rangle$ is the nonequilibrium spin
density at the $i$-th atomic plane calculated with the Kubo-Bastin formula
Bastin  _et al._ (1971):

$\langle\bm{S}^{i}(\varepsilon)\rangle=-2\text{Im}\int_{-\infty}^{\varepsilon}d\varepsilon^{\prime}f(\varepsilon^{\prime})\times\\\
\left(\text{Tr}\langle\bm{S}^{i}\delta(\hat{H}-\varepsilon^{\prime})\partial_{\varepsilon^{\prime}}{G_{\varepsilon^{\prime}}}^{+}(\hat{\bm{J}}\cdot\bm{E})\rangle\right),$
(1)

where $\bm{S}^{i}$ is the vector of spin density operators at the $i$-th
atomic plane, $\bm{E}$ is the electric field, $\hat{\bm{J}}$ is the current
density operator $\hat{\bm{J}}=-\frac{ei}{\hbar}\left[\hat{H},\bm{R}\right]$
with $\hat{\bm{R}}$ the position operator and $\hat{H}$ the Hamiltonian of the
system, and $G_{\varepsilon}^{+}$ and $\delta(H-\varepsilon)$ the retarded
Green’s and spectral functions are approximated using the kernel-polynomial
method João  _et al._ (2020); Fan  _et al._ (2021) as implemented in the
LSQUANT toolkit lsq . We used $1024$ Chebyshev moments with the Jackson kernel
to obtain an energy resolution of $\delta\varepsilon=33$ meV and fully exploit
the symmetries of the system, which allows considering sizes of $256\times
256$ unit cells with $34$ orbitals each, amounting to over $2$ millions of
atomic orbitals with $\sim 3.067\times 10^{9}$ hopping terms per unit cell
achieving DFT-level precision (See SM Sup ).

Figure 2: (a) Layer-projected nonequilibrium $S_{y}$ densities as a function
of the energy for the cases with $\Delta_{0}=0$ (dotted line) and
$\Delta_{0}=100\text{ meV}/\hbar$ (solid lines) with $\beta\to\infty$. The
colors indicate the $i=$top (red) and $i=$bottom (blue) contributions for the
case with $\Delta_{0}\neq 0$. Inset: Layer-projected nonequilibrium spin
densities for a wider energy range. (b) Torque efficiency $\xi_{x}$ for
$\Delta_{0}=100\text{ meV}/\hbar$ as a function of the energy. The colors
signal the speed of the decay of the effective exchange interaction with the
vertical distance.

Orbital and spin-momentum locking. FIG. 1 (a) shows a perfect agreement
between the DFT energy bands (red dots) against those computed using the PAO
Hamiltonian (blue solid lines) within the considered energy range. In the
inset of FIG. 1 (a), we show the crystalline structure of PtSe2 (left)
together with the electric dipole field (right) computed directly from the DFT
as the partial derivative of the planar average of the potential energy versus
the vertical distance within the simulation box. We found a finite and sizable
field located at both Se atomic planes (which are inversion partners) with
opposite values. Moreover, the fields vanish on the Pt ion as a consequence of
the inversion symmetry of the system. Panels FIG. 1 (b-c) present the orbital
and spin textures nearby the Fermi contour defined by the energy $E_{F}=-0.05$
eV, respectively. The orbital (spin) textures are defined as the Fermi surface
average of the orbital (spin) operators
($\langle\mathcal{O}_{n}(\bm{k})\rangle=-4K_{B}T\sum_{n}{f^{\prime}_{n\bm{k}}}\langle{\psi_{\bm{k}}}^{n}|\mathcal{O}|{\psi_{\bm{k}}}^{n}\rangle$),
where $K_{B}$ is the Boltzmann constant, $T$ is the temperature,
$f^{\prime}_{n\bm{k}}$ is the derivative of the Fermi-Dirac distribution and
$|\psi_{\bm{k}}^{n}\rangle$ is the eigenstate of the $n$-th band evaluated at
$\bm{k}$. The arrows in these figures indicate the $x$ and $y$ components of
the orbital (spin) textures while the color represents their out-of-plane
components as quantified by the color bars. FIG. 1 (b) shows that within our
energy range, the bands display orbital-momentum locking. The orbital textures
are reversed at opposite Se planes (see SM Sup ), which is related to the
internal symmetries of the system. Indeed, $1T$ monolayers of PtSe2 belong to
the $D_{3d}$ point group so that the chalcogen sublayers are rotated
$180^{\circ}$ to each other, yielding an equal distribution of charges across
the Se-Pt-Se atomic layers. Thus, this charge distribution and the absence of
a horizontal mirror plane allow the existence of the vertical dipole fields
(inset of FIG. 1 (a)), which are similar to the ones observed in layers of
bulk HfS2 Clark  _et al._ (2022). These fields induce a local out-of-the-plane
asymmetry and separate the manifold of $p$ orbitals of the chalcogen atoms in
two irreducible representations (Irrep). One of these representations is one-
dimensional, and it is related to the $p_{z}$ orbitals, which are moved to
lower energies, whereas the other is a $2\times 2$ Irrep spanned by linear
combinations of the $p_{x}$ and $p_{y}$ orbitals and are the main constituents
of the top valence bands at $\Gamma$. Since the latter Irrep is $2\times 2$,
in close analogy with spin, the top valence bands near the Brillouin zone are
mappable into orbital angular momentum (OAM) pseudospinors. In this picture,
it becomes clear that the hybridizations with the $p_{z}$ orbitals occurring
far from $\Gamma$ are translated as an effective coupling between these OAM
pseudospinors, which consequently leads to the emergence of the orbital Rashba
effect Park  _et al._ (2012); Go  _et al._ (2021). This mechanism explaining
the appearance of orbital textures is similar to the one presented in Ref.[Han
_et al._ , 2022]. However, here the global inversion symmetry in the system
imposes that the orbital (spin) textures on the two atomic chalcogen planes
are oppositely aligned and degenerate in reciprocal space, leading to the
vanishing of the absolute spin and OAM. Such type of orbital textures and
their real-space configuration can therefore form even in the absence of spin-
orbit coupling (See SM Sup ), reinforcing their crystal field origin. Finally,
FIG. 1 (c) shows that when the spin-orbit coupling is present, such orbital
textures are imprinted into the spin features, inducing spin-momentum locking,
which has the particularity to be opposite at each chalcogen atomic plane.
This is in agreement with recent experimental findings by Refs. [Yao  _et al._
, 2017] and [Clark  _et al._ , 2022], which have revealed the presence of
hidden spin-textures at the surfaces of PtSe2 and HfS2.

Atomic-plane localized spin-orbit torque. The dominant mechanism for SOT in
conventional 2D materials is the REE. It relies on the RE that emerges in
systems with broken inversion symmetry, typically due to the presence of an
interface, which splits the energy bands of opposite chirality and generates a
nonequilibrium spin density due to the differences in the Fermi contour. Here
for PtSe2, we demonstrate that spin-momentum locking emerges naturally from
the interplay between the SOC and the crystal fields. However, momentum-space
degeneracy enforced by the inversion symmetry of the system prevents the
formation of a net nonequilibrium spin densities. Nonetheless, practical 2D
SOT devices will be typically based on PtSe2/Ferromagnetic (FM)
heterostructures. Such stacking produces two effects at the origin of SOT,
namely (I) a built-in electric field induced by the presence of the FM and the
substrate, and responsible for the spin-splitting (Stark effect); and (II) an
asymmetric electronic coupling between the closest and farthest Se-planes with
respect to the FM. Generally, (I) is expected to dominate most of the
conventional SOT phenomenology in _centrosymmetric_ systems. However, the
complex crystalline structure and the real-space localization of orbital and
spin textures observed in 1T PtSe2 and other 1T TMDs suggest that the
asymmetric coupling between the chalcogen sublayers and the FM will disrupt
the compensation of the angular momentum textures, resulting in sizable-spin-
to-charge conversion and favoring the appearance of new SOT components. To
fully quantify the resulting contribution of (II), driven by such localized
nature of the interaction between the exchange coupling field and the orbital
and spin textures in this system, we included some phenomenological position-
dependent exchange term in our Hamiltonian as:

Figure 3: Total and extrinsic torque efficiencies $\xi_{x}$ and
${\xi_{x}}^{ext}$, as a function of the energy for the case with
$\Delta_{0}=100$ meV$/\hbar$ and $\beta\to\infty$ for negative and positive
fields. Inset: Comparison between the torque efficiencies in the absence of
external electric fields and with $\bm{E}=\pm 300$ mV/$\AA$, the labels refer
to the dominant contribution for SOT. The colors indicate the value of the
out-of-plane electric field.

$H_{xc}(\beta)=\sum_{i}\Delta_{0}\exp\left(-\beta\left(\frac{z_{i}-z_{top}}{z_{bot}}\right)^{2}\right)\cdot{S_{z}^{i}},$
(2)

where $S_{z}^{i}$ is the $z$ component of the spin operator of the $i$-atomic
plane, $\Delta_{0}=100\text{ meV}/\hbar$ is the intensity of the exchange
field, $\beta$ is a phenomenological decay factor of the proximity-induced
exchange interaction, $z_{top}$ and $z_{bot}$ are the positions of the top and
bottom Se atomic planes, respectively.

We first analyze the effect associated with the contribution (II) on the spin-
to-charge conversion and on the torque generation in the system. FIG. 2 (a)
compares the energy-resolved nonequilibrium $S_{y}$ spin density projected at
both chalcogen planes in the cases with (considering $\beta\to\infty$) and
without exchange coupling. From the figure, it is clear that the coupling term
breaks the inversion symmetry and disrupts the compensation of the
nonequilibrium spin densities of the top and bottom Se planes. Following this,
a layer-dependent energy shift occurs near the band gap and signals the real-
space decoupling of the energy states of both layers in this energy region.
FIG. 2 (a) inset, present the overall layer-dependent nonequilibrium $S_{y}$
spin densities for both cases. Qualitatively, the spin-to-charge conversion in
this system is not altered by the inclusion of the exchange term at the top
Se-plane, indicating that the resilience of these layer-localized spin-orbital
textures is due to the dominating energy scale set by the crystal field. Due
to the concealed nature of these spin textures and their Fermi-surface
dependence, we define a specific figure of merit; the torque efficiency as
$\mathbf{\xi}(\beta)=\sum_{i}\Delta_{xc}(\beta)\times\mathbf{S}^{i}(\beta)/\Delta_{0}\sum_{i}|\mathbf{S}^{i}(\beta)|$,
which quantifies the fraction of the total spin angular momentum that
participates in the torque generation irrespective of the specific details
associated to the bandstructure of the system.

Fig. 2(b) shows $\xi_{x}$ associated with the torque component along the
x-direction. One observes that $\xi_{x}$ increases with increasing $\beta$,
reaching values as large as $\xi_{x}\simeq 0.5$ over most of the considered
energy range. This enhancement of $\xi_{x}$ is related to the suppression of
the coupling between the bottom Se-plane and the FM, which leaves the top Se-
plane as the sole contributor to the torque intensity. In the vicinity of the
charge neutrality point, $\xi_{x}$ exceeds $0.6$, indicating the predominance
of the contribution coming from the top Se plane to the total current-induced
nonequilibrium spin densities. Additionally, near the charge neutrality point,
$\xi_{x}$ exhibits a dip that occurs for all the values of $\beta$. This
reduction is related to the predominance of contributions to the spin-to-
charge conversion coming from the bottom Se and Pt atoms. In contrast, the
$y$-component of the $\xi_{y}$ (shown in SM Sup ) is only sizable near the
band gap edge and inverts its value for large enough $\beta$, owing to the
suppression of the SOT contributions from the bottom Se and Pt atoms.

_Electrical tailoring of spin-orbit torque._ We further include the effect of
an external electric field in our simulations to deepen the analysis of the
interplay between the contributions (I) and (II) to the spin-orbit torque
components. FIG. 3 (a) shows the $\xi_{x}$ torque efficiency for various
negative and positive out-of-the-plane electric fields for large $\beta$. To
facilitate the comparison with the efficiencies portrayed in Fig. 2, we
compute the torque efficiency using the total spin angular momentum in the
system for the case without external electric fields as a reference to
quantify the additional contributions to the torque coming from (I). Upon
inspection, one can identify two regions where the effects operate
differently. Away from the energy gap (shown as (I) in the inset), both field
configurations produce similar changes in the torque efficiency but with
opposite signs. This sign dependence shows that contributions (I) dominate the
torque generation in this region. Conversely, near the charge neutrality point
(shaded area labeled as (II) on the inset), the combined effect of (I) and
(II) modulates the total torque efficiency. FIG. 3 (b) illustrates this more
clearly. Here we defined the extrinsic efficiency as
${\xi_{x}}^{ext}=\xi_{x}(E)-\xi_{x}(E=0)$, which captures solely the effects
associated with the external fields. This figure shows that for negative
fields, (I) and (II) have opposite signs, in the limit of large negative
fields, (I) dominates and defines the sign of the torques in the system. In
contrast, for positive fields, (I) and (II) work cooperatively, and for large
field amplitudes, the torque efficiency reaches approximately $\sim\times 3$
the efficiency without an external field. Thus, this electrical tunability
creates opportunities for electrical control of SOT in compact devices.

In conclusion, by combining first-principles calculations, tight-binding
models, and large-scale quantum transport simulations, we uncovered the
orbital origin of the hidden spin textures in inversion symmetric TMD and
analyzed their real-space characteristics. Using PtSe2 as a prototypical
member of the 1T TMD family, we demonstrated that the overall efficiency in
generating spin-orbit torques in these systems is substantially increased by
the real-space proximity interactions breaking inversion symmetry between the
two chalcogen atoms (which are inversion partners). Our results suggest that
other members of the 1T TMD family, such as HfS2, PtTe2 and PdSe2 could be
good candidates for exploiting hidden spin-orbital textures and their real-
space localization in torque applications. This discovered phenomenon,
evidencing the importance of the cooperative interplay between the orbital and
spin degrees of freedom, paves the way to more systematic real-space control
of spin-orbit torques for developing novel spin-based technologies.

###### Acknowledgements.

We acknowledge discussions with A.I. Figueroa, J.F. Sierra, S.O. Valenzuela,
and T.G. Rappoport. L.M.C. acknowledges funding from Ministerio de Ciencia e
Innovación de España under grant No. PID2019-106684GB-I00 / AEI /
10.13039/501100011033, FJC2021-047300-I, financiado por
MCIN/AEI/10.13039/501100011033 and the European Union
“NextGenerationEU”/PRTR.". J.H.G. acknowledge funding from the European Union
(ERC, AI4SPIN, 101078370). S.R and J.H.G, acknowledge funding from MCIN/AEI
/10.13039/501100011033 and European Union "NextGenerationEU/PRTR” under grant
PCI2021-122035-2A-2a and funding from the European Union’s Horizon 2020
research and innovation programme under grant agreement No 881603. ICN2 is
funded by the CERCA Programme/Generalitat de Catalunya and supported by the
Severo Ochoa Centres of Excellence programme, Grant CEX2021-001214-S, funded
by MCIN/AEI/10.13039.501100011033.

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