Representations of Quantum Affine Algebras in their $R$-Matrix Realization

Representations of Quantum Affine Algebras
in their $\boldsymbol{R}$-Matrix Realization††This paper is a contribution to
the Special Issue on Representation Theory and Integrable Systems in honor of
Vitaly Tarasov on the 60th birthday and Alexander Varchenko on the 70th
birthday. The full collection is available at
https://www.emis.de/journals/SIGMA/Tarasov-Varchenko.html

Naihuan JING †, Ming LIU ‡ and Alexander MOLEV §

N. Jing, M. Liu and A. Molev

† Department of Mathematics, North Carolina State University, Raleigh, NC
27695, USA<EMAIL_ADDRESS>
‡ School of Mathematics, South China University of Technology, Guangzhou,
510640, China<EMAIL_ADDRESS>
§ School of Mathematics and Statistics, University of Sydney, NSW 2006,
Australia<EMAIL_ADDRESS>
Received August 19, 2020, in final form December 25, 2020; Published online
December 28, 2020

We use the isomorphisms between the $R$-matrix and Drinfeld presentations of
the quantum affine algebras in types $B$, $C$ and $D$ produced in our previous
work to describe finite-dimensional irreducible representations in the
$R$-matrix realization. We also review the isomorphisms for the Yangians of
these types and use Gauss decomposition to establish an equivalence of the
descriptions of the representations in the $R$-matrix and Drinfeld
presentations of the Yangians.

$R$-matrix presentation; Drinfeld polynomials; highest weight representation;
Gauss decomposition

17B37

##  1 Introduction

The Yangians and quantum affine algebras associated with simple Lie algebras
comprise two remarkable families of infinite-dimensional quantum groups, as
introduced by V. Drinfeld [9] and M. Jimbo [17]. Both families have since
found numerous connections with many areas in mathematics and physics, they
possess rich and versatile representation theory.

Finite-dimensional irreducible representations of the Yangians were classified
by Drinfeld in his paper [11] by relying on the pioneering work by V. Tarasov
[27, 28]. The classification in [11] uses a new presentation of the Yangians
and quantum affine algebras which is now often referred to as the Drinfeld
presentation and which was used by V. Chari and A. Pressley to classify
finite-dimensional irreducible representations of quantum affine algebras [6,
Chapter 12].

A different kind of presentations of these algebras known as $R$-matrix
presentations goes back to the work of the Leningrad school of L. Faddeev;
see, e.g., [12, 23, 25, 26]. In accordance to Drinfeld [10], such
presentations can be produced from the universal $R$-matrix associated with a
quantum group and they can be associated with arbitrary finite-dimensional
representations. This approach was developed further in a more recent work
[29].

Explicit isomorphisms between the $R$-matrix and Drinfeld presentations of the
Yangians in type $A$ were given in the original work [11], while detailed
proofs were produced by J. Brundan and A. Kleshchev [5]. An analogous
isomorphism for the type $A$ quantum affine algebras is due to J. Ding and I.
Frenkel [8].

Despite their importance in representation theory and applications, such
isomorphisms had remained unknown beyond type $A$ until recent work [15, 19,
20, 21], where they were produced for the remaining classical types $B$, $C$
and $D$.

Our goal in this paper is to give a brief review of Yangians and quantum
affine algebras in types $B$, $C$ and $D$ and apply the isomorphisms to
describe finite-dimensional irreducible representations of these algebras in
their $R$-matrix realization.

In the case of Yangians, such a description was already given in [1], so that
the isomorphisms connect two sides of the representation theory and explain
how additional symmetries of the representation parameters arise from the
Gauss decomposition of the generator matrices. However, for the $R$-matrix
realization of the quantum affine algebras the isomorphisms are essential to
get a parametrization of their finite-dimensional irreducible representations.

A key step in our arguments relies on a consistency property for the
triangular decomposition of the algebra and the Gauss decomposition of the
generator matrices. The property ensures that the upper triangular Gaussian
generators can be expressed in terms of the ‘simple root generators’ implying
that the annihilation properties of the highest vectors agree in both
presentations of the algebras. Its Yangian version is well-known in type $A$
(see, e.g., [24, Section 3.1]), while proofs in the remaining classical types
follow from [19, Lemma 5.15]. This property has also been known for the
quantum affine algebras of type $A$ [22], so that the isomorphism of [8]
connects the descriptions of the representations given in [6, Section 12.2]
and [14]. Similar to the approach of [22], we establish the consistency
property in Section 3.2.4 in types $B$, $C$ and $D$ by deriving it from the
defining relations. An alternative way can rely on explicit formulas for the
universal $R$-matrices which, however, leads to more involved calculations.

In Appendix A we give a modified version of the Yangian isomorphisms produced
in [19], which is based on the opposite Gauss decomposition of the generator
matrix. This version can be applied to make an alternative connection between
the parameters of representations in the two realizations of the Yangians.

##  2 Representations of Yangians of types $\boldsymbol{B}$, $\boldsymbol{C}$
and $\boldsymbol{D}$

###  2.1 Classification results

We will denote by $\mathfrak{g}$ one of the simple Lie algebras of type
$B_{n}$, n or $D_{n}$. That is, $\mathfrak{g}=\mathfrak{o}_{N}$ is either the
orthogonal Lie algebra with $N=2n+1$ and $N=2n$, or
$\mathfrak{g}=\mathfrak{sp}_{N}$ is the symplectic Lie algebra with $N=2n$.
Choose simple roots in the form

$\displaystyle\alpha_{i}=\epsilon_{i}-\epsilon_{i+1}\qquad\text{for}\quad
i=1,\dots,n-1,$ (2.1)

and

$\displaystyle\alpha_{n}=\begin{cases}\epsilon_{n}&\text{if}\
\mathfrak{g}=\mathfrak{o}_{2n+1},\\\ 2\hskip 1.0pt\epsilon_{n}&\text{if}\
\mathfrak{g}=\mathfrak{sp}_{2n},\\\ \epsilon_{n-1}+\epsilon_{n}&\text{if}\
\mathfrak{g}=\mathfrak{o}_{2n},\end{cases}$

where $\epsilon_{1},\dots,\epsilon_{n}$ is an orthonormal basis of a Euclidian
space with the inner product $(\cdot\,,\cdot)$. The Cartan matrix $[A_{ij}]$
for $\mathfrak{g}$ is given by

$\displaystyle
A_{ij}=\frac{2(\alpha_{i},\alpha_{j})}{(\alpha_{i},\alpha_{i})}.$ (2.2)

We will use the notation

$\displaystyle r_{i}=\tfrac{1}{2}\hskip 1.0pt(\alpha_{i},\alpha_{i}),\qquad
i=1,\dots,n.$ (2.3)

As defined in [11], the Drinfeld Yangian ${\rm Y}^{D}(\mathfrak{g})$ is
generated by elements $\kappa_{i\hskip 1.0ptr}$, $\xi_{i\hskip 1.0ptr}^{+}$
and $\xi_{i\hskip 1.0ptr}^{-}$ with $i=1,\dots,n$ and $r=0,1,\dots$ subject to
the defining relations

$\displaystyle[\kappa_{i\hskip 1.0ptr},\kappa_{j\hskip 1.0pts}]=0,$
$\displaystyle\big{[}\xi_{i\hskip 1.0ptr}^{+},\xi_{j\hskip
1.0pts}^{-}\big{]}=\delta_{ij}\,\kappa_{i\,r+s},$
$\displaystyle\big{[}\kappa_{i\hskip 1.0pt0},\xi_{j\hskip
1.0pts}^{\pm}\big{]}=\pm\,(\alpha_{i},\alpha_{j})\,\xi_{j\hskip
1.0pts}^{\pm},$ $\displaystyle\big{[}\kappa_{i\,r+1},\xi_{j\hskip
1.0pts}^{\pm}\big{]}-\big{[}\kappa_{i\hskip
1.0ptr},\xi_{j\,s+1}^{\pm}\big{]}=\pm\,\frac{(\alpha_{i},\alpha_{j})}{2}\big{(}\kappa_{i\hskip
1.0ptr}\,\xi_{j\hskip 1.0pts}^{\pm}+\xi_{j\hskip
1.0pts}^{\pm}\,\kappa_{i\hskip 1.0ptr}\big{)},$
$\displaystyle\big{[}\xi_{i\,r+1}^{\pm},\xi_{j\hskip
1.0pts}^{\pm}\big{]}-\big{[}\xi_{i\hskip
1.0ptr}^{\pm},\xi_{j\,s+1}^{\pm}\big{]}=\pm\,\frac{(\alpha_{i},\alpha_{j})}{2}\big{(}\xi_{i\hskip
1.0ptr}^{\pm}\,\xi_{j\hskip 1.0pts}^{\pm}+\xi_{j\hskip
1.0pts}^{\pm}\,\xi_{i\hskip 1.0ptr}^{\pm}\big{)},$
$\displaystyle\sum_{p\in\mathfrak{S}_{m}}\big{[}\xi_{i\hskip
1.0ptr_{p(1)}}^{\pm},\big{[}\xi_{i\hskip
1.0ptr_{p(2)}}^{\pm},\dots\big{[}\xi_{i\hskip
1.0ptr_{p(m)}}^{\pm},\xi_{j\hskip 1.0pts}^{\pm}\big{]}\dots\big{]}\big{]}=0,$

where the last relation holds for all $i\neq j$, and we denoted $m=1-a_{ij}$.

By the results of [11] (they apply to any simple Lie algebra $\mathfrak{g}$),
every finite-dimensional irreducible representation of the algebra ${\rm
Y}^{D}(\mathfrak{g})$ is generated by a nonzero vector $\zeta$ (called the
highest vector) which is annihilated by all $\xi_{i\hskip 1.0ptr}^{+}$ and is
a simultaneous eigenvector for all $\kappa_{i\hskip 1.0ptr}$ so that

$\displaystyle\kappa_{i\hskip 1.0ptr}\zeta=d_{i\hskip 1.0ptr}\hskip
1.0pt\zeta,\qquad d_{i\hskip 1.0ptr}\in\mathbb{C}\hskip 1.0pt.$

Furthermore, there exist unique monic polynomials $P_{1}(u),\dots,P_{n}(u)$ in
$u$ such that

$\displaystyle 1+\sum_{r=0}^{\infty}d_{i\hskip 1.0ptr}\hskip
1.0ptu^{-r-1}=\frac{P_{i}(u+r_{i})}{P_{i}(u)},\qquad i=1,\dots,n.$

Equivalently, every finite-dimensional irreducible representation of the
algebra ${\rm Y}^{D}(\mathfrak{g})$ is generated by a nonzero vector
$\zeta^{\prime}$ (the highest vector with respect to the opposite triangular
decomposition) which is annihilated by all $\xi_{i\hskip 1.0ptr}^{-}$ and is a
simultaneous eigenvector for all $\kappa_{i\hskip 1.0ptr}$ so that

$\displaystyle\kappa_{i\hskip 1.0ptr}\zeta^{\prime}=d^{\,\prime}_{i\hskip
1.0ptr}\hskip 1.0pt\zeta^{\prime},\qquad d^{\,\prime}_{i\hskip
1.0ptr}\in\mathbb{C}\hskip 1.0pt.$

Furthermore, there exist unique monic polynomials $Q_{1}(u),\dots,Q_{n}(u)$ in
$u$ such that

$\displaystyle 1+\sum_{r=0}^{\infty}d^{\,\prime}_{i\hskip 1.0ptr}\hskip
1.0ptu^{-r-1}=\frac{Q_{i}(u)}{Q_{i}(u+r_{i})},\qquad i=1,\dots,n.$ (2.4)

All possible $n$-tuples of monic polynomials $(P_{1}(u),\dots,P_{n}(u))$ and
$(Q_{1}(u),\dots,Q_{n}(u))$ arise in this way. The equivalence of the two
parametrizations is seen by using the automorphism of the algebra ${\rm
Y}^{D}(\mathfrak{g})$ defined by

$\displaystyle\xi_{i\hskip 1.0ptr}^{+}\mapsto(-1)^{r+1}\hskip
1.0pt\xi_{i\hskip 1.0ptr}^{-},\qquad\xi_{i\hskip
1.0ptr}^{-}\mapsto(-1)^{r+1}\hskip 1.0pt\xi_{i\hskip
1.0ptr}^{+},\qquad\kappa_{i\hskip 1.0ptr}\mapsto(-1)^{r+1}\hskip
1.0pt\kappa_{i\hskip 1.0ptr}.$

###  2.2 Gaussian generators and isomorphisms

Introduce the following elements of the endomorphism algebra
${\rm{End}\,}\mathbb{C}\hskip 1.0pt^{N}\otimes{\rm{End}\,}\mathbb{C}\hskip
1.0pt^{N}$:

$\displaystyle P=\sum_{i,j=1}^{N}e_{ij}\otimes e_{ji}\qquad\text{and}\qquad
Q=\sum_{i,j=1}^{N}\varepsilon_{i}\hskip 1.0pt\varepsilon_{j}\,e_{ij}\otimes
e_{i^{\prime}j^{\prime}},$

where $e_{ij}\in{\rm{End}\,}\mathbb{C}\hskip 1.0pt^{N}$ are the matrix units,
and we use the notation $i^{\prime}=N+1-i$ and set $\varepsilon_{i}\equiv 1$
in the orthogonal case, and

$\displaystyle\varepsilon_{i}=\begin{cases}\phantom{-}1&\text{for}\
i=1,\dots,n,\\\ -1&\text{for}\ i=n+1,\dots,2n,\end{cases}$

in the symplectic case. Set

$\displaystyle\kappa=\begin{cases}N/2-1&\text{in the orthogonal case,}\\\
N/2+1&\text{in the symplectic case}.\end{cases}$

Following [30], consider the $R$-matrix $R(u)$

$\displaystyle R(u)=1-\frac{P}{u}+\frac{Q}{u-\kappa}.$

The extended Yangian ${\rm X}(\mathfrak{g})$ is defined as a unital
associative algebra with generators $t_{ij}^{(r)}$, where $1\leqslant
i,j\leqslant N$ and $r=1,2,\dots$, satisfying certain quadratic relations.
Introduce the formal series

$\displaystyle
t_{ij}(u)=\delta_{ij}+\sum_{r=1}^{\infty}t_{ij}^{(r)}\,u^{-r}\in{\rm
X}(\mathfrak{g})\big{[}\big{[}u^{-1}\big{]}\big{]}$

and set

$\displaystyle T(u)=\sum_{i,j=1}^{N}e_{ij}\otimes
t_{ij}(u)\in{\rm{End}\,}\mathbb{C}\hskip 1.0pt^{N}\otimes{\rm
X}(\mathfrak{g})\big{[}\big{[}u^{-1}\big{]}\big{]}.$

Consider the tensor product algebra ${\rm{End}\,}\mathbb{C}\hskip
1.0pt^{N}\otimes{\rm{End}\,}\mathbb{C}\hskip 1.0pt^{N}\otimes{\rm
X}(\mathfrak{g})$ and introduce the series with coefficients in this algebra
by

$\displaystyle T_{1}(u)=\sum\limits_{i,j=1}^{N}e_{ij}\otimes 1\otimes
t_{ij}(u)\qquad\text{and}\qquad T_{2}(u)=\sum\limits_{i,j=1}^{N}1\otimes
e_{ij}\otimes t_{ij}(u).$

The defining relations for the algebra ${\rm X}(\mathfrak{g})$ are then
written in the form

$\displaystyle R(u-v)\,T_{1}(u)\,T_{2}(v)=T_{2}(v)\,T_{1}(u)\,R(u-v).$

The Yangian ${\rm Y}(\mathfrak{g})$ is defined as the subalgebra of ${\rm
X}(\mathfrak{g})$ which consists of the elements stable under the
automorphisms

$\displaystyle\mu_{f}\colon\ T(u)\mapsto f(u)\,T(u),$

for all series $f(u)=1+f_{1}u^{-1}+f_{2}u^{-2}+\cdots$ with
$f_{i}\in\mathbb{C}\hskip 1.0pt$. Equivalently, ${\rm Y}(\mathfrak{g})$ is
isomorphic to the quotient of the algebra ${\rm X}(\mathfrak{g})$ by the
relation

$\displaystyle T^{\hskip 1.0pt{\rm t}}(u+\kappa)\,T(u)=1,$ (2.5)

where ${\rm t}$ denotes the matrix transposition with $e_{ij}^{{\rm
t}}=\varepsilon_{i}\,\varepsilon_{j}\,e_{j^{\prime},i^{\prime}}$.

####  2.2.1 Isomorphisms

Apply the Gauss decomposition to the matrix $T(u)$,

$\displaystyle T(u)=F(u)\,H(u)\,E(u),$ (2.6)

where $F(u)$, $H(u)$ and $E(u)$ are uniquely determined matrices of the form

$\displaystyle F(u)=\begin{bmatrix}1&0&\dots&0\,\\\ f_{21}(u)&1&\dots&0\\\
\vdots&\vdots&\ddots&\vdots\\\ f_{N1}(u)&f_{N2}(u)&\dots&1\end{bmatrix},\qquad
E(u)=\begin{bmatrix}\,1&e_{12}(u)&\dots&e_{1N}(u)\,\\\
\,0&1&\dots&e_{2N}(u)\\\ \vdots&\vdots&\ddots&\vdots\\\
0&0&\dots&1\end{bmatrix},$

and $H(u)={\rm diag}\,[h_{1}(u),\dots,h_{N}(u)]$. Define the series with
coefficients in ${\rm Y}(\mathfrak{g})$ by

$\displaystyle\kappa_{i}(u)=h_{i}(u-(i-1)/2)^{-1}\,h_{i+1}(u-(i-1)/2)$

for $i=1,\dots,n-1$, and

$\displaystyle\kappa_{n}(u)=\begin{cases}h_{n}(u-(n-1)/2)^{-1}\,h_{n+1}(u-(n-1)/2)&\text{for}\
\mathfrak{o}_{2n+1},\\\ \hskip
1.0pth_{n}(u-n/2)^{-1}\,h_{n+1}(u-n/2)&\text{for}\ \mathfrak{sp}_{2n},\\\
h_{n-1}(u-(n-2)/2)^{-1}\,h_{n+1}(u-(n-2)/2)&\text{for}\
\mathfrak{o}_{2n}.\end{cases}$

Furthermore, set

$\displaystyle\xi_{i}^{+}(u)=f_{i+1\,i}(u-(i-1)/2),\qquad\xi_{i}^{-}(u)=e_{i\,i+1}(u-(i-1)/2)$

for $i=1,\dots,n-1$,

$\displaystyle\xi_{n}^{+}(u)=\begin{cases}f_{n+1\,n}(u-(n-1)/2)&\text{for}\
\mathfrak{o}_{2n+1},\\\ f_{n+1\,n}(u-n/2)&\text{for}\ \mathfrak{sp}_{2n},\\\
f_{n+1\,n-1}(u-(n-2)/2)&\text{for}\ \mathfrak{o}_{2n}\end{cases}$

and

$\displaystyle\xi_{n}^{-}(u)=\begin{cases}e_{n\,n+1}(u-(n-1)/2)&\text{for}\
\mathfrak{o}_{2n+1},\\\ \frac{1}{2}e_{n\,n+1}(u-n/2)&\text{for}\
\mathfrak{sp}_{2n},\\\ e_{n-1\,n+1}(u-(n-2)/2)&\text{for}\
\mathfrak{o}_{2n}.\end{cases}$

Introduce elements of ${\rm Y}(\mathfrak{g})$ by the respective expansions
into power series in $u^{-1}$,

$\displaystyle\kappa_{i}(u)=1+\sum_{r=0}^{\infty}\kappa_{i\hskip
1.0ptr}\,u^{-r-1}\qquad\text{and}\qquad\xi_{i}^{\pm}(u)=\sum_{r=0}^{\infty}\xi_{i\hskip
1.0ptr}^{\pm}\,u^{-r-1}$

for $i=1,\dots,n$. According to [19, Main Theorem], the mapping which sends
the generators $\kappa_{i\hskip 1.0ptr}$ and $\xi_{i\hskip 1.0ptr}^{\pm}$ of
${\rm Y}^{D}(\mathfrak{g})$ to the elements of ${\rm Y}(\mathfrak{g})$ with
the same names defines an isomorphism ${\rm Y}^{D}(\mathfrak{g})\cong{\rm
Y}(\mathfrak{g})$.

####  2.2.2 Central elements of the extended Yangian

A presentation of the extended Yangian ${\rm X}(\mathfrak{g})$ in terms of the
Gaussian generators is given in [19, Theorem 5.14].111Note that the formulas
in [19, equations (5.4) and (5.47)] should be corrected by swapping the order
of the factors on their right hand sides. By [19, Theorem 5.8], all
coefficients of the series

$\displaystyle
z(u)=\begin{cases}\displaystyle\prod_{i=1}^{n}h_{i}(u+\kappa-i)^{-1}\hskip
1.0pt\prod_{i=1}^{n}h_{i}(u+\kappa-i+1)\cdot h_{n+1}(u)\hskip
1.0pth_{n+1}(u-1/2)&\text{if}\ N=2n+1,\vspace{1mm}\\\
\displaystyle\prod_{i=1}^{n-1}h_{i}(u+\kappa-i)^{-1}\hskip
1.0pt\prod_{i=1}^{n}h_{i}(u+\kappa-i+1)\cdot h_{n+1}(u)&\text{if}\
N=2n,\end{cases}$

belong to the center $\mathcal{C}$ of the extended Yangian ${\rm
X}(\mathfrak{g})$. Moreover, these coefficients generate the center and we
have the tensor product decomposition

$\displaystyle{\rm X}(\mathfrak{g})\cong{\rm
Y}(\mathfrak{g})\otimes\mathcal{C}.$

The following identity holds in ${\rm X}(\mathfrak{g})$:

$\displaystyle T^{\hskip 1.0pt{\rm t}}(u+\kappa)\,T(u)=z(u),$

so that the Yangian ${\rm Y}(\mathfrak{g})$ is isomorphic to the quotient of
${\rm X}(\mathfrak{g})$ by the relation $z(u)=1$. We will record the relations
which follow from the arguments in [19, Section 5.3].

######  Lemma 2.1.

In the algebra ${\rm X}(\mathfrak{g})$ we have

$\displaystyle h_{1}(u+\kappa)\hskip 1.0pth_{1^{\prime}}(u)=z(u),$
$\displaystyle h_{i}(u+\kappa-i)\hskip
1.0pth_{i^{\prime}}(u)=h_{i+1}(u+\kappa-i)\hskip 1.0pth_{(i+1)^{\prime}}(u),$
(2.7)

where $i=1,\dots,n$ for $N=2n+1$, and $i=1,\dots,n-1$ for $N=2n$.

###  2.3 Highest weight representations

######  Definition 2.2.

A representation $V$ of the algebra ${\rm Y}(\mathfrak{g})$ (or ${\rm
X}(\mathfrak{g})$) is called a highest weight representation if there exists a
nonzero vector $\zeta\in V$ such that $V$ is generated by $\zeta$ and the
following relations hold:

$\displaystyle t_{ij}(u)\,\zeta=0\qquad$ $\displaystyle\text{for}\quad
1\leqslant i<j\leqslant N,\qquad\text{and}$ $\displaystyle
t_{ii}(u)\,\zeta=\lambda_{i}(u)\,\zeta\qquad$ $\displaystyle\text{for}\quad
i=1,\dots,N,$

for some formal series $\lambda_{i}(u)\in 1+u^{-1}\hskip 1.0pt\mathbb{C}\hskip
1.0pt\big{[}\big{[}u^{-1}\big{]}\big{]}$. The vector $\zeta$ is called the
highest vector of the representation $V$.

The following classification theorem for finite-dimensional irreducible
representations of the Yangians in types $B$, $C$ and $D$ was proved in [1] in
terms of their $R$-matrix presentation. We will use the isomorphisms of [19]
which we recalled in Section 2.2.1, to make an explicit connection between
this theorem and the results of [11]. Note that such a connection was already
established in [15], where isomorphisms between three presentations of the
orthogonal and symplectic Yangians were constructed. However, those results
did not use the Gaussian presentation which we will rely on in our arguments.

######  Theorem 2.3.

  1. $1.$

Any finite-dimensional irreducible representation of the algebra ${\rm
Y}(\mathfrak{g})$ is a highest weight representation. Its parameters satisfy
the relations

$\displaystyle\frac{\lambda_{i}(u)}{\lambda_{i+1}(u)}=\frac{P_{i}(u+1)}{P_{i}(u)},\qquad
i=1,\dots,n-1,$

and

$\displaystyle\frac{\lambda_{n}(u)}{\lambda_{n+1}(u)}$
$\displaystyle=\frac{P_{n}(u+1/2)}{P_{n}(u)}\qquad$ $\displaystyle\text{for
type $B_{n}$},$ $\displaystyle\frac{\lambda_{n}(u)}{\lambda_{n+1}(u)}$
$\displaystyle=\frac{P_{n}(u+2)}{P_{n}(u)}\qquad$ $\displaystyle\text{for type
$C_{n}$},$ $\displaystyle\frac{\lambda_{n-1}(u)}{\lambda_{n+1}(u)}$
$\displaystyle=\frac{P_{n}(u+1)}{P_{n}(u)}\qquad$ $\displaystyle\text{for type
$D_{n}$},$

for some monic polynomials $P_{i}(u)$ in $u$.

  2. $2.$

Every $n$-tuple $(P_{1}(u),\dots,P_{n}(u))$ of monic polynomials in $u$ arises
in this way.

  3. $3.$

The series $\lambda_{i}(u)$ satisfy the relations

$\displaystyle\lambda_{i}(u+\kappa-i)\hskip
1.0pt\lambda_{i^{\prime}}(u)=\lambda_{i+1}(u+\kappa-i)\hskip
1.0pt\lambda_{(i+1)^{\prime}}(u),$ (2.8)

where $i=0,1,\dots,n$ for $N=2n+1$, and $i=0,1,\dots,n-1$ for $N=2n$, and we
set $\lambda_{0}(u)=\lambda_{0^{\prime}}(u):=1$.

###### Proof.

Using the isomorphism ${\rm Y}^{D}(\mathfrak{g})\cong{\rm Y}(\mathfrak{g})$
and the classification results recalled in Section 2.1, we find that any
finite-dimensional irreducible representation $V$ of the algebra ${\rm
Y}(\mathfrak{g})$ in types $B_{n}$ and $C_{n}$ is generated by a vector
$\zeta^{\prime}$ such that

$\displaystyle e_{i,i+1}(u)\hskip 1.0pt\zeta^{\prime}=0\qquad$
$\displaystyle\text{for}\quad i=1,\dots,n,$ (2.9) $\displaystyle
h_{i}(u)\,\zeta^{\prime}=\lambda_{i}(u)\,\zeta^{\prime}\qquad$
$\displaystyle\text{for}\quad i=1,\dots,n+1,$ (2.10)

for some formal series $\lambda_{i}(u)\in 1+u^{-1}\mathbb{C}\hskip
1.0pt\big{[}\big{[}u^{-1}\big{]}\big{]}$. For type $D_{n}$, the same
conditions hold, except that relation (2.9) with $i=n$ should be replaced with
$e_{n-1,n+1}(u)\hskip 1.0pt\zeta^{\prime}=0$. Indeed, for all types, relation
(2.9) is clear from the definition of the highest vector $\zeta^{\prime}$,
while (2.10) follows from the condition that $\zeta^{\prime}$ is an
eigenvector for all series $\kappa_{i}(u)$ with $i=1,\dots,n$ and $z(u)\hskip
1.0pt\zeta^{\prime}=\zeta^{\prime}$. Now, Lemma 2.1 and [19, Lemma 5.15] imply
that

$\displaystyle e_{ij}(u)\hskip 1.0pt\zeta^{\prime}$ $\displaystyle=0\qquad$
$\displaystyle\text{for}\quad i<j$ $\displaystyle h_{i}(u)\,\zeta^{\prime}$
$\displaystyle=\lambda_{i}(u)\,\zeta^{\prime}\qquad$
$\displaystyle\text{for}\quad i=1,\dots,N,$

for certain formal series $\lambda_{i}(u)\in 1+u^{-1}\mathbb{C}\hskip
1.0pt\big{[}\big{[}u^{-1}\big{]}\big{]}$ satisfying identities (2.8). Finally,
note that the values of the parameters (2.3) are found by $r_{i}=1$ for
$i=1,\dots,n-1$, while $r_{n}=1/2$ for type $B_{n}$, $r_{n}=2$ for type
$C_{n}$, and $r_{n}=1$ for type $D_{n}$, so that conditions in Part 1 of the
theorem follow from (2.4). ∎

######  Corollary 2.4.

All statements of Theorem 2.3 hold in the same form for the extended Yangian
${\rm X}(\mathfrak{g})$, except that the value $i=0$ is excluded for the
conditions (2.8).

###### Proof.

The proof of the theorem obviously extends to the algebra ${\rm
X}(\mathfrak{g})$. Relation (2.8) with $i=0$ is now replaced by the property
that the series $z(u)$ acts in the highest weight representation as
multiplication by $\lambda_{1}(u+\kappa)\hskip 1.0pt\lambda_{1^{\prime}}(u)$.
∎

######  Remark 2.5.

It is clear that the arguments used in the proof of the theorem can be
reversed, so that the classification theorem for the Yangian representations
in types $B$, $C$ and $D$ proved in [1] implies the corresponding results of
[11].

##  3 Representations of quantum affine algebras

###  3.1 Classification results

We will suppose that $q$ is a nonzero complex number which is not a root of
unity and set $q_{i}=q^{r_{i}}$ for $i=1,\dots,n$, with $r_{i}$ defined in
(2.3). The Cartan matrix of the simple Lie algebra $\mathfrak{g}$ of type
$B_{n}$, $C_{n}$ or $D_{n}$ is given by (2.2). We will use the standard
notation

$\displaystyle[k]_{q}=\frac{q^{k}-q^{-k}}{q-q^{-1}}$

for a nonnegative integer $k$, together with

$\displaystyle[k]_{q}!=\prod_{s=1}^{k}[s]_{q}\qquad\text{and}\qquad\begin{bmatrix}k\\\
r\end{bmatrix}_{q}=\frac{[k]_{q}!}{[r]_{q}!\,[k-r]_{q}!}.$

The quantum affine algebra ${\rm U}_{q}(\widehat{\mathfrak{g}})$ (with the
trivial central charge) in its Drinfeld presentation is the associative
algebra with generators $x_{i,m}^{\pm}$, $a_{i,l}$ and $k_{i}^{\pm 1}$ for
$i=1,\dots,n$ and $m,l\in\mathbb{Z}\hskip 1.0pt$ with $l\neq 0$, subject to
the following defining relations:

$\displaystyle k_{i}k_{j}=k_{j}k_{i},\qquad
k_{i}\,a_{j,l}=a_{j,l}\,k_{i},\qquad a_{i,m}\hskip 1.0pta_{j,l}=a_{j,l}\hskip
1.0pta_{i,m},$ $\displaystyle k_{i}\,x_{j,m}^{\pm}\,k_{i}^{-1}=q_{i}^{\pm
A_{ij}}x_{j,m}^{\pm},\qquad[a_{i,m},x_{j,l}^{\pm}]=\pm\frac{[mA_{ij}]_{q_{i}}}{m}\,x^{\pm}_{j,m+l},$
$\displaystyle x^{\pm}_{i,m+1}x^{\pm}_{j,l}-q_{i}^{\pm
A_{ij}}x^{\pm}_{j,l}x^{\pm}_{i,m+1}=q_{i}^{\pm
A_{ij}}x^{\pm}_{i,m}x^{\pm}_{j,l+1}-x^{\pm}_{j,l+1}x^{\pm}_{i,m},$
$\displaystyle\big{[}x_{i,m}^{+},x_{j,l}^{-}\big{]}=\delta_{ij}\,\frac{\psi_{i,m+l}-\varphi_{i,m+l}}{q_{i}-q_{i}^{-1}},$
$\displaystyle\sum_{\pi\in\mathfrak{S}_{r}}\sum_{l=0}^{r}(-1)^{l}\begin{bmatrix}r\\\
l\end{bmatrix}_{q_{i}}x^{\pm}_{i,s_{\pi(1)}}\cdots
x^{\pm}_{i,s_{\pi(l)}}x^{\pm}_{j,m}x^{\pm}_{i,s_{\pi(l+1)}}\cdots
x^{\pm}_{i,s_{\pi(r)}}=0,\qquad i\neq j,$

where in the last relation we set $r=1-A_{ij}$. The elements $\psi_{i,m}$ and
$\varphi_{i,-m}$ with $m\in\mathbb{Z}\hskip 1.0pt_{+}$ are defined by

$\displaystyle\psi_{i}(u):=\sum_{m=0}^{\infty}\psi_{i,m}u^{-m}=k_{i}\,\exp\left(\big{(}q_{i}-q_{i}^{-1}\big{)}\sum_{s=1}^{\infty}a_{i,s}u^{-s}\right),$
$\displaystyle\varphi_{i}(u):=\sum_{m=0}^{\infty}\varphi_{i,-m}u^{m}=k_{i}^{-1}\exp\left({-}\big{(}q_{i}-q_{i}^{-1}\big{)}\sum_{s=1}^{\infty}a_{i,-s}u^{s}\right),$

whereas $\psi_{i,m}=\varphi_{i,-m}=0$ for $m<0$.

######  Remark 3.1.

For the Lie algebras $\mathfrak{g}$ of types $C_{n}$ and $D_{n}$, it will be
convenient to work with an extended quantum algebra obtained by adjoining the
square roots $k_{n}^{\pm 1/2}$ and $(k_{n-1}k_{n})^{\pm 1/2}$, respectively.
Accordingly, we need to add the defining relations

$\displaystyle k_{n}^{1/2}\hskip 1.0ptx_{j,m}^{\pm}\hskip
1.0ptk_{n}^{-1/2}=q^{\pm A_{nj}}x_{j,m}^{\pm}$

for type $C_{n}$, and

$\displaystyle(k_{n-1}k_{n})^{1/2}\hskip 1.0ptx_{j,m}^{\pm}\hskip
1.0pt(k_{n-1}k_{n})^{-1/2}=q^{\pm(A_{n-1,j}+A_{nj})/2}x_{j,m}^{\pm}$

for type $D_{n}$, while the new elements commute with all the remaining
generators.

As explained in [6, Section 12.2], the algebra ${\rm
U}_{q}(\widehat{\mathfrak{g}})$ admits a family of sign automorphisms such
that the composition of any finite-dimensional irreducible representation with
a suitable automorphism of this kind is isomorphic to a type $\mathbf{1}$
representation. Such a representation is generated by a vector $\zeta$ which
is annihilated by all $x_{i,m}^{+}$ and is a simultaneous eigenvector for all
$k_{i}$ and $a_{i,l}$. Furthermore, if the series
$\Phi_{i}(u)\in\mathbb{C}\hskip 1.0pt[[u]]$ and
$\Psi_{i}(u)\in\mathbb{C}\hskip 1.0pt\big{[}\big{[}u^{-1}\big{]}\big{]}$ are
defined by

$\displaystyle\varphi_{i}(u)\hskip 1.0pt\zeta=\Phi_{i}(u)\hskip
1.0pt\zeta\qquad\text{and}\qquad\psi_{i}(u)\hskip 1.0pt\zeta=\Psi_{i}(u)\hskip
1.0pt\zeta,$

then there exist unique polynomials $P_{1}(u),\dots,P_{n}(u)$ in $u$ all with
constant term $1$ such that

$\displaystyle\Phi_{i}(u)=q_{i}^{{-}\deg P_{i}}\,\frac{P_{i}\big{(}u\hskip
1.0ptq_{i}^{2}\big{)}}{P_{i}(u)}=\Psi_{i}(u),\qquad i=1,\dots,n,$

where the equalities are understood for the expansions of the rational
functions in $u$ as series in $u$ and $u^{-1}$, respectively.222The roles of
$u$ and $u^{-1}$ are swapped in our notation as compared to [6, Section 12.2].
Every $n$-tuple of polynomials $(P_{1}(u),\dots,P_{n}(u))$ in $u$, where each
$P_{i}(u)$ has constant term $1$, arises in this way.

The above classification of finite-dimensional irreducible representations of
${\rm U}_{q}(\widehat{\mathfrak{g}})$ is valid for any simple Lie algebra
$\mathfrak{g}$. Note also that the corresponding results of [6, Section 12.2]
apply to centrally extended algebras ${\rm U}_{q}(\widehat{\mathfrak{g}})$,
which show that the central element acts trivially in any finite-dimensional
representation. For this reason we only consider the quotients of the quantum
affine algebras by the relation specifying the value of the central element as
equal to $1$.

###  3.2 Gaussian generators and isomorphisms

To define $R$-matrix realizations of the quantum affine algebras, introduce
elements of the endomorphism algebra ${\rm{End}\,}\mathbb{C}\hskip
1.0pt^{N}\otimes{\rm{End}\,}\mathbb{C}\hskip 1.0pt^{N}$ by

$\displaystyle P=\sum_{i,j=1}^{N}e_{ij}\otimes e_{ji}\qquad\text{and}\qquad
Q=\sum_{i,j=1}^{N}q^{\bar{\imath}-\bar{\jmath}}\,\varepsilon_{i}\hskip
1.0pt\varepsilon_{j}\,e_{i^{\prime}j^{\prime}}\otimes e_{ij},$

where $i^{\prime}=N+1-i$, as before, and

$\displaystyle\big{(}\,\overline{1},\overline{2},\dots,\overline{N}\,\big{)}=\begin{cases}\big{(}n-\frac{1}{2},\dots,\frac{3}{2},\frac{1}{2},0,-\frac{1}{2},-\frac{3}{2},\dots,-n+\frac{1}{2}\big{)}&\quad\text{for}\quad\mathfrak{g}=\mathfrak{o}_{2n+1},\\\
(n,\dots,2,1,-1,-2,\dots,-n)&\quad\text{for}\quad\mathfrak{g}=\mathfrak{sp}_{2n},\\\
(n-1,\dots,1,0,0,-1,\dots,-n+1)&\quad\text{for}\quad\mathfrak{g}=\mathfrak{o}_{2n}.\end{cases}$

Furthermore, introduce the $R$-matrix by

$\displaystyle R=q\,\sum_{i=1,i\neq i^{\prime}}^{N}e_{ii}\otimes
e_{ii}+e_{n+1,n+1}\otimes e_{n+1,n+1}+\sum_{i\neq j,j^{\prime}}e_{ii}\otimes
e_{jj}+q^{-1}\sum_{i\neq i^{\prime}}e_{ii}\otimes e_{i^{\prime}i^{\prime}}$
$\displaystyle\hphantom{R=}{}+\big{(}q-q^{-1}\big{)}\sum_{i<j}e_{ij}\otimes
e_{ji}-\big{(}q-q^{-1}\big{)}\sum_{i>j}q^{\bar{\imath}-\bar{\jmath}}\,\varepsilon_{i}\hskip
1.0pt\varepsilon_{j}\,e_{i^{\prime}j^{\prime}}\otimes e_{ij},$

where the second term $e_{n+1,n+1}\otimes e_{n+1,n+1}$ should be omitted if
$N=2n$. This formula goes back to [2, 3, 18, 26], which appeared along with
the spectral parameter-dependent $R$-matrix

$\displaystyle R(u)=(u-1)R+\big{(}q-q^{-1}\big{)}\left(P-\frac{u\hskip
1.0pt\xi-\xi}{u-\xi}\,Q\right),$ (3.1)

where

$\displaystyle\xi=q^{-2\kappa}=\begin{cases}q^{-N+2}&\text{if}\
\mathfrak{g}=\mathfrak{o}_{N},\\\ q^{-N-2}&\text{if}\
\mathfrak{g}=\mathfrak{sp}_{2n}.\end{cases}$

The corresponding two-parameter $R$-matrix $R(u,v)=v\hskip 1.0ptR(u/v)$ can be
written as

$\displaystyle R(u,v)=\sum_{i,j=1}^{N}\hskip 1.0pt\big{(}u\hskip
1.0ptq^{\delta_{ij}}-v\hskip 1.0ptq^{-\delta_{ij}}\big{)}\hskip
1.0pte_{ii}\otimes e_{jj}+\big{(}q-q^{-1}\big{)}\hskip
1.0pt\sum_{i,j=1}^{N}(u\,\delta_{i<j}+v\,\delta_{i>j})\,e_{ij}\otimes e_{ji}$
$\displaystyle\hphantom{R(u,v)=}{}-\,\frac{u-v}{u-v\hskip
1.0pt\xi}\,\sum_{i,j=1}^{N}d_{ij}(u,v)\,e_{i^{\prime}j^{\prime}}\otimes
e_{ij},$ (3.2)

where $\delta_{i<j}$ or $\delta_{i>j}$ equals $1$ if the subscript inequality
holds, and $0$ otherwise, and

$\displaystyle d_{ij}(u,v)=\begin{cases}\big{(}q-q^{-1}\big{)}\hskip
1.0pt\xi\hskip 1.0ptv\hskip
1.0ptq^{\bar{\imath}-\bar{\jmath}}\,\varepsilon_{i}\hskip
1.0pt\varepsilon_{j}&\text{for}\ i<j,\\\ \big{(}q-q^{-1}\big{)}\hskip
1.0ptu\hskip 1.0ptq^{\bar{\imath}-\bar{\jmath}}\,\varepsilon_{i}\hskip
1.0pt\varepsilon_{j}&\text{for}\ i>j,\\\ \big{(}1-q^{-1}\big{)}(u+v\hskip
1.0ptq\hskip 1.0pt\xi)&\text{for}\ i=j,\ i\neq i^{\prime},\\\
\big{(}1-q^{-1}\big{)}(u\hskip 1.0ptq+v\hskip 1.0pt\xi)&\text{for}\
i=j=i^{\prime}.\end{cases}$ (3.3)

Observe that by setting $d_{ij}(u,v)\equiv 0$ we recover the trigonometric
$R$-matrix for type $A$. Note also that if $N=2n$, then the equality
$i=i^{\prime}$ is impossible, so that the last case in the definition of
$d_{ij}(u,v)$ does not occur.

The quantum affine algebra ${\rm U}^{R}_{q}(\widehat{\mathfrak{g}})$ (with the
trivial central charge) is generated by the elements ${l}^{\pm}_{ij}[\mp m]$
with $1\leqslant i,j\leqslant N$ and $m\in\mathbb{Z}\hskip 1.0pt_{+}$ subject
to the following defining relations. We have333The condition $i<j$ was
erroneously replaced by the opposite inequality in [20, 21].

$\displaystyle{l}_{ij}^{+}[0]={l}^{-}_{ji}[0]=0\quad\text{for}\quad
i<j\qquad\text{and}\qquad{l}^{+}_{ii}[0]\,{l}_{ii}^{-}[0]={l}_{ii}^{-}[0]\,{l}^{+}_{ii}[0]=1,$
(3.4)

while the remaining relations will be written in terms of the formal power
series

$\displaystyle{l}^{\pm}_{ij}(u)=\sum_{m=0}^{\infty}{l}^{\pm}_{ij}[\mp
m]\,u^{\pm m}$ (3.5)

combined into the respective matrices

$\displaystyle{L}^{\pm}(u)=\sum\limits_{i,j=1}^{N}e_{ij}\otimes{l}_{ij}^{\pm}(u)\in{\rm{End}\,}\mathbb{C}\hskip
1.0pt^{N}\otimes{\rm
U}^{R}_{q}(\widehat{\mathfrak{g}})\big{[}\big{[}u,u^{-1}\big{]}\big{]}.$ (3.6)

Consider the tensor product algebra ${\rm{End}\,}\mathbb{C}\hskip
1.0pt^{N}\otimes{\rm{End}\,}\mathbb{C}\hskip 1.0pt^{N}\otimes{\rm
U}^{R}_{q}(\widehat{\mathfrak{g}})$ and introduce the series with coefficients
in this algebra by

$\displaystyle{L}^{\pm}_{1}(u)=\sum\limits_{i,j=1}^{N}e_{ij}\otimes
1\otimes{l}_{ij}^{\pm}(u)\qquad\text{and}\qquad{L}^{\pm}_{2}(u)=\sum\limits_{i,j=1}^{N}1\otimes
e_{ij}\otimes{l}_{ij}^{\pm}(u).$ (3.7)

The defining relations then take the form

$\displaystyle{R}(u,v)L^{\pm}_{1}(u)L^{\pm}_{2}(v)=L^{\pm}_{2}(v)L^{\pm}_{1}(u){R}(u,v),$
(3.8)
$\displaystyle{R}(u,v)L^{+}_{1}(u)L^{-}_{2}(v)=L^{-}_{2}(v)L^{+}_{1}(u){R}(u,v),$
(3.9)

together with the relations

$\displaystyle{L}^{\pm}(u)D{L}^{\pm}(u\hskip 1.0pt\xi)^{{\rm t}}D^{-1}=1,$
(3.10)

where $D$ is the diagonal matrix

$\displaystyle D={\rm
diag}\,\big{[}q^{\overline{1}},\dots,q^{\overline{N}}\hskip 1.0pt\big{]}.$

####  3.2.1 Isomorphisms

By applying the Gauss decomposition to ${L}^{+}(u)$ and ${L}^{-}(u)$ introduce
matrices

$\displaystyle F^{\pm}(u)=\begin{bmatrix}1&0&\dots&0\,\\\
f^{\pm}_{21}(u)&1&\dots&0\\\ \vdots&\vdots&\ddots&\vdots\\\
f^{\pm}_{N\,1}(u)&f^{\pm}_{N\,2}(u)&\dots&1\end{bmatrix},\qquad
E^{\pm}(u)=\begin{bmatrix}\,1&e^{\pm}_{12}(u)&\dots&e^{\pm}_{1\,N}(u)\,\\\
\,0&1&\dots&e^{\pm}_{2\,N}(u)\\\ \vdots&\vdots&\ddots&\vdots\\\
0&0&\dots&1\end{bmatrix},$

and $H^{\pm}(u)={\rm
diag}\,\big{[}h^{\pm}_{1}(u),\dots,h^{\pm}_{N}(u)\big{]}$, such that

$\displaystyle L^{\pm}(u)=F^{\pm}(u)H^{\pm}(u)E^{\pm}(u).$ (3.11)

Their entries are found by the quasideterminant formulas [13]:

$\displaystyle h^{\pm}_{i}(u)=\begin{vmatrix}l^{\pm}_{1\hskip
1.0pt1}(u)&\dots&l^{\pm}_{1\,i-1}(u)&l^{\pm}_{1\hskip 1.0pti}(u)\\\
\vdots&\ddots&\vdots&\vdots\\\
l^{\pm}_{i-1\,1}(u)&\dots&l^{\pm}_{i-1\,i-1}(u)&l^{\pm}_{i-1\,i}(u)\\\\[1.99997pt]
l^{\pm}_{i\hskip 1.0pt1}(u)&\dots&l^{\pm}_{i\,i-1}(u)&\boxed{l^{\pm}_{i\hskip
1.0pti}(u)}\\\ \end{vmatrix},\qquad i=1,\dots,N,$ (3.12)

whereas

$\displaystyle
e^{\pm}_{ij}(u)=h^{\pm}_{i}(u)^{-1}\,\begin{vmatrix}l^{\pm}_{1\hskip
1.0pt1}(u)&\dots&l^{\pm}_{1\,i-1}(u)&l^{\pm}_{1\,j}(u)\\\
\vdots&\ddots&\vdots&\vdots\\\
l^{\pm}_{i-1\,1}(u)&\dots&l^{\pm}_{i-1\,i-1}(u)&l^{\pm}_{i-1\,j}(u)\\\\[1.99997pt]
l^{\pm}_{i\hskip 1.0pt1}(u)&\dots&l^{\pm}_{i\,i-1}(u)&\boxed{l^{\pm}_{i\hskip
1.0ptj}(u)}\\\ \end{vmatrix}$ (3.13)

and

$\displaystyle f^{\pm}_{ji}(u)=\begin{vmatrix}l^{\pm}_{1\hskip
1.0pt1}(u)&\dots&l^{\pm}_{1\,i-1}(u)&l^{\pm}_{1\hskip 1.0pti}(u)\\\
\vdots&\ddots&\vdots&\vdots\\\
l^{\pm}_{i-1\,1}(u)&\dots&l^{\pm}_{i-1\,i-1}(u)&l^{\pm}_{i-1\,i}(u)\\\\[1.99997pt]
l^{\pm}_{j\,1}(u)&\dots&l^{\pm}_{j\,i-1}(u)&\boxed{l^{\pm}_{j\hskip
1.0pti}(u)}\\\ \end{vmatrix}\,h^{\pm}_{i}(u)^{-1}$

for $1\leqslant i<j\leqslant N$. Set

$\displaystyle X^{+}_{i}(u)=e^{+}_{i,i+1}(u)-e_{i,i+1}^{-}(u),\qquad
X^{-}_{i}(u)=f^{+}_{i+1,i}(u)-f^{-}_{i+1,i}(u),$

for $i=1,\dots,n-1$, and

$\displaystyle
X^{+}_{n}(u)=\begin{cases}e^{+}_{n,n+1}(u)-e_{n,n+1}^{-}(u)&\text{for types
$B_{n}$ and $C_{n}$},\\\ e^{+}_{n-1,n+1}(u)-e_{n-1,n+1}^{-}(u)&\text{for type
$D_{n}$},\end{cases}$ $\displaystyle
X^{-}_{n}(u)=\begin{cases}f^{+}_{n+1,n}(u)-f^{-}_{n+1,n}(u)&\text{for types
$B_{n}$ and $C_{n}$},\\\ f^{+}_{n+1,n-1}(u)-f^{-}_{n+1,n-1}(u)&\text{for type
$D_{n}$}.\end{cases}$

Combine the generators $x^{\pm}_{i,m}$ of the algebra ${\rm
U}_{q}(\widehat{\mathfrak{g}})$ into the series

$\displaystyle x^{\pm}_{i}(u)=\sum_{m\in\mathbb{Z}\hskip
1.0pt}x^{\pm}_{i,m}\,u^{-m}.$

By the Main Theorems of [20] and [21], the maps

$\displaystyle
x^{\pm}_{i}(u)\mapsto\big{(}q_{i}-q_{i}^{-1}\big{)}^{-1}X^{\pm}_{i}\big{(}uq^{i}\big{)},$
$\displaystyle\psi_{i}(u)\mapsto
h^{-}_{i+1}\big{(}uq^{i}\big{)}\,h^{-}_{i}\big{(}uq^{i}\big{)}^{-1},$
$\displaystyle\varphi_{i}(u)\mapsto
h^{+}_{i+1}\big{(}uq^{i}\big{)}\,h^{+}_{i}\big{(}uq^{i}\big{)}^{-1},$ (3.14)

for $i=1,\dots,n-1$, and

$\displaystyle
x^{\pm}_{n}(u)\mapsto\begin{cases}\big{(}q_{n}-q_{n}^{-1}\big{)}^{-1}[2]_{q_{n}}^{-1/2}X^{\pm}_{n}\big{(}uq^{n}\big{)}&\text{for
type $B_{n}$},\\\
\big{(}q_{n}-q_{n}^{-1}\big{)}^{-1}X^{\pm}_{n}\big{(}uq^{n+1}\big{)}&\text{for
type $C_{n}$},\\\
\big{(}q_{n}-q_{n}^{-1}\big{)}^{-1}X^{\pm}_{n}\big{(}uq^{n-1}\big{)}&\text{for
type $D_{n}$},\end{cases}$
$\displaystyle\psi_{n}(u)\mapsto\begin{cases}h^{-}_{n+1}\big{(}uq^{n}\big{)}\,h^{-}_{n}\big{(}uq^{n}\big{)}^{-1}&\text{for
type $B_{n}$},\\\
h^{-}_{n+1}\big{(}uq^{n+1}\big{)}\,h^{-}_{n}\big{(}uq^{n+1}\big{)}^{-1}&\text{for
type $C_{n}$},\\\
h^{-}_{n+1}\big{(}uq^{n-1}\big{)}\,h^{-}_{n-1}\big{(}uq^{n-1}\big{)}^{-1}&\text{for
type $D_{n}$},\end{cases}$
$\displaystyle\varphi_{n}(u)\mapsto\begin{cases}h^{+}_{n+1}\big{(}uq^{n}\big{)}\,h^{+}_{n}\big{(}uq^{n}\big{)}^{-1}&\text{for
type $B_{n}$},\\\
h^{+}_{n+1}\big{(}uq^{n+1}\big{)}\,h^{+}_{n}\big{(}uq^{n+1}\big{)}^{-1}&\text{for
type $C_{n}$},\\\
h^{+}_{n+1}\big{(}uq^{n-1}\big{)}\,h^{+}_{n-1}\big{(}uq^{n-1}\big{)}^{-1}&\text{for
type $D_{n}$},\end{cases}$

define an isomorphism ${\rm U}_{q}(\widehat{\mathfrak{g}})\to{\rm
U}^{R}_{q}(\widehat{\mathfrak{g}})$.

####  3.2.2 Central elements of the extended quantum affine algebra

The extended quantum affine algebra is defined by the same presentation as the
algebra ${\rm U}^{R}_{q}(\widehat{\mathfrak{g}})$, except that the relation
(3.10) is omitted. It was shown in [20, 21], that this algebra can be
explicitly described in terms of the Gaussian generators by producing complete
sets of relations. We will denote the extended algebra by ${\rm U}^{{\rm
ext}}_{q}(\widehat{\mathfrak{g}})$ and identify its $R$-matrix and Gaussian
presentations.

The maps described above can be understood as an embedding $\iota\colon{\rm
U}_{q}(\widehat{\mathfrak{g}})\hookrightarrow{\rm U}^{{\rm
ext}}_{q}(\widehat{\mathfrak{g}})$ so that we can regard ${\rm
U}_{q}(\widehat{\mathfrak{g}})$ as a subalgebra of ${\rm U}^{{\rm
ext}}_{q}(\widehat{\mathfrak{g}})$. This subalgebra can also be described with
the use of the multiplication automorphisms

$\displaystyle\mu_{f}\colon\ {\rm U}^{{\rm
ext}}_{q}(\widehat{\mathfrak{g}})\to{\rm U}^{{\rm
ext}}_{q}(\widehat{\mathfrak{g}}),\qquad L^{\pm}(u)\mapsto f^{\pm}(u)\hskip
1.0ptL^{\pm}(u),$ (3.15)

where

$\displaystyle f^{\pm}(u)=\sum_{m=0}^{\infty}f^{\pm}[\mp m]\hskip 1.0ptu^{\pm
m},\qquad f^{\pm}[\mp m]\in\mathbb{C}\hskip 1.0pt,\qquad f^{+}[0]\hskip
1.0ptf^{-}[0]=1.$ (3.16)

Namely, the image of ${\rm U}_{q}(\widehat{\mathfrak{g}})$ under the embedding
$\iota$ consists of the elements in ${\rm U}^{{\rm
ext}}_{q}(\widehat{\mathfrak{g}})$ which are fixed by all automorphisms of the
form (3.15).

All coefficients of the series $z^{\pm}(u)$ given by

$\displaystyle
z^{\pm}(u)=\begin{cases}\displaystyle\prod_{i=1}^{n}{h}^{\pm}_{i}\big{(}u\xi
q^{2i}\big{)}^{-1}\prod_{i=1}^{n}{h}^{\pm}_{i}\big{(}u\xi
q^{2i-2}\big{)}\,\cdot{h}^{\pm}_{n+1}(u)\,{h}^{\pm}_{n+1}(uq)&\text{for}\
N=2n+1,\\\ \displaystyle\prod_{i=1}^{n-1}{h}^{\pm}_{i}\big{(}u\xi
q^{2i}\big{)}^{-1}\prod_{i=1}^{n}{h}^{\pm}_{i}\big{(}u\xi
q^{2i-2}\big{)}\cdot{h}^{\pm}_{n+1}(u)&\text{for}\ N=2n,\end{cases}$

belong to the center of the algebra ${\rm U}^{{\rm
ext}}_{q}(\widehat{\mathfrak{g}})$. If $N=2n$ then the constant terms of the
series $z^{\pm}(u)$ are the central elements
$z^{\pm}[0]=l^{\pm}_{nn}[0]\,l_{n^{\prime}n^{\prime}}^{\pm}[0]$. In this case
we will extend this algebra by adjoining the square roots $z^{\pm}[0]^{1/2}$.
Then in all three cases there exist power series $\zeta^{\pm}(u)$ with
coefficients in the center of ${\rm U}^{{\rm
ext}}_{q}(\widehat{\mathfrak{g}})$ such that
$\zeta^{\pm}(u)\,\zeta^{\pm}(u\hskip 1.0pt\xi)=z^{\pm}(u)$. Under the
automorphism (3.15) we have

$\displaystyle\mu_{f}\colon\ \zeta^{\pm}(u)\mapsto f^{\pm}(u)\hskip
1.0pt\zeta^{\pm}(u).$

This implies that the coefficients of the entries of the matrices
$\zeta^{\pm}(u)^{-1}\hskip 1.0ptL^{\pm}(u)$ belong to the subalgebra ${\rm
U}_{q}(\widehat{\mathfrak{g}})\subset{\rm U}^{{\rm
ext}}_{q}(\widehat{\mathfrak{g}})$. Therefore, if $\mathcal{C}$ denotes the
subalgebra of ${\rm U}^{{\rm ext}}_{q}(\widehat{\mathfrak{g}})$ generated by
the coefficients of the series $\zeta^{\pm}(u)$, then we have the tensor
product decomposition

$\displaystyle{\rm U}^{{\rm ext}}_{q}(\widehat{\mathfrak{g}})={\rm
U}_{q}(\widehat{\mathfrak{g}})\otimes\mathcal{C},$

assuming that the algebra ${\rm U}_{q}(\widehat{\mathfrak{g}})$ is extended by
adjoining square roots in types $C_{n}$ and $D_{n}$ as in Remark 3.1. In the
algebra ${\rm U}^{{\rm ext}}_{q}(\widehat{\mathfrak{g}})$ we have

$\displaystyle{L}^{\pm}(u)D{L}^{\pm}(u\,\xi)^{{\rm t}}D^{-1}=z^{\pm}(u),$
(3.17)

so that the subalgebra ${\rm U}_{q}(\widehat{\mathfrak{g}})$ can also be
regarded as the quotient of ${\rm U}^{{\rm ext}}_{q}(\widehat{\mathfrak{g}})$
by the relations $z^{\pm}(u)=1$. The following relations are implied by
(3.17), and they were essentially derived in Section 4.5 of [20] and [21].

######  Lemma 3.2.

In the algebra ${\rm U}^{{\rm ext}}_{q}(\widehat{\mathfrak{g}})$ we have

$\displaystyle h^{\pm}_{1}(u\xi)\hskip
1.0pth^{\pm}_{1^{\prime}}(u)=z^{\pm}(u),$ $\displaystyle h^{\pm}_{i}(u\xi
q^{2i})\hskip 1.0pth^{\pm}_{i^{\prime}}(u)=h^{\pm}_{i+1}(u\xi q^{2i})\hskip
1.0pth^{\pm}_{(i+1)^{\prime}}(u),$

where $i=1,\dots,n$ for $N=2n+1$, and $i=1,\dots,n-1$ for $N=2n$.

######  Remark 3.3.

Note that for the parameters $d_{ij}(u,v)$ defined in (3.3) we have
$d_{j^{\prime}i^{\prime}}(u,v)=d_{ij}(u,v)$. Therefore the $R$-matrix (3.2)
possesses the symmetry property

$\displaystyle R^{T_{1}T_{2}}(u,v)=R_{21}(u,v),$ (3.18)

where $R_{21}(u,v)=PR(u,v)P$, while $T$ denotes the standard matrix
transposition with $e_{ij}^{T}=e_{ji}$ and $T_{a}$ is the partial
transposition applied to the $a$-th copy of the endomorphism algebra
${\rm{End}\,}\mathbb{C}\hskip 1.0pt^{N}$. We can use the $R$-matrix
$R_{21}(u,v)$ instead of $R(u,v)$ to define the extended quantum affine
algebra $\widetilde{\rm U}^{{\rm ext}}_{q}(\widehat{\mathfrak{g}})$ in a way
similar to ${\rm U}^{{\rm ext}}_{q}(\widehat{\mathfrak{g}})$, by using the
relations

$\displaystyle{R}_{21}(u,v)\widetilde{L}^{\pm}_{1}(u)\widetilde{L}^{\pm}_{2}(v)=\widetilde{L}^{\pm}_{2}(v)\widetilde{L}^{\pm}_{1}(u){R}_{21}(u,v),$
$\displaystyle{R}_{21}(u,v)\widetilde{L}^{+}_{1}(u)\widetilde{L}^{-}_{2}(v)=\widetilde{L}^{-}_{2}(v)\widetilde{L}^{+}_{1}(u){R}_{21}(u,v),$

where we impose the opposite zero mode conditions

$\displaystyle{\widetilde{l}}_{ij}^{+}[0]={\widetilde{l}}^{-}_{ji}[0]=0\quad\text{for}\quad
i>j\qquad\text{and}\qquad{\widetilde{l}}^{+}_{ii}[0]\,{\widetilde{l}}_{ii}^{-}[0]={\widetilde{l}}_{ii}^{-}[0]\,{\widetilde{l}}^{+}_{ii}[0]=1.$

The symmetry property (3.18) implies that the mapping

$\displaystyle\widetilde{L}^{\pm}(u)\mapsto L^{\pm}(u)^{T}$

defines an anti-isomorphism $\widetilde{\rm U}^{{\rm
ext}}_{q}(\widehat{\mathfrak{g}})\to{\rm U}^{{\rm
ext}}_{q}(\widehat{\mathfrak{g}})$. Using the definition of quasideterminants,
we obtain the following formulas for the images of the respective Gaussian
generators

$\displaystyle\widetilde{h}^{\pm}_{i}(u)\mapsto
h^{\pm}_{i}(u),\qquad\widetilde{e}^{\,\pm}_{ij}(u)\mapsto
f^{\pm}_{ji}(u)\qquad\text{and}\qquad\widetilde{f}^{\,\pm}_{ij}(u)\mapsto
e^{\pm}_{ji}(u).$

####  3.2.3 Hopf algebra structure

The quantum affine algebra ${\rm U}^{{\rm ext}}_{q}(\widehat{\mathfrak{g}})$
possesses a Hopf algebra structure defined by the coproduct

$\displaystyle\Delta\colon\
l^{\pm}_{ij}(u)\mapsto\sum_{k=1}^{N}l^{\pm}_{ik}(u)\otimes l^{\pm}_{kj}(u),$
(3.19)

the antipode

$\displaystyle S\colon\ L^{\pm}(u)\mapsto L^{\pm}(u)^{-1}$ (3.20)

and the counit

$\displaystyle\varepsilon\colon\ L^{\pm}(u)\mapsto 1.$ (3.21)

######  Proposition 3.4.

In the Hopf algebra ${\rm U}^{{\rm ext}}_{q}(\widehat{\mathfrak{g}})$ we have

$\displaystyle\Delta\colon\ z^{\pm}(u)\mapsto z^{\pm}(u)\otimes z^{\pm}(u)$

and

$\displaystyle S\colon\ z^{\pm}(u)\mapsto z^{\pm}(u)^{-1}.$ (3.22)

In particular, ${\rm U}_{q}(\widehat{\mathfrak{g}})$ is a Hopf subalgebra of
${\rm U}^{{\rm ext}}_{q}(\widehat{\mathfrak{g}})$. Moreover,

$\displaystyle S^{2}\colon\ L^{\pm}(u)\mapsto\frac{z^{\pm}(u)}{z^{\pm}(u\hskip
1.0pt\xi)}\hskip 1.0ptL^{\pm}\big{(}u\hskip 1.0pt\xi^{2}\big{)}.$ (3.23)

###### Proof.

The formulas for the images of the series $z^{\pm}(u)$ under the maps $\Delta$
and $S$ follow easily from the definition of $z^{\pm}(u)$ and the Hopf algebra
axioms. For the proof of (3.23) use the relation

$\displaystyle L^{\pm}(u)^{-1}=z^{\pm}(u)^{-1}\hskip 1.0ptD{L}^{\pm}(u\hskip
1.0pt\xi)^{{\rm t}}D^{-1}$

and apply (3.22). Furthermore, for the power series $\zeta^{\pm}(u)$ we have

$\displaystyle\Delta\colon\
\zeta^{\pm}(u)\mapsto\zeta^{\pm}(u)\otimes\zeta^{\pm}(u),$

and so

$\displaystyle\Delta\big{(}{\rm
U}_{q}(\widehat{\mathfrak{g}})\big{)}\subset{\rm
U}_{q}(\widehat{\mathfrak{g}})\otimes{\rm
U}_{q}(\widehat{\mathfrak{g}}),\qquad S\big{(}{\rm
U}_{q}(\widehat{\mathfrak{g}})\big{)}\subset{\rm
U}_{q}(\widehat{\mathfrak{g}}),$

thus proving that ${\rm U}_{q}(\widehat{\mathfrak{g}})$ is a Hopf subalgebra
of ${\rm U}^{{\rm ext}}_{q}(\widehat{\mathfrak{g}})$. ∎

####  3.2.4 Consistency with the triangular decomposition

Denote by ${\rm U}_{q}(\widehat{\mathfrak{g}})^{+}$ (respectively, ${\rm
U}_{q}(\widehat{\mathfrak{g}})^{-}$) the subalgebra of ${\rm
U}_{q}(\widehat{\mathfrak{g}})$ generated by the elements $x_{i,m}^{+}$
(respectively, $x_{i,m}^{-}$), and denote by ${\rm
U}_{q}(\widehat{\mathfrak{g}})^{0}$ the subalgebra generated by $k_{i}^{\pm
1}$ and $a_{i,l}$ together with the additional elements in types $C_{n}$ and
$D_{n}$, introduced in Remark 3.1. The multiplication map provides the
triangular decomposition isomorphism

$\displaystyle{\rm U}_{q}(\widehat{\mathfrak{g}})^{-}\otimes{\rm
U}_{q}(\widehat{\mathfrak{g}})^{0}\otimes{\rm
U}_{q}(\widehat{\mathfrak{g}})^{+}\cong{\rm U}_{q}(\widehat{\mathfrak{g}}),$

as proved in [4]; see also [16] for a generalization to quantum affinizations
of symmetrizable Kac–Moody algebras.

Here we aim to establish a key property of the Gauss decomposition by showing
that it is consistent with the triangular decomposition (see Proposition 3.8
below). We will rely on a few relations for the Gaussian generators described
in the following lemmas.

We will use a standard notation $[x,y]_{q}=x\hskip 1.0pty-q\hskip 1.0pty\hskip
1.0ptx$ for $q$-commutators and begin by proving some $q$-commutator formulas;
cf. [22, Lemma 5.6].

######  Lemma 3.5.

For any $k<i<j<k^{\prime}$ such that $i\neq j^{\prime}$ we have

$\displaystyle\big{[}e^{\pm}_{k\hskip 1.0pti}(u),e^{-}_{i\hskip
1.0ptj}[0]\big{]}_{q}=\big{(}1-q^{2}\big{)}\,e^{\pm}_{k\hskip 1.0ptj}(u).$

###### Proof.

Due to the consistency property of Gauss decompositions for subalgebras as
stated in Proposition 4.2 in [20] and [21], we may assume without loss of
generality, that $k=1$. By taking matrix entries in (3.8) and (3.9), write the
defining relations in terms of the series $l^{\pm}_{ij}(u)$ to get

$\displaystyle\big{(}u\hskip 1.0ptq^{\delta_{ij}}-v\hskip
1.0ptq^{-\delta_{ij}}\big{)}\,l^{\pm}_{ia}(u)\,l^{\pm}_{jb}(v)+\big{(}q-q^{-1}\big{)}\,(u\,\delta_{i<j}+v\,\delta_{i>j})\,l^{\pm}_{ja}(u)\,l^{\pm}_{ib}(v)$
$\displaystyle\qquad\quad{}-\delta_{i\hskip
1.0ptj^{\prime}}\,\frac{u-v}{u-v\hskip
1.0pt\xi}\,\sum_{k=1}^{N}\,d_{jk}(u,v)\hskip
1.0ptl^{\pm}_{k^{\prime}a}(u)\,l^{\pm}_{kb}(v)$
$\displaystyle\qquad{}=\big{(}u\hskip 1.0ptq^{\delta_{ab}}-v\hskip
1.0ptq^{-\delta_{ab}}\big{)}\,l^{\pm}_{jb}(v)\,l^{\pm}_{ia}(u){}+\big{(}q-q^{-1}\big{)}\,(u\,\delta_{a>b}+v\,\delta_{a<b})\,l^{\pm}_{ja}(v)\,l^{\pm}_{ib}(u)$
$\displaystyle\qquad\quad{}-\delta_{a\hskip
1.0ptb^{\prime}}\,\frac{u-v}{u-v\hskip
1.0pt\xi}\,\sum_{c=1}^{N}\,d_{cb}(u,v)\hskip
1.0ptl^{\pm}_{jc}(v)\,l^{\pm}_{ic^{\prime}}(u),$

and

$\displaystyle\big{(}u\hskip 1.0ptq^{\delta_{ij}}-v\hskip
1.0ptq^{-\delta_{ij}}\big{)}\,l^{+}_{ia}(u)\,l^{-}_{jb}(v)+\big{(}q-q^{-1}\big{)}\,(u\,\delta_{i<j}+v\,\delta_{i>j})\,l^{+}_{ja}(u)\,l^{-}_{ib}(v)$
$\displaystyle\qquad\quad{}-\delta_{i\hskip
1.0ptj^{\prime}}\,\frac{u-v}{u-v\hskip
1.0pt\xi}\,\sum_{k=1}^{N}\,d_{jk}(u,v)\hskip
1.0ptl^{+}_{k^{\prime}a}(u)\,l^{-}_{kb}(v)$
$\displaystyle\qquad{}=\big{(}u\hskip 1.0ptq^{\delta_{ab}}-v\hskip
1.0ptq^{-\delta_{ab}}\big{)}\,l^{-}_{jb}(v)\,l^{+}_{ia}(u)+\big{(}q-q^{-1}\big{)}\,(u\,\delta_{a>b}+v\,\delta_{a<b})\,l^{-}_{ja}(v)\,l^{+}_{ib}(u)$
$\displaystyle\qquad\quad{}-\delta_{a\hskip
1.0ptb^{\prime}}\,\frac{u-v}{u-v\hskip
1.0pt\xi}\,\sum_{c=1}^{N}\,d_{cb}(u,v)\hskip
1.0ptl^{-}_{jc}(v)\,l^{+}_{ic^{\prime}}(u),$

where $d_{ij}(u,v)$ are defined in (3.3). If $1<i<j$ and $j\neq i^{\prime}$,
then they give

$\displaystyle(u-v)\hskip 1.0ptl^{\pm}_{1\hskip 1.0pti}(u)\hskip
1.0ptl^{-}_{i\hskip 1.0ptj}(v)+\big{(}q-q^{-1}\big{)}\,u\hskip
1.0ptl^{\pm}_{i\hskip 1.0pti}(u)\,l^{-}_{1\hskip 1.0ptj}(v)$
$\displaystyle\qquad{}=(u-v)\hskip 1.0ptl^{-}_{i\hskip
1.0ptj}(v)\,l^{\pm}_{1\hskip 1.0pti}(u)+\big{(}q-q^{-1}\big{)}\,v\hskip
1.0ptl^{-}_{i\hskip 1.0pti}(v)\,l^{\pm}_{1\hskip 1.0ptj}(u).$

By comparing the coefficients of $v$ on both sides, we come to the relations

$\displaystyle l^{\pm}_{1\hskip 1.0pti}(u)\hskip 1.0ptl^{-}_{i\hskip
1.0ptj}[0]=l^{-}_{i\hskip 1.0ptj}[0]\,l^{\pm}_{1\hskip
1.0pti}(u)-\big{(}q-q^{-1}\big{)}\hskip 1.0ptl^{-}_{i\hskip
1.0pti}[0]\,l^{\pm}_{1\hskip 1.0ptj}(u).$ (3.24)

Similarly, assuming that $1<i\leqslant m$ and $m\neq 1^{\prime}$, we get

$\displaystyle(u-v)\hskip 1.0ptl^{\pm}_{1\hskip 1.0pt1}(u)\hskip
1.0ptl^{-}_{i\hskip 1.0ptm}(v)+\big{(}q-q^{-1}\big{)}\,u\hskip
1.0ptl^{\pm}_{i\hskip 1.0pt1}(u)\,l^{-}_{1\hskip 1.0ptm}(v)$
$\displaystyle\qquad{}=(u-v)\hskip 1.0ptl^{-}_{i\hskip
1.0ptm}(v)\,l^{\pm}_{1\hskip 1.0pt1}(u)+\big{(}q-q^{-1}\big{)}\,v\hskip
1.0ptl^{-}_{i\hskip 1.0pt1}(v)\,l^{\pm}_{1\hskip 1.0ptm}(u),$

which yields

$\displaystyle l^{\pm}_{1\hskip 1.0pt1}(u)\hskip 1.0ptl^{-}_{i\hskip
1.0ptm}[0]=l^{-}_{i\hskip 1.0ptm}[0]\,l^{\pm}_{1\hskip 1.0pt1}(u).$ (3.25)

Since $l^{\pm}_{1\hskip 1.0pti}(u)=l^{\pm}_{1\hskip 1.0pt1}(u)\hskip
1.0pte^{\pm}_{1\hskip 1.0pti}(u)$, together with relation (3.24) this implies

$\displaystyle e^{\pm}_{1\hskip 1.0pti}(u)\hskip 1.0ptl^{-}_{i\hskip
1.0ptj}[0]=l^{-}_{i\hskip 1.0ptj}[0]\,e^{\pm}_{1\hskip
1.0pti}(u)-\big{(}q-q^{-1}\big{)}\hskip 1.0ptl^{-}_{i\hskip
1.0pti}[0]\,e^{\pm}_{1\hskip 1.0ptj}(u).$ (3.26)

Applying the defining relations again, for any $1<i<1^{\prime}$ we get

$\displaystyle(u-v)\hskip 1.0ptl^{\pm}_{1\hskip 1.0pti}(u)\hskip
1.0ptl^{-}_{i\hskip 1.0pti}(v)+\big{(}q-q^{-1}\big{)}\,u\hskip
1.0ptl^{\pm}_{i\hskip 1.0pti}(u)\,l^{-}_{1\hskip 1.0pti}(v)=\big{(}u\hskip
1.0ptq-v\hskip 1.0ptq^{-1}\big{)}\hskip 1.0ptl^{-}_{i\hskip
1.0pti}(v)\,l^{\pm}_{1\hskip 1.0pti}(u).$

Comparing the coefficients of $v$ on both sides, we come to

$\displaystyle l^{\pm}_{1\hskip 1.0pti}(u)\hskip 1.0ptl^{-}_{i\hskip
1.0pti}[0]=q^{-1}\hskip 1.0ptl^{-}_{i\hskip 1.0pti}[0]\,l^{\pm}_{1\hskip
1.0pti}(u)$

which implies

$\displaystyle e^{\pm}_{1\hskip 1.0pti}(u)\hskip 1.0ptl^{-}_{i\hskip
1.0pti}[0]=q^{-1}\hskip 1.0ptl^{-}_{i\hskip 1.0pti}[0]\,e^{\pm}_{1\hskip
1.0pti}(u).$ (3.27)

Hence, using this together with the decomposition $l^{-}_{i\hskip
1.0ptj}[0]=l^{-}_{i\hskip 1.0pti}[0]e^{-}_{ij}[0]$, we derive from (3.26) that

$\displaystyle q^{-1}\hskip 1.0pte^{\pm}_{1\hskip 1.0pti}(u)\hskip
1.0pte^{-}_{i\hskip 1.0ptj}[0]=e^{-}_{i\hskip 1.0ptj}[0]\,e^{\pm}_{1\hskip
1.0pti}(u)-\big{(}q-q^{-1}\big{)}\,e^{\pm}_{1\hskip 1.0ptj}(u).$

We can write this as the $q$-commutator relation

$\displaystyle\big{[}e^{\pm}_{1\hskip 1.0pti}(u),e^{-}_{i\hskip
1.0ptj}[0]\big{]}_{q}=\big{(}1-q^{2}\big{)}\,e^{\pm}_{1\hskip 1.0ptj}(u),$

as required. ∎

######  Lemma 3.6.

For $n\geqslant 2$ we have

$\displaystyle\big{[}e^{\pm}_{1\hskip
1.0pt2^{\prime}}(u),e^{-}_{2^{\prime}1^{\prime}}[0]\big{]}=\big{(}1-q^{2}\big{)}\,\big{(}e^{\pm}_{1\hskip
1.0pt1^{\prime}}(u)+e^{\pm}_{1\hskip 1.0pt2}(u)e^{\pm}_{1\hskip
1.0pt2^{\prime}}(u)\big{)}.$

###### Proof.

The defining relations give

$\displaystyle(u-v)\,l^{\pm}_{11}(u)\,l^{-}_{2^{\prime}1^{\prime}}(v)+\big{(}q-q^{-1}\big{)}\,u\,l^{\pm}_{2^{\prime}1}(u)\,l^{-}_{11^{\prime}}(v)$
$\displaystyle{}=(u-v)\,l^{-}_{2^{\prime}1^{\prime}}(v)\,l^{\pm}_{11}(u)+\big{(}q-q^{-1}\big{)}\,v\,l^{-}_{2^{\prime}1}(v)\,l^{\pm}_{11^{\prime}}(u)-\frac{u-v}{u-v\hskip
1.0pt\xi}\,\sum_{c=1}^{N}\,d_{c\hskip 1.0pt1^{\prime}}(u,v)\hskip
1.0ptl^{-}_{2^{\prime}c}(v)\,l^{\pm}_{1\hskip 1.0ptc^{\prime}}(u).$

By comparing the coefficients of $v$ on both sides we come to the relation

$\displaystyle l^{-}_{2^{\prime}1^{\prime}}[0]\hskip 1.0ptl^{\pm}_{1\hskip
1.0pt1}(u)=q^{-1}\hskip 1.0ptl^{\pm}_{1\hskip 1.0pt1}(u)\hskip
1.0ptl^{-}_{2^{\prime}1^{\prime}}[0]-\big{(}q-q^{-1}\big{)}\hskip
1.0ptl^{-}_{2^{\prime}2^{\prime}}[0]\,l^{\pm}_{1\hskip 1.0pt2}(u).$ (3.28)

On the other hand, taking $i=2^{\prime}$ and $j=1^{\prime}$ in (3.24) we get

$\displaystyle l^{\pm}_{1\hskip 1.0pt2^{\prime}}(u)\hskip
1.0ptl^{-}_{2^{\prime}1^{\prime}}[0]=l^{-}_{2^{\prime}1^{\prime}}[0]\,l^{\pm}_{1\hskip
1.0pt2^{\prime}}(u)-\big{(}q-q^{-1}\big{)}\hskip
1.0ptl^{-}_{2^{\prime}2^{\prime}}[0]\,l^{\pm}_{1\hskip 1.0pt1^{\prime}}(u).$

Write $l^{\pm}_{1\hskip 1.0pti}(u)=l^{\pm}_{1\hskip 1.0pt1}(u)\hskip
1.0pte^{\pm}_{1\hskip 1.0pti}(u)$ for $i=1^{\prime},2^{\prime}$ and use (3.28)
together with (3.25) to bring this to the form

$\displaystyle e^{\pm}_{1\hskip 1.0pt2^{\prime}}(u)\hskip
1.0ptl^{-}_{2^{\prime}1^{\prime}}[0]=q^{-1}\hskip
1.0ptl^{-}_{2^{\prime}1^{\prime}}[0]\,e^{\pm}_{1\hskip
1.0pt2^{\prime}}(u)-\big{(}q-q^{-1}\big{)}\hskip
1.0ptl^{-}_{2^{\prime}2^{\prime}}[0]\,\big{(}e^{\pm}_{1\hskip
1.0pt1^{\prime}}(u)+e^{\pm}_{1\hskip 1.0pt2}(u)e^{\pm}_{1\hskip
1.0pt2^{\prime}}(u)\big{)}.$

Finally, write
$l^{-}_{2^{\prime}1^{\prime}}[0]=l^{-}_{2^{\prime}2^{\prime}}[0]\hskip
1.0pte^{-}_{2^{\prime}1^{\prime}}[0]$ and apply (3.27) with $i=2^{\prime}$. ∎

######  Lemma 3.7.

For any $i<j<i^{\prime}$ we have the relations

$\displaystyle e^{\pm}_{ij}\big{(}u\xi
q^{2i}\big{)}=q^{\bar{\jmath}-\bar{\imath}+1}\hskip
1.0pt\sum_{s=0}^{j-i-1}\hskip
1.0pt(-1)^{s+1}\sum_{j^{\prime}=a_{0}<a_{1}<\dots<a_{s+1}=i^{\prime}}e^{\pm}_{a_{0}a_{1}}(u)\hskip
1.0pte^{\pm}_{a_{1}a_{2}}(u)\cdots e^{\pm}_{a_{s}a_{s+1}}(u).$

###### Proof.

We will rely on Proposition 4.2 in [20] and [21], which allows us to reduce
the proof to the case $i=1$. Working in the algebra ${\rm U}^{{\rm
ext}}_{q}(\widehat{\mathfrak{g}})$, write (3.17) in the form

$\displaystyle D{L}^{\pm}(u\,\xi)^{{\rm t}}D^{-1}={L}^{\pm}(u)^{-1}z^{\pm}(u)$

and take the $(j^{\prime},1^{\prime})$ entries on both sides. Using the Gauss
decompositions (3.11) and

$\displaystyle L^{\pm}(u)^{-1}=E^{\pm}(u)^{-1}H^{\pm}(u)^{-1}F^{\pm}(u)^{-1},$

we obtain

$\displaystyle q^{\bar{1}-\bar{\jmath}}\,h^{\pm}_{1}(u\hskip 1.0pt\xi)\hskip
1.0pte^{\pm}_{1j}(u\hskip
1.0pt\xi)=\widehat{e}^{\,\pm}_{j^{\prime}1^{\prime}}(u)\hskip
1.0pth^{\pm}_{1^{\prime}}(u)^{-1}\hskip 1.0ptz^{\pm}(u),$

where we used the notation $\widehat{e}^{\,\pm}_{ij}(u)$ for the $(i,j)$ entry
of the matrix $E^{\pm}(u)^{-1}$. Now apply (3.2) to replace
$h^{\pm}_{1^{\prime}}(u)^{-1}\hskip 1.0ptz^{\pm}(u)$ with $h^{\pm}_{1}(u\hskip
1.0pt\xi)$ and note that

$\displaystyle h^{\pm}_{1}(u\hskip 1.0pt\xi)\hskip 1.0pte^{\pm}_{1j}(u\hskip
1.0pt\xi)=q^{-1}\hskip 1.0pte^{\pm}_{1j}\big{(}u\hskip 1.0pt\xi\hskip
1.0ptq^{2}\big{)}\hskip 1.0pth^{\pm}_{1}(u\hskip 1.0pt\xi).$ (3.29)

Indeed, by the defining relations,

$\displaystyle\big{(}u\hskip 1.0ptq-v\hskip 1.0ptq^{-1}\big{)}\hskip
1.0ptl^{\pm}_{1\hskip 1.0pt1}(u)\hskip 1.0ptl^{\pm}_{1\hskip
1.0ptj}(v)=(u-v)\hskip 1.0ptl^{\pm}_{1\hskip 1.0ptj}(v)\,l^{\pm}_{1\hskip
1.0pt1}(u)+\big{(}q-q^{-1}\big{)}\,v\hskip 1.0ptl^{\pm}_{1\hskip
1.0pt1}(v)\,l^{\pm}_{1\hskip 1.0ptj}(u).$

By setting $v=u\hskip 1.0ptq^{2}$ we get

$\displaystyle l^{\pm}_{1\hskip 1.0ptj}(u\hskip 1.0ptq^{2})\,l^{\pm}_{1\hskip
1.0pt1}(u)=q\,l^{\pm}_{1\hskip 1.0pt1}\big{(}u\hskip
1.0ptq^{2}\big{)}\,l^{\pm}_{1\hskip 1.0ptj}(u).$

Since $l^{\pm}_{1\hskip 1.0ptj}(u)=h^{\pm}_{1}(u)\hskip 1.0pte^{\pm}_{1\hskip
1.0ptj}(u)$, (3.29) follows. It remains to apply the formula

$\displaystyle\widehat{e}^{\,\pm}_{j^{\prime}1^{\prime}}(u)=\sum_{s=0}^{j-2}\hskip
1.0pt(-1)^{s+1}\sum_{j^{\prime}=a_{0}<a_{1}<\dots<a_{s+1}=1^{\prime}}e^{\pm}_{a_{0}a_{1}}(u)\hskip
1.0pte^{\pm}_{a_{1}a_{2}}(u)\cdots e^{\pm}_{a_{s}a_{s+1}}(u)$

for the $(j^{\prime},1^{\prime})$ entries of the inverse matrix
$E^{\pm}(u)^{-1}$. ∎

######  Proposition 3.8.

The images of the coefficients of all series $e^{\pm}_{ij}(u)$ (respectively,
$f^{\pm}_{ji}(u)$) with $1\leqslant i<j\leqslant N$ under the isomorphism
${\rm U}^{R}_{q}(\widehat{\mathfrak{g}})\to{\rm
U}_{q}(\widehat{\mathfrak{g}})$ belong to the subalgebra ${\rm
U}_{q}(\widehat{\mathfrak{g}})^{+}$ (respectively, ${\rm
U}_{q}(\widehat{\mathfrak{g}})^{-}$). The images of the coefficients of all
series $h^{\pm}_{i}(u)$ with $i=1,\dots,N$ belong to the subalgebra ${\rm
U}_{q}(\widehat{\mathfrak{g}})^{0}$.

###### Proof.

We will identify the coefficients of all the series with their images under
the isomorphism ${\rm U}^{R}_{q}(\widehat{\mathfrak{g}})\to{\rm
U}_{q}(\widehat{\mathfrak{g}})$. We start by verifying the claim for the
series $h^{+}_{i}(u)$; the argument for $h^{-}_{i}(u)$ is exactly the same. We
have

$\displaystyle h^{+}_{i}(u)=\varphi_{1}\big{(}u\hskip 1.0ptq^{-1}\big{)}\hskip
1.0pt\varphi_{2}\big{(}u\hskip
1.0ptq^{-2}\big{)}\cdots\varphi_{i-1}\big{(}u\hskip 1.0ptq^{-i+1}\big{)}\hskip
1.0pth^{+}_{1}(u),\qquad i=2,\dots,n.$

This relation is also valid for $i=n+1$ in type $B_{n}$, whereas

$\displaystyle h^{+}_{n+1}(u)=\varphi_{n}\big{(}u\hskip
1.0ptq^{-n-1}\big{)}\hskip 1.0pth^{+}_{n}(u)\qquad\text{and}\qquad
h^{+}_{n+1}(u)=\varphi_{n}\big{(}u\hskip 1.0ptq^{-n+1}\big{)}\hskip
1.0pth^{+}_{n-1}(u)$

for types $C_{n}$ and $D_{n}$, respectively. Now substitute these expressions
into the formula for the series $z^{+}(u)$ given in Section 3.2.2, so that the
relation $z^{+}(u)=1$ would give an equation for the coefficients of the
series $h^{+}_{1}(u)$. For type $B_{n}$ it reads

$\displaystyle h^{+}_{1}(u)\hskip
1.0pth^{+}_{1}(u\xi)\prod_{i=1}^{n}\varphi_{i}\big{(}uq^{-i}\big{)}\hskip
1.0pt\varphi_{i}\big{(}u\xi q^{i}\big{)}=1.$

Hence, the constant term $h^{+}_{1,0}$ of $h^{+}_{1}(u)$ is found from
$h^{+\,2}_{1,0}=k_{1}^{2}\cdots k_{n}^{2}$ which together with Lemma 3.2
implies that all coefficients of the series $h^{+}_{i}(u)$ with $i=1,\dots,N$
belong to the subalgebra ${\rm U}_{q}(\widehat{\mathfrak{g}})^{0}$. The same
conclusion is reached for types $C_{n}$ and $D_{n}$ from the respective
formulas

$\displaystyle h^{+}_{1}(u)\hskip
1.0pth^{+}_{1}(u\xi)\prod_{i=1}^{n-1}\varphi_{i}\big{(}uq^{-i}\big{)}\hskip
1.0pt\varphi_{i}\big{(}u\xi q^{i}\big{)}\cdot\varphi_{n}\big{(}u\hskip
1.0ptq^{-n-1}\big{)}=1$

and

$\displaystyle h^{+}_{1}(u)\hskip
1.0pth^{+}_{1}(u\xi)\prod_{i=1}^{n-2}\varphi_{i}\big{(}uq^{-i}\big{)}\hskip
1.0pt\varphi_{i}\big{(}u\xi q^{i}\big{)}\cdot\varphi_{n-1}\big{(}u\hskip
1.0ptq^{-n+1}\big{)}\hskip 1.0pt\varphi_{n}\big{(}u\hskip 1.0pt\xi\hskip
1.0ptq^{n-1}\big{)}=1,$

which imply the relations $h^{+\,2}_{1,0}=k_{1}^{2}\cdots k_{n-1}^{2}k_{n}$
and $h^{+\,2}_{1,0}=k_{1}^{2}\cdots k_{n-2}^{2}\hskip 1.0ptk_{n-1}k_{n}$,
respectively, for the constant terms.

Now turn to the series $e^{\pm}_{ij}(u)$ and use induction on $n$. By
Proposition 4.2 in [20] and [21], the coefficients of these series with
$2\leqslant i<j\leqslant 2^{\prime}$ can be regarded as elements of the
algebra ${\rm U}_{q}(\widehat{\mathfrak{g}^{\prime}})$ associated with the Lie
algebra $\mathfrak{g}^{\prime}$ of rank $n-1$, which belongs to the same type
as $\mathfrak{g}$. These coefficients coincide with the generators obtained
from the Gauss decompositions of the associated $L$-operators, so that we can
apply the induction hypothesis to suppose that the coefficients of all series
$e^{\pm}_{ij}(u)$ with $2\leqslant i<j\leqslant 2^{\prime}$ belong to the
subalgebra ${\rm U}_{q}(\widehat{\mathfrak{g}})^{+}$.

On the other hand, the coefficients of the series $e^{\pm}_{12}(u)$ also
belong to the subalgebra ${\rm U}_{q}(\widehat{\mathfrak{g}})^{+}$, and so by
applying Lemma 3.5 we derive that the coefficients of the series
$e^{\pm}_{1j}(u)$ with $j=2,\dots,2^{\prime}$ belong to ${\rm
U}_{q}(\widehat{\mathfrak{g}})^{+}$, assuming that $n\geqslant 2$ for type
$B_{n}$, and $n\geqslant 3$ for types $C_{n}$ and $D_{n}$. Furthermore, by
Lemma 3.7 we have $e^{-}_{2^{\prime}1^{\prime}}[0]=-e^{-}_{12}[0]$ so that
applying Lemma 3.6 we may conclude that the required property is shared by the
series $e^{\pm}_{11^{\prime}}(u)$. The induction step is completed by another
application of Lemma 3.7, which implies that the coefficients of the series
$e^{\pm}_{j^{\prime}1^{\prime}}(u)$ with $j=2,\dots,2^{\prime}$ also belong to
${\rm U}_{q}(\widehat{\mathfrak{g}})^{+}$.

It remains to verify the induction base for the Lie algebras
$\mathfrak{g}=\mathfrak{o}_{3},\mathfrak{sp}_{4}$ and $\mathfrak{o}_{4}$. In
the case $\mathfrak{g}=\mathfrak{o}_{3}$ it follows from the identity

$\displaystyle e^{\pm}_{12}(u)^{2}=-\big{(}q^{1/2}+q^{-1/2}\big{)}\hskip
1.0pte^{\pm}_{11^{\prime}}(u),$

which is a particular case of [21, Lemma 4.9] obtained by taking the residue
at $u=q^{-2}v$ in the second formula. In the case
$\mathfrak{g}=\mathfrak{sp}_{4}$ it is sufficient to verify the identity

$\displaystyle\big{[}e^{\pm}_{1\hskip 1.0pt2}(u),e^{-}_{2\hskip
1.0pt2^{\prime}}[0]\big{]}_{q^{2}}=\big{(}1-q^{4}\big{)}\,e^{\pm}_{1\hskip
1.0pt2^{\prime}}(u).$ (3.30)

By the defining relations,

$\displaystyle(u-v)\,l^{\pm}_{12}(u)\,l^{-}_{22^{\prime}}(v)+\big{(}q-q^{-1}\big{)}\,u\,l^{\pm}_{22}(u)\,l^{-}_{12^{\prime}}(v)$
$\displaystyle{}=(u-v)\,l^{-}_{22^{\prime}}(v)\,l^{\pm}_{12}(u)+\big{(}q-q^{-1}\big{)}\,v\,l^{-}_{22}(v)\,l^{\pm}_{12^{\prime}}(u)-\frac{u-v}{u-v\hskip
1.0pt\xi}\,\sum_{c=1}^{4}\,d_{c\hskip 1.0pt2^{\prime}}(u,v)\hskip
1.0ptl^{-}_{2c}(v)\,l^{\pm}_{1\hskip 1.0ptc^{\prime}}(u).$

Comparing the coefficients of $v$ on both sides we derive

$\displaystyle
l^{\pm}_{12}(u)\,l^{-}_{22^{\prime}}[0]=q\,l^{-}_{22^{\prime}}[0]\,l^{\pm}_{12}(u)+\big{(}q^{-1}-q^{3}\big{)}\,l^{-}_{22}[0]\,l^{\pm}_{12^{\prime}}(u).$

Relation (3.30) now follows from (3.25) and (3.27).

The particular case $\mathfrak{g}=\mathfrak{o}_{4}$ was considered in [21,
Section 4.4]; it was shown therein that $e^{\pm}_{22^{\prime}}(u)=0$, which is
sufficient to complete the argument in this case. These relations essentially
refer to the algebra ${\rm U}^{R}_{q}(\widehat{\mathfrak{g}})$ with
$\mathfrak{g}=\mathfrak{o}_{2}$, whose definition extends to the abelian Lie
algebra $\mathfrak{o}_{2}$. In this case we have $\xi=1$ and the defining
relations give444This corrects the formula used in the proof of [21, Lemma
4.13].

$\displaystyle\big{(}u\hskip 1.0ptq-v\hskip 1.0ptq^{-1}\big{)}\hskip
1.0ptl^{\pm}_{11^{\prime}}(u)\hskip 1.0ptl^{\pm}_{11}(v)=\big{(}u\hskip
1.0ptq^{-1}-v\hskip 1.0ptq\big{)}\hskip 1.0ptl^{\pm}_{11}(v)\hskip
1.0ptl^{\pm}_{11^{\prime}}(u).$

By setting $v=u\hskip 1.0ptq^{2}$ we derive that $l^{\pm}_{11^{\prime}}(u)=0$
and hence $e^{\pm}_{11^{\prime}}(u)=0$.

The argument for the series $f^{\pm}_{ji}(u)$ is quite similar to that for
$e^{\pm}_{ij}(u)$. As an alternative approach, one can work with the quantum
affine algebra defined with the $R$-matrix $R_{21}(u,v)$ used instead of
$R(u,v)$, and modify the above calculations for the series $e^{\pm}_{ij}(u)$
accordingly; then apply the anti-isomorphism introduced in Remark 3.3. ∎

###  3.3 Highest weight representations

######  Definition 3.9.

A representation $V$ of the algebra ${\rm U}^{R}_{q}(\widehat{\mathfrak{g}})$
(or the extended quantum affine algebra ${\rm U}^{{\rm
ext}}_{q}(\widehat{\mathfrak{g}})$) is called a highest weight representation
if there exists a nonzero vector $\zeta\in V$ such that $V$ is generated by
$\zeta$ and the following relations hold:

$\displaystyle l^{\pm}_{ij}(u)\,\zeta=0\qquad$ $\displaystyle\text{for}\quad
1\leqslant i<j\leqslant N,\qquad\text{and}$ $\displaystyle
l^{\pm}_{ii}(u)\,\zeta=\lambda^{\pm}_{i}(u)\,\zeta\qquad$
$\displaystyle\text{for}\quad i=1,\dots,N,$

for some formal series $\lambda^{+}_{i}(u)\in\mathbb{C}\hskip 1.0pt[[u]]$ and
$\lambda^{-}_{i}(u)\in\mathbb{C}\hskip
1.0pt\big{[}\big{[}u^{-1}\big{]}\big{]}$. The vector $\zeta$ is called the
highest vector of $V$.

Note that by (3.4), the product of the constant terms of $\lambda^{+}_{i}(u)$
and $\lambda^{-}_{i}(u)$ must equal $1$.

######  Proposition 3.10.

In terms of the Gaussian generators, the conditions in Definition 3.9 are
equivalent to the following:

$\displaystyle e^{\pm}_{ij}(u)\,\zeta=0\qquad$ $\displaystyle\text{for}\quad
1\leqslant i<j\leqslant N,\qquad\text{and}$ $\displaystyle
h^{\pm}_{i}(u)\,\zeta=\lambda^{\pm}_{i}(u)\,\zeta\qquad$
$\displaystyle\text{for}\quad i=1,\dots,N.$

###### Proof.

This is immediate from the Gauss decomposition formulas (3.11)–(3.13). ∎

The algebra ${\rm U}^{R}_{q}(\widehat{\mathfrak{g}})$ (as well as ${\rm
U}^{{\rm ext}}_{q}(\widehat{\mathfrak{g}})$) admits a family of automorphisms

$\displaystyle L^{+}(u)\mapsto\mathcal{S}L^{+}(u)\qquad\text{and}\qquad
L^{-}(u)\mapsto\mathcal{S}^{-1}L^{-}(u),$ (3.31)

parameterized by invertible diagonal matrices $\mathcal{S}={\rm
diag}\,[\sigma_{1},\dots,\sigma_{N}]$ satisfying the conditions

$\displaystyle\mathcal{S}^{\,{\rm
t}}=\mathcal{S}^{-1}=\mathcal{S}\qquad\text{or}\qquad\mathcal{S}^{\,{\rm
t}}=\mathcal{S}^{-1}=-\mathcal{S}.$

The second condition can occur only for $N=2n$. When applied to the isomorphic
algebra ${\rm U}_{q}(\widehat{\mathfrak{g}})\cong{\rm
U}^{R}_{q}(\widehat{\mathfrak{g}})$, the automorphisms (3.31) correspond to
the sign automorphisms considered in [6, Section 12.2.B].

######  Theorem 3.11.

  1. $1.$

Any finite-dimensional irreducible representation of the algebra ${\rm
U}^{R}_{q}(\widehat{\mathfrak{g}})$ is a highest weight representation. Up to
twisting this representation with a suitable automorphism (3.31), its
parameters satisfy the relations

$\displaystyle\frac{\lambda^{+}_{i}(u)}{\lambda^{+}_{i+1}(u)}=q^{\deg
P_{i}}\,\frac{P_{i}\big{(}uq^{-2}\big{)}}{P_{i}(u)}=\frac{\lambda^{-}_{i}(u)}{\lambda^{-}_{i+1}(u)},\qquad
i=1,\dots,n-1,$ (3.32)

and

$\displaystyle\frac{\lambda^{+}_{n}(u)}{\lambda^{+}_{n+1}(u)}=q^{\frac{1}{2}\deg
P_{n}}\,\frac{P_{n}\big{(}uq^{-1}\big{)}}{P_{n}(u)}=\frac{\lambda^{-}_{n}(u)}{\lambda^{-}_{n+1}(u)}\qquad$
$\displaystyle\text{for type $B_{n}$},$
$\displaystyle\frac{\lambda^{+}_{n}(u)}{\lambda^{+}_{n+1}(u)}=q^{2\deg
P_{n}}\,\frac{P_{n}\big{(}uq^{-4}\big{)}}{P_{n}(u)}=\frac{\lambda^{-}_{n}(u)}{\lambda^{-}_{n+1}(u)}\qquad$
$\displaystyle\text{for type $C_{n}$},$
$\displaystyle\frac{\lambda^{+}_{n-1}(u)}{\lambda^{+}_{n+1}(u)}=q^{\deg
P_{n}}\,\frac{P_{n}\big{(}uq^{-2}\big{)}}{P_{n}(u)}=\frac{\lambda^{-}_{n-1}(u)}{\lambda^{-}_{n+1}(u)}\qquad$
$\displaystyle\text{for type $D_{n}$},$

for some polynomials $P_{i}(u)$ in $u$, all with constant term $1$, where the
first and second equalities are regarded in $\mathbb{C}\hskip 1.0pt[[u]]$ and
$\mathbb{C}\hskip 1.0pt\big{[}\big{[}u^{-1}\big{]}\big{]}$, respectively.

  2. $2.$

Every $n$-tuple $(P_{1}(u),\dots,P_{n}(u))$, where each $P_{i}(u)$ is a
polynomial in $u$ with constant term $1$, arises in this way.

  3. $3.$

The series $\lambda^{\pm}_{i}(u)$ satisfy the relations

$\displaystyle\lambda^{\pm}_{i}\big{(}u\xi q^{2i}\big{)}\hskip
1.0pt\lambda^{\pm}_{i^{\prime}}(u)=\lambda^{\pm}_{i+1}\big{(}u\xi
q^{2i}\big{)}\hskip 1.0pt\lambda^{\pm}_{(i+1)^{\prime}}(u),$ (3.33)

where $i=0,1,\dots,n$ for $N=2n+1$, and $i=0,1,\dots,n-1$ for $N=2n$, and we
set $\lambda^{\pm}_{0}(u)=\lambda^{\pm}_{0^{\prime}}(u):=1$.

###### Proof.

Using the isomorphism ${\rm U}_{q}(\widehat{\mathfrak{g}})\cong{\rm
U}^{R}_{q}(\widehat{\mathfrak{g}})$ and the classification results recalled in
Section 3.1, we find that any type $\mathbf{1}$ finite-dimensional irreducible
representation $V$ of the algebra ${\rm U}_{q}(\widehat{\mathfrak{g}})$ is
generated by a vector $\zeta$ such that

$\displaystyle e^{\pm}_{i,i+1}(u)\hskip 1.0pt\zeta=0\qquad$
$\displaystyle\text{for}\quad i=1,\dots,n,$ (3.34) $\displaystyle
h^{\pm}_{i}(u)\,\zeta=\lambda^{\pm}_{i}(u)\,\zeta\qquad$
$\displaystyle\text{for}\quad i=1,\dots,n+1,$

for some formal series $\lambda^{+}_{i}(u)\in\mathbb{C}\hskip 1.0pt[[u]]$ and
$\lambda^{-}_{i}(u)\in\mathbb{C}\hskip
1.0pt\big{[}\big{[}u^{-1}\big{]}\big{]}$, where for type $D_{n}$ relation
(3.34) with $i=n$ should be replaced with $e^{\pm}_{n-1,n+1}(u)\hskip
1.0pt\zeta=0$. Proposition 3.8 implies that $V$ is a highest weight
representation of ${\rm U}^{R}_{q}(\widehat{\mathfrak{g}})$. Relations (3.32)
follow from the results of [6, Section 12.2]. They can also be derived by
considering a subalgebra ${\rm U}_{i}\subset{\rm
U}^{R}_{q}(\widehat{\mathfrak{g}})$ associated with the $i$-th simple root of
$\mathfrak{g}$ such that ${\rm U}_{i}\cong{\rm
U}_{q}\big{(}\widehat{\mathfrak{sl}}_{2}\big{)}$. The cyclic span ${\rm
U}_{i}\zeta$ is a finite-dimensional ${\rm
U}_{q}(\widehat{\mathfrak{sl}}_{2})$-module which yields the required
conditions on the series $\lambda^{\pm}_{i}(u)$; see [14, Section 3.1] for
more details on this approach going back to [28].

Part 2 of the theorem follows from the classification results of [6, Section
12.2]. The proof relies on the Hopf algebra structure on ${\rm
U}^{R}_{q}(\widehat{\mathfrak{g}})$ introduced in Section 3.2.3. It is implied
by a well-known property of tensor products of representations. Namely,
suppose that $V$ and $W$ are finite-dimensional irreducible representations of
${\rm U}^{R}_{q}(\widehat{\mathfrak{g}})$ with respective highest vectors
$\zeta$ and $\eta$, with the parameter series
$\big{(}\lambda^{\pm}_{1}(u),\dots,\lambda^{\pm}_{N}(u)\big{)}$ and
$\big{(}\mu^{\pm}_{1}(u),\dots,\mu^{\pm}_{N}(u)\big{)}$. The coproduct formula
(3.19) implies that the irreducible quotient $X$ of the cyclic span

$\displaystyle{\rm U}^{R}_{q}(\widehat{\mathfrak{g}})(\zeta\otimes\eta)\subset
V\otimes W$

is a highest weight representation with the parameter series
$\big{(}\lambda^{\pm}_{1}(u)\mu^{\pm}_{1}(u),\dots,\lambda^{\pm}_{N}(u)\mu^{\pm}_{N}(u)\big{)}$.
Therefore, if $V$ and $W$ are associated with the respective $n$-tuples of
polynomials

$\displaystyle(P_{1}(u),\dots,P_{n}(u))\qquad\text{and}\qquad(Q_{1}(u),\dots,Q_{n}(u)),$

then by the formulas of Part 1, the representation $X$ is associated with the
$n$-tuple

$\displaystyle\big{(}P_{1}(u)Q_{1}(u),\dots,P_{n}(u)Q_{n}(u)\big{)}.$

Hence, to complete the proof of Part 2, one only needs to produce a finite-
dimensional irreducible representation to each $n$-tuple of the form

$\displaystyle\big{(}1,\dots,1,au+1,1,\dots,1\big{)},\qquad
a\in\mathbb{C}\hskip 1.0pt.$

The existence of such fundamental representations of the quantum affine
algebra was established in [7]. Part 3 of the theorem is immediate from Lemma
3.2. ∎

######  Corollary 3.12.

All statements of Theorem 3.11 hold in the same form for the algebra ${\rm
U}^{{\rm ext}}_{q}(\widehat{\mathfrak{g}})$, except that the value $i=0$ is
excluded for the conditions (3.33).

###### Proof.

The arguments used in the proof of the theorem equally apply to the
representations of the algebra ${\rm U}^{{\rm
ext}}_{q}(\widehat{\mathfrak{g}})$. Relation (3.33) with $i=0$ is now replaced
by the property that the series $z^{\pm}(u)$ acts in the highest weight
representation as multiplication by $\lambda^{\pm}_{1}(u\xi)\hskip
1.0pt\lambda^{\pm}_{1^{\prime}}(u)$. ∎

####  3.3.1 Representations of quantum affine algebras in type
$\boldsymbol{A}$

Corresponding versions of the classification results described in Theorem 3.11
for the quantum affine algebras in type $A$ are well-known. They can be
derived in the same way as for the types $B$, $C$ and $D$ from the isomorphism
between the $R$-matrix and Drinfeld presentations of the quantum affine
algebra ${\rm U}_{q}(\widehat{\mathfrak{g}})$ for
$\mathfrak{g}=\mathfrak{sl}_{n}$ constructed in [8]. On the other hand, an
independent proof of the classification theorem for the quantum affine
algebras in the $R$-matrix presentation, which we state below, was given in
[14] following the original approach of [28].

The quantum affine algebra ${\rm U}_{q}(\widehat{\mathfrak{sl}}_{n})$ (with
the trivial central charge) is defined by the formulas of Section 3.1, where
the simple roots are chosen in the form (2.1). Finite-dimensional irreducible
representations of ${\rm U}_{q}(\widehat{\mathfrak{sl}}_{n})$ are described by
the results recalled in that section.

Consider the $R$-matrix defined by

$\displaystyle R_{A}(u)=\sum_{i,j=1}^{n}\hskip 1.0pt\big{(}u\hskip
1.0ptq^{\delta_{ij}}-q^{-\delta_{ij}}\big{)}\hskip 1.0pte_{ii}\otimes
e_{jj}+\big{(}q-q^{-1}\big{)}\hskip
1.0pt\sum_{i,j=1}^{n}(u\,\delta_{i<j}+\delta_{i>j})\,e_{ij}\otimes e_{ji}.$

The quantum affine algebra ${\rm
U}^{R}_{q}\big{(}\widehat{\mathfrak{gl}}_{n}\big{)}$ (with the trivial central
charge) is generated by elements ${l}^{\pm}_{ij}[\mp m]$ with $1\leqslant
i,j\leqslant n$ and $m\in\mathbb{Z}\hskip 1.0pt_{+}$ subject to the defining
relations described in (3.4)–(3.9), where the parameter $N$ is replaced by $n$
and the $R$-matrix (3.1) is replaced by $R_{A}(u)$. The Hopf algebra structure
on ${\rm U}^{R}_{q}\big{(}\widehat{\mathfrak{gl}}_{n}\big{)}$ is described by
the maps (3.19), (3.20) and (3.21). The algebra ${\rm
U}^{R}_{q}\big{(}\widehat{\mathfrak{sl}}_{n}\big{)}$ is a Hopf subalgebra of
${\rm U}^{R}_{q}\big{(}\widehat{\mathfrak{gl}}_{n}\big{)}$ which consists of
all elements which are stable with respect to all automorphisms

$\displaystyle L^{\pm}(u)\mapsto f^{\pm}(u)\hskip 1.0ptL^{\pm}(u),$

where $f^{\pm}(u)$ are arbitrary series satisfying conditions (3.16). By the
results of [8] the maps (3.14) define a Hopf algebra isomorphism ${\rm
U}_{q}\big{(}\widehat{\mathfrak{sl}}_{n}\big{)}\to{\rm
U}^{R}_{q}\big{(}\widehat{\mathfrak{sl}}_{n}\big{)}$, where the Gaussian
generators are defined by (3.11). The algebra ${\rm
U}^{R}_{q}\big{(}\widehat{\mathfrak{sl}}_{n}\big{)}$ (as well as ${\rm
U}^{R}_{q}\big{(}\widehat{\mathfrak{gl}}_{n}\big{)}$) admits a family of
automorphisms (3.31) parameterized by diagonal matrices $\mathcal{S}={\rm
diag}\,[\sigma_{1},\dots,\sigma_{n}]$ such that $\sigma_{i}=\pm 1$ for all
$i$.

Highest weight representations of the algebra ${\rm
U}^{R}_{q}(\widehat{\mathfrak{gl}}_{n})$ are defined in the same way as in
Definition 3.9. By restrictions, we get highest weight representations of the
subalgebra ${\rm U}^{R}_{q}\big{(}\widehat{\mathfrak{sl}}_{n}\big{)}$.
Proposition 3.10 holds in the same form. The following theorem is contained in
[14, Theorem 3.6]. The proof of Theorem 3.11 applied to the quantum affine
algebras of type $A$ provides another derivation of that result.

######  Theorem 3.13.

  1. $1.$

Any finite-dimensional irreducible representation of the algebra ${\rm
U}^{R}_{q}\big{(}\widehat{\mathfrak{gl}}_{n}\big{)}$ (and ${\rm
U}^{R}_{q}\big{(}\widehat{\mathfrak{sl}}_{n}\big{)}$) is a highest weight
representation. Up to twisting this representation with a suitable
automorphism (3.31), its parameters satisfy the relations

$\displaystyle\frac{\lambda^{+}_{i}(u)}{\lambda^{+}_{i+1}(u)}=q^{\deg
P_{i}}\,\frac{P_{i}\big{(}uq^{-2}\big{)}}{P_{i}(u)}=\frac{\lambda^{-}_{i}(u)}{\lambda^{-}_{i+1}(u)},\qquad
i=1,\dots,n-1,$

for some polynomials $P_{i}(u)$ in $u$, all with constant term $1$, where the
equalities are understood for the expansions of the rational function in $u$
as a series in $u$ and $u^{-1}$, respectively.

  2. $2.$

Every $n$-tuple of polynomials $(P_{1}(u),\dots,P_{n-1}(u))$ in $u$, where
each $P_{i}(u)$ has constant term $1$, arises in this way.

####  3.3.2 Outlook

The isomorphisms of [20, 21] do not rely on any Poincaré–Birkhoff–Witt-type
theorem for the algebra ${\rm U}^{R}_{q}(\widehat{\mathfrak{g}})$. Therefore,
such a theorem is implied by the results of Beck [4]. In particular, by
Proposition 3.8, the image of the basis of the subalgebra ${\rm
U}_{q}(\widehat{\mathfrak{g}})^{+}$ under the isomorphism ${\rm
U}_{q}(\widehat{\mathfrak{g}})\to{\rm U}^{R}_{q}(\widehat{\mathfrak{g}})$ is a
basis of the subalgebra ${\rm U}^{R}_{q}(\widehat{\mathfrak{g}})^{+}$
generated by the coefficients of all series $e^{\pm}_{ij}(u)$. However, a
precise expression of the basis elements in terms of these coefficients is
unknown.

Note also that the version of the Poincaré–Birkhoff–Witt theorem for the
algebra ${\rm U}_{q}\big{(}\widehat{\mathfrak{gl}}_{N}\big{)}$ given in [14,
Corollary 2.13] does not immediately extend to types $B$, $C$ and $D$, because
of more complicated defining relations on the generators $l^{\pm}_{ij}[m]$. It
would be interesting to find an alternative form of the quadratic relations
for the generator series to lead to such a version.

##  Appendix A Modified isomorphism for Yangians

Here we give a modified version of the isomorphism produced in [19, Main
Theorem] which can be used to establish a different correspondence between the
parameters of representations in the two realizations of the Yangians; cf.
Section 2.2.1. An analogous isomorphism was used for type $A$ in [24, Section
3.1]; see also [5] for an isomorphism with an opposite presentation of the
Yangian ${\rm Y}(\mathfrak{sl}_{N})$. The new version is based on the
alternative Gauss decomposition of the matrix $T(u)$, defined by

$\displaystyle T(u)=\overline{E}(u)\,\overline{H}(u)\,\overline{F}(u),$ (A.1)

where $\overline{E}(u)$, $\overline{H}(u)$ and $\overline{F}(u)$ are uniquely
determined matrices of the form

$\displaystyle\overline{E}(u)=\begin{bmatrix}\,1&\bar{e}_{12}(u)&\dots&\bar{e}_{1N}(u)\,\\\
\,0&1&\dots&\bar{e}_{2N}(u)\\\ \vdots&\vdots&\ddots&\vdots\\\
0&0&\dots&1\end{bmatrix},\qquad\overline{F}(u)=\begin{bmatrix}1&0&\dots&0\,\\\
\bar{f}_{21}(u)&1&\dots&0\\\ \vdots&\vdots&\ddots&\vdots\\\
\bar{f}_{N1}(u)&\bar{f}_{N2}(u)&\dots&1\end{bmatrix},$

and $\overline{H}(u)={\rm
diag}\,\big{[}\bar{h}_{1}(u),\dots,\bar{h}_{N}(u)\big{]}$. Define the series
with coefficients in ${\rm Y}(\mathfrak{g})$ by

$\displaystyle\kappa_{i}(u)=\bar{h}_{i}(u+(i-1)/2)\,\bar{h}_{i+1}(u+(i-1)/2)^{-1}$

for $i=1,\dots,n-1$, and

$\displaystyle\kappa_{n}(u)=\begin{cases}\bar{h}_{n}(u+(n-1)/2)\,\bar{h}_{n+1}(u+(n-1)/2)^{-1}&\text{for}\
\mathfrak{o}_{2n+1},\\\ \hskip
1.0pt\bar{h}_{n}(u+n/2)\,\bar{h}_{n+1}(u+n/2)^{-1}&\text{for}\
\mathfrak{sp}_{2n},\\\
\bar{h}_{n-1}(u+(n-2)/2)\,\bar{h}_{n+1}(u+(n-2)/2)^{-1}&\text{for}\
\mathfrak{o}_{2n}.\end{cases}$

Furthermore, set

$\displaystyle\xi_{i}^{+}(u)=\bar{e}_{i\,i+1}\big{(}u+(i-1)/2\big{)},\qquad\xi_{i}^{-}(u)=\bar{f}_{i+1\,i}\big{(}u+(i-1)/2\big{)}$

for $i=1,\dots,n-1$,

$\displaystyle\xi_{n}^{+}(u)=\begin{cases}\bar{e}_{n\,n+1}(u+(n-1)/2\big{)}&\text{for}\
\mathfrak{o}_{2n+1},\\\ \bar{e}_{n\,n+1}\big{(}u+n/2\big{)}&\text{for}\
\mathfrak{sp}_{2n},\\\ \bar{e}_{n-1\,n+1}\big{(}u+(n-2)/2\big{)}&\text{for}\
\mathfrak{o}_{2n}\end{cases}$

and

$\displaystyle\xi_{n}^{-}(u)=\begin{cases}\bar{f}_{n+1\,n}(u+(n-1)/2)&\text{for}\
\mathfrak{o}_{2n+1},\\\ \frac{1}{2}\bar{f}_{n+1\,n}(u+n/2)&\text{for}\
\mathfrak{sp}_{2n},\\\ \bar{f}_{n+1\,n-1}(u+(n-2)/2)&\text{for}\
\mathfrak{o}_{2n}.\end{cases}$

Introduce elements of ${\rm Y}(\mathfrak{g})$ by the respective expansions
into power series in $u^{-1}$,

$\displaystyle\kappa_{i}(u)=1+\sum_{r=0}^{\infty}\kappa_{i\hskip
1.0ptr}\,u^{-r-1}\qquad\text{and}\qquad\xi_{i}^{\pm}(u)=\sum_{r=0}^{\infty}\xi_{i\hskip
1.0ptr}^{\pm}\,u^{-r-1}$

for $i=1,\dots,n$.

######  Theorem A.1.

The mapping which sends the generators $\kappa_{i\hskip 1.0ptr}$ and
$\xi_{i\hskip 1.0ptr}^{\pm}$ of ${\rm Y}^{D}(\mathfrak{g})$ to the elements of
${\rm Y}(\mathfrak{g})$ with the same names defines an isomorphism ${\rm
Y}^{D}(\mathfrak{g})\cong{\rm Y}(\mathfrak{g})$.

###### Proof.

We will derive this result from [19, Main Theorem] as recalled in Section
2.2.1, by taking the composition of the isomorphism ${\rm
Y}^{D}(\mathfrak{g})\to{\rm Y}(\mathfrak{g})$ constructed therein, with
certain automorphisms of the algebra ${\rm Y}(\mathfrak{g})$. We have the
following quasideterminant formulas for the entries of the matrices
$\overline{H}(u)$, $\overline{E}(u)$ and $\overline{F}(u)$ [13]:

$\displaystyle\bar{h}_{i}(u)=\begin{vmatrix}\boxed{t_{i\hskip
1.0pti}(u)}&t_{i\,i+1}(u)&\dots&t_{i\hskip 1.0ptN}(u)\\\
t_{i+1\,i}(u)&t_{i+1\,i+1}(u)&\dots&t_{i+1\hskip 1.0ptN}(u)\\\
\vdots&\vdots&\ddots&\vdots\\\ t_{N\hskip
1.0pti}(u)&t_{N\,i+1}(u)&\dots&t_{N\hskip 1.0ptN}(u)\end{vmatrix},\qquad
i=1,\dots,N,$ (A.2)

whereas

$\displaystyle\bar{e}_{ij}(u)=\begin{vmatrix}\boxed{t_{i\hskip
1.0ptj}(u)}&t_{i\,j+1}(u)&\dots&t_{i\hskip 1.0ptN}(u)\\\
t_{j+1\,j}(u)&t_{j+1\,j+1}(u)&\dots&t_{j+1\hskip 1.0ptN}(u)\\\
\vdots&\vdots&\ddots&\vdots\\\ t_{N\hskip
1.0ptj}(u)&t_{N\,j+1}(u)&\dots&t_{N\hskip
1.0ptN}(u)\end{vmatrix}\,h_{j}(u)^{-1}$ (A.3)

and

$\displaystyle\bar{f}_{ji}(u)=h_{j}(u)^{-1}\,\begin{vmatrix}\boxed{t_{j\hskip
1.0pti}(u)}&t_{j\,j+1}(u)&\dots&t_{j\hskip 1.0ptN}(u)\\\
t_{j+1\,i}(u)&t_{j+1\,j+1}(u)&\dots&t_{j+1\hskip 1.0ptN}(u)\\\
\vdots&\vdots&\ddots&\vdots\\\ t_{N\hskip
1.0pti}(u)&t_{N\,j+1}(u)&\dots&t_{N\hskip 1.0ptN}(u)\end{vmatrix}$ (A.4)

for $1\leqslant i<j\leqslant N$. The Gaussian generators $h_{i}(u)$,
$e_{ij}(u)$ and $f_{ji}(u)$ are defined by using the decomposition (2.6) dual
to (A.1), and are given by the respective dual quasideterminant formulas; see
[19, Section 4]. On the other hand, the mapping

$\displaystyle\varsigma\colon\ t_{ij}(u)\mapsto
t_{i^{\prime}j^{\prime}}(u),\qquad 1\leqslant i,j\leqslant N,$ (A.5)

defines an involutive automorphism of the algebra ${\rm Y}(\mathfrak{g})$.
Since quasideterminants are unchanged under permutations of rows or columns,
we find that the images of the Gaussian generators are given by

$\displaystyle\varsigma\colon\ h_{i}(u)\mapsto\bar{h}_{i^{\prime}}(u),\qquad
e_{ij}(u)\mapsto\bar{f}_{i^{\prime}j^{\prime}}(u),\qquad
f_{ji}(u)\mapsto\bar{e}_{j^{\prime}i^{\prime}}(u).$

Furthermore, the unitary condition (2.5) implies symmetry relations for the
Gaussian generators which were described in [19, Section 5.3].555The formulas
in [19, Proposition 5.7] concerning the Lie algebra $\mathfrak{o}_{2n+1}$
should be corrected as follows: $e_{n+1\,n+2}(u)=-e_{n}(u-1/2)$ and
$f_{n+2\,n+1}(u)=-f_{n}(u-1/2)$. They are given in (2.7), and for
$i=1,\dots,n-1$ we have

$\displaystyle
e_{(i+1)^{\prime}\,i^{\prime}}(u)=-e_{i\,i+1}(u+\kappa-i)\qquad\text{and}\qquad
f_{i^{\prime}\,(i+1)^{\prime}}(u)=-f_{i+1\,i}(u+\kappa-i),$

with some additional type-specific relations. These relations allow us to
express the images of the generators of ${\rm Y}^{D}(\mathfrak{g})$ under the
composition of the isomorphism ${\rm Y}^{D}(\mathfrak{g})\to{\rm
Y}(\mathfrak{g})$ provided by [19, Main Theorem] with the automorphism (A.5),
in terms of the Gaussian generators (A.2)–(A.4). The formulas given in the
statement of the theorem are obtained by taking further compositions with the
shift automorphism $T(u)\mapsto T(u+\kappa-1)$ and with the automorphism which
multiplies all generators $\xi_{i\hskip 1.0ptr}^{\pm}$ by $-1$, while leaving
all $\kappa_{i\,r}$ unchanged. ∎

### Acknowledgements

We acknowledge the support of the Australian Research Council, grant
DP180101825.

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