On the R-matrix realization of Yangians and their representations

Following the approach of Ding and Frenkel [Comm. Math. Phys. 156 (1993), 277-300] for type $A$, we showed in our previous work [J. Math. Phys. 61 (2020), 031701, 41 pages] that the Gauss decomposition of the generator matrix in the $R$-matrix presentation of the quantum affine algebra yields the Drinfeld generators in all classical types. Complete details for type $C$ were given therein, while the present paper deals with types $B$ and $D$. The arguments for all classical ty...

For a large class of finite W algebras, the defining relations of a Yangian are proved to be satisfied. Therefore such finite W algebras appear as realisations of Yangians. This result is useful to determine properties of such W algebra representations.

We continue the study of finite-dimensional irreducible representations of twisted Yangians associated to symmetric pairs of types B, C and D, with focus on those of types BI, CII and DI. After establishing that, for all twisted Yangians of these types, the highest weight of such a module necessarily satisfies a certain set of relations, we classify the finite-dimensional irreducible representations of twisted Yangians for the pairs $(\mathfrak{so}_N,\mathfrak{so}_{N-2} \oplu...

The twisted q-Yangians are coideal subalgebras of the quantum affine algebra associated with gl(N). We prove a classification theorem for finite-dimensional irreducible representations of the twisted q-Yangians associated with the symplectic Lie algebras sp(2n). The representations are parameterized by their highest weights or by their Drinfeld polynomials. In the simplest case of sp(2) we give an explicit description of all the representations as tensor products of evaluatio...

We give explicit realizations of irreducible representations of the Yangian of the general linear Lie algebra and of its twisted analogues, corresponding to symplectic and orthogonal Lie algebras. In particular, we develop the fusion procedure for twisted Yangians. For the non-twisted Yangian, this procedure goes back to the works of Cherednik.

On the basis of `$RTT=TTR$' formalism, we introduce the quantum double of the Yangian $Y_{\hbar}(\gtg)$ for $\gtg=\gtgl_N,\gtsl_N$ with a central extension. The Gauss decomposition of T-matrices gives us the so-called Drinfel'd generators. Using these generators, we present some examples of both finite and infinite dimensional representations that are quite natural deformations of the corresponding affine counterpart.

Using recent results of P. Etingof and A. Varchenko on the Classical Dynamical Yang-Baxter equation, we reduce the classification of dynamical r-matrices r on a commutative subalgebra l of a Lie algebra g to a purely algebraic problem under some technical conditions on the symmetric part of r. Using this, we then classify all non skew-symmetric dynamical r-matrices when g is a simple Lie algebra and l a commutative subalgebra containing a regular semisimple element. This part...

We present a connection between W-algebras and Yangians, in the case of gl(N) algebras, as well as for twisted Yangians and/or super-Yangians. This connection allows to construct an R-matrix for the W-algebras, and to classify their finite-dimensional irreducible representations. We illustrate it in the framework of nonlinear Schroedinger equation in 1+1 dimension.

Studying the algebraic structure of the double ${\cal D}Y(g)$ of the yangian $Y(g)$ we present the triangular decomposition of ${\cal D}Y(g)$ and a factorization for the canonical pairing of the yangian with its dual inside ${\cal D}Y(g)$. As a consequence we obtain an explicit formula for the universal R-matrix $R$ of ${\cal D}Y(g)$ and demonstrate how it works in evaluation representations of $Y(sl_2)$. We interprete one-dimensional factor arising in concrete representation...

We study in detail the structure of the Yangian Y(gl(N)) and of some new Yangian-type algebras called twisted Yangians. The algebra Y(gl(N)) is a `quantum' deformation of the universal enveloping algebra U(gl(N)[x]), where gl(N)[x] is the Lie algebra of gl(N)-valued polynomial functions. The twisted Yangians are quantized enveloping algebras of certain twisted Lie algebras of polynomial functions which are naturally associated to the B, C, and D series of the classical Lie al...