Tridiagonal pairs and the quantum affine algebra $${\boldmath U_q({\widehat {sl}}_2)}$$

Abstract

Let K denote an algebraically closed field and let q denote a nonzero scalar in K that is not a root of unity. Let V denote a vector space over K with finite positive dimension and let A, A * denote a tridiagonal pair on V . Let θ 0 , θ 1 , . . . , θ d (resp. θ * 0 , θ * 1 , . . . , θ * d ) denote a standard ordering of the eigenvalues of A (resp. A * ). We assume there exist nonzero scalars a, a * in K such that θ i = aq 2i−d and θ * i = a * q d−2i for 0 ≤ i ≤ d. We display two irreducible U q ( sl 2 )-module structures on V and discuss how these are related to the actions of A and A * .

A family of tridiagonal pairs and related symmetric functions

“…Having these equations, we can now consider the action of W (N +1) 1 on the eigenvectors, using (17). Let us first consider the domain j ∈ {1, ..., N n }:…”

Theq-Tetrahedron Algebra and Its Finite Dimensional Irreducible Modules

“…The x ij satisfy Definition 6.1(ii) by [19,Theorems 7.1,10.1,10.2]. The x ij satisfy Definition 6.1(iii) by [19,Theorem 12.1]. We have now shown that the transformations x ij satisfy the defining relations for ⊠ q .…”

“…The x ij satisfy Definition 6.1(i) by the construction. The x ij satisfy Definition 6.1(ii) by [19,Theorems 7.1,10.1,10.2]. The x ij satisfy Definition 6.1(iii) by [19,Theorem 12.1].…”

“…In Sections 10-18 we describe the finite dimensional irreducible ⊠ q -modules. In Section 19 we mention some open problems.…”

Higher Order Relations for ADE-Type Generalized q-Onsager Algebras

“…For g = a (1) n , it can be understood as a q−deformation of the sl n+1 -Onsager algebra introduced by Uglov and Ivanov [UI]. By analogy with the sl 2 case [B,IT2], an algebra homomorphism from O q ( g) to a coideal subalgebra of the Drinfeld-Jimbo [Dr,J1] quantum universal enveloping algebra U q ( g) is known [BB] (see also [Ko]). Realizations in terms of finite dimensional quantum algebras can be also considered (see for instance [BF]): using either the coideal subalgebras of U q (g) studied by Letzter [Le] or the non-standard U ′ q (so n ) introduced by Klimyk, Gavrilik and Iorgov [GI, Klim].…”