Drinfel'd doubles and Lusztig's symmetries of two-parameter quantum groups

Abstract

We find the defining structures of two-parameter quantum groups U r,s (g) corresponding to the orthogonal and the symplectic Lie algebras, which are realized as Drinfel'd doubles. We further investigate the environment conditions upon which the Lusztig's symmetries exist between (U r,s (g), , ) and its associated object (U s −1 ,r −1 (g), | ).

Notes on 2-parameter quantum groups I

“…Recently, Hu et al continued this project (see [11,12] In this note, we give a simpler definition for a class of two-parameter quantum groups U r,s (g) associated to semisimple Lie algebras in terms of the Euler form (or say, Ringel form). As in [5,9,11,12,16,17], these quantum groups also possess the Drinfel'd double structures and the triangular decompositions (see Section 2). As a main point of this note, we show that the positive parts of quantum groups under consideration are 2-cocycle deformations of each other as Q + -graded associative Calgebras if the parameters satisfy certain conditions (see Section 3).…”

On the centre of two-parameter quantum groups Ur,s(g) for type Hu1, Shi2 2014Journal of Geometry and PhysicsSelf Cite

“…Since then, a systematic study of the two-parameter quantum groups has been going on, see, for instance, ([BW2], [BW3]) for type A; ([BGH1], [BGH2]) for types B, C, D; [HS], [BH] for types G 2 , E 6 , E 7 , E 8 . For a unified definition, see ([HP1], [HP2]).…”

“…Definition of U r,s (so 2n+1 ). The definition of two-parameters quantum groups of type B was given in [BGH1]. Here K = Q(r, s) is a subfield of C with r, s ∈ C, r 2 +s 2 = 1, r = s. Φ is the root system of so 2n+1 with Π a base of simple roots, which is a finite subset of a Euclidean space E = R n with an inner product (, ).…”

Convex PBW-type Lyndon basis and restricted two-parameter quantum groups of typeG2

“…Since then, a systematic study of the 2-parameter quantum groups of type A has been developed by Benkart and Witherspoon and their collaborators Kang and Lee; see [Benkart et al 2003;2006;Benkart and Witherspoon 2004a;2004b]. In 2004, Bergeron, Gao and Hu [2006] defined the 2-parameter quantum groups U r,s (g) (in the sense of Benkart and Witherspoon) for g = so 2n+1 , so 2n and sp 2n , which are realized as Drinfel'd doubles, and described weight modules in the category ᏻ when r s −1 is not a root of unity; see [Bergeron et al 2007]. Afterwards, Hu and his collaborators continued to develop the corresponding theory for exceptional types G and E and the affine cases; see [Hu and Shi 2007;Bai and Hu 2008;Hu and Zhang 2006].…”