Two-Parameter Quantum Groups and Drinfel'd Doubles

Abstract

Abstract. We investigate two-parameter quantum groups corresponding to the general linear and special linear Lie algebras gl n and sl n . We show that these quantum groups can be realized as Drinfel'd doubles of certain Hopf subalgebras with respect to Hopf pairings. Using the Hopf pairing, we construct a corresponding R-matrix and a quantum Casimir element. We discuss isomorphisms among these quantum groups and connections with multiparameter quantum groups.

Pointed Hopf Algebras with Triangular Decomposition

“…The first class we discuss are the multiparameter quantum groups U λ,p (gl n ) introduced by [19] (adapting the presentation in [15]). They are asymmetric braided Drinfeld doubles, which is a generalization of the result of [14] showing that two-parameter quantum groups are Drinfeld doubles. In Section 5.2 we bring results of [31] on growth condition (finite Gelfand-Kirillov dimension) and classification of Nichols algebras from [9] into the picture.…”

“…It was shown in [24] that U q (g) in fact is a braided Drinfeld double (which is referred to as a double bosonization there). It was proved in [14] that also two-parameter quantum groups are Drinfeld doubles.…”

“…We will provide a variation of the presentation of [11,15] The result that the multiparameter quantum group U λ,p (gl n ) is the asymmetric braided Drinfeld double U kG (F, E) can be seen as a generalization of the result in [14] where the two-parameter quantum groups were shown to be Drinfeld doubles.…”

On the centre of two-parameter quantum groups

“…As shown in [4,5], these quantum groups are Drinfel'd doubles and have an R-matrix. They are related to the down-up algebras in [2,3] and to the multi-parameter quantum groups of Chin and Musson [8] and Dobrev and Parashar [10].…”

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

“…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]).…”