Multiparameter Quantum Supergroups, Deformations and Specializations

TL;DR

This paper develops a multiparameter framework for quantum supergroups, exploring their deformations, semiclassical limits, and stability under various transformations, thereby extending the theory of quantum superalgebras.

Contribution

It introduces a multiparameter version of quantum supergroups, analyzes their deformation stability, and establishes the compatibility of quantization with deformation processes.

Findings

Family of multiparameter quantum supergroups is stable under twist and 2-cocycle deformations.

Semiclassical limits are multiparameter Lie superbialgebras.

Quantization commutes with deformation for these structures.

Abstract

In this paper we introduce a multiparameter version of the quantum universal enveloping superalgebras introduced by Yamane in [H. Yamane, "Quantized enveloping algebras associated to simple Lie superalgebras and their universal $R$-matrices", Publ. Res. Inst. Math. Sci. 30 (1994), no. 1, 15-87]. For these objects we consider: - (1) their deformations by twist and by 2-cocycle (both of "toral type"); in particular, we prove that this family is stable under both types of deformations; - (2) their semiclassical limits, which are multiparameter Lie superbialgebras; - (3) the deformations by twist and by 2-cocycle (of "toral type") of these multiparameter Lie superbialgebras: in particular, we prove that this family is stable under these deformations, and that "quantization commutes with deformation".