On crystal bases of two-parameter $(v, t)$-quantum groups

TL;DR

This paper develops crystal bases for two-parameter quantum groups and their modules, demonstrating their equivalence with geometrically constructed canonical bases up to a 2-cocycle deformation, extending Kashiwara's approach.

Contribution

It introduces the construction of crystal bases for two-parameter quantum groups and proves their equivalence with geometric canonical bases.

Findings

Crystal bases constructed for two-parameter quantum algebras.

Global crystal basis coincides with Fan and Li's canonical basis up to a 2-cocycle.

Extension of Kashiwara's algebraic approach to two-parameter setting.

Abstract

Following Kashiwara's algebraic approach in one-parameter case, we construct crystal bases for two-parameter quantum algebras and for their integrable modules. We also show that the global crystal basis coincides with the canonical basis geometrically constructed by Fan and Li up to a 2-cocycle deformation.