Some remarks on quantized Lie superalgebras of classical type

TL;DR

This paper explores the quantization of classical Lie superalgebras, extending known isomorphisms to arbitrary Cartan matrices, and demonstrates deformation of modules and a super version of the Drinfeld-Kohno Theorem.

Contribution

It extends the isomorphism between Drinfeld-Jimbo superalgebras and Etingof-Kazhdan quantizations to all Cartan matrices and establishes module deformation and a super Drinfeld-Kohno Theorem.

Findings

Isomorphism extended to arbitrary Cartan matrices.

All highest weight modules can be deformed to superalgebra modules.

Established a super version of the Drinfeld-Kohno Theorem.

Abstract

In this paper we use the Etingof-Kazhdan quantization of Lie bi-superalgebras to investigate some interesting questions related to Drinfeld-Jimbo type superalgebra associated to a Lie superalgebra of classical type. It has been shown that the D-J type superalgebra associated to a Lie superalgebra of type A-G, with the distinguished Cartan matrix, is isomorphic to the E-K quantization of the Lie superalgebra. The first main result in the present paper is to extend this to arbitrary Cartan matrices. This paper also contains two other main results: 1) a theorem stating that all highest weight modules of a Lie superalgebra of type A-G can be deformed to modules over the corresponding D-J type superalgebra and 2) a super version of the Drinfeld-Kohno Theorem.