Etingof–Kazhdan quantization of Lie superbialgebras

Abstract

For every semi-simple Lie algebra g one can construct the Drinfeld–Jimbo algebra U h DJ ( g ) . This algebra is a deformation Hopf algebra defined by generators and relations. To study the representation theory of U h DJ ( g ) , Drinfeld used the KZ-equations to construct a quasi-Hopf algebra A g . He proved that particular categories of modules over the algebras U h DJ ( g ) and A g are tensor equivalent. Analogous constructions of the algebras U h DJ ( g ) and A g exist in the case when g is a Lie superalgebra of type A-G. However, Drinfeld's proof of the above equivalence of categories does not generalize to Lie superalgebras. In this paper, we will discuss an alternate proof for Lie superalgebras of type A-G. Our proof utilizes the Etingof–Kazhdan quantization of Lie (super)bialgebras. It should be mentioned that the above equivalence is very useful. For example, it has been used in knot theory to relate quantum group invariants and the Kontsevich integral.