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Nathan Geer

Quantization of Lie superalgebras

### Résumé

For every semi-simple Lie algebra one can construct the Drinfeld-Jimbo
algebra U. This algebra is a deformation Hopf algebra defined by
generators and relations. To study the representation theory of U,
Drinfeld used the KZ-equations to construct a quasi-Hopf algebra A.
He proved that particular categories of modules over the algebras U
and A are tensor equivalent. Analogous constructions of the algebras
U and A exist for basic Lie superalgebras. However, Drinfeld's proof
of the above equivalence of categories does not generalize to Lie
superalgebras. In this talk, we will discuss an alternate proof for
basic Lie superalgebras. 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.