Quantum Groups, Deformations, and Geometry

A Conference in Memory of Joseph Donin

Andrea Appel - Cohomological properties of universal relative twists

In 1995, Donin and Shnider provided a new approach to the quantization of a complex semisimple Lie algebra g, using cohomological methods to directly construct a twist, i.e. an element which reduces the non-trivial associator in the Drinfeld category of Ug to the trivial associativity constraint. Their methods were later used by Toledano Laredo to prove existence and properties of relatives twists, which, in the Tannakian formalism, can be thought of as tensor structures on restriction functors. Motivated by the theory of quasi-Coxeter structures on Kac--Moody algebras, I will discuss the existence and the uniqueness properties of universal relative twists in the framework of PROP categories, generalizing the results of Etingof--Kazhdan and Enriquez for the standard twists. The talk is based on a joint work with V. Toledano Laredo.