2020 Thematic Program (I)
Lie algebras of vector fields on algebraic varieties
December 16-20, 2020
W303 School of Mathematics, Sichuan University
We are going to discuss the representation theory of Lie algebras of polynomial vector fields on algebraic varieties. Classical examples include Virasoro and Witt algebras which correspond to the case of the torus. These algebras play an important role in different areas of Mathematics and Physics. In recent years there has been a growing interest to the study of representation of Lie algebras of derivations of rings of functions on more general algebraic varieties, this will be the focus of our lectures.
In the first lecture we will discuss Lie algebras of differential operators and derivation algebras and consider examples. In the second lecture we focus on representation theory of Virasoro and Witt algebras, in particular on the classification of irreducible Harish-Chandra modules for these algebras based on the results of O. Mathieu and joint results with Y. Billig. In the third lecture we will establish the simplicity criterion for the Lie algebras of vector fields and consider examples. Finally, in the fourth lecture we will discuss new constructions of Rudakov and Gauge representations in the case of arbitrary varieties based on the results of Y. Billig, J. Nilsen, A. Zaidan.