# VINBERG DISTINGUISHED LECTURE SERIES

## Monday, April 24, 2023.

#### 8:00 AM (Los Angeles), 11:00 AM (Toronto, New York), 5:00 PM (Zurich, Paris) 6:00 PM (Tel-Aviv)

# Victor Kac

## MIT

# Exceptional de Rham complexes

## Abstract

A celebrated theorem of E. Cartan says that a simple infinite-dimensional Lie algebra of vector fields on a finite-dimensional manifold is isomorphic to one of the four series of Lie algebras: of all vector fields, of divergence zero vector fields on a manifold with a volume form, of Hamiltonian vector fields on a symplectic manifold, and of contact vector fields on a contact manifold. For each of them there is the associated de Rham complex, consisting of “degenerate” modules over these Lie algebras. In my talk I will explain an extension of this theorem to Lie superalgebras of vector fields on a finite-dimensional supermanifold. It turns out that the theory is much richer: there are, up to an isomorphism, 10 series and 5 exceptional infinite-dimensional Lie superalgebras of vector fields on a finite-dimensional supermanifold. For the latter the degenerate modules can be arranged in an infinite number of complexes, called the exceptional de Rham complexes.