About:

ICMS Mathematics Colloquium: Exceptional de Rham Complexes

Prof Victor Kac (he/him), MIT

About the talk:

By a celebrated theorem of E. Cartan, a simple infinite-dimensional Lie algebra of vector fields on a finite-dimensional manifold is isomorphic to one of the following Lie algebras: all vector fields, divergence zero vector fields on a manifold with a volume form, Hamiltonian vector fields on a symplectic manifold, and contact vector fields on a contact manifold. For each of these there is the associated de Rham complex, consisting of "degenerate" modules over these Lie algebras. With the advent of supersymmetry theories in physics the problem of extending Cartan's classification to supermanifolds naturally arose. In this talk Victor explained the extension of Cartan's theorem to Lie superalgebras of vector fields on a finite-dimensional supermanifold. He further went on to discuss how it turned out that the theory is much richer: there are, up to an isomorphism, 10 series and 5 exceptional simple infinite-dimensional Lie superalgebras of vector fields on a finite-dimensional supermanifold.He sketched a proof of this theorem. Furthermore, it tuned out that for the exceptional Lie superalgebras the degenerate modules can be neatly arranged in an infinite number of complexes, called the exceptional de Rham complexes.

About the speaker:

Professor Victor Kac received his A.B. in 1965 and his PhD in 1968 from Moscow State University. Ernest Vinberg was his scientific advisor. Professor Kac taught at the Moscow Institute of Electronic Machine Building in 1968-1976, emigrating to the US and joining MIT mathematics faculty in 1977 (professor 1981).