Abstract

The famous theorem of E. Cartan (1909) states that a simple infinite-dimensional Lie algebra of vector fields in a formal neighborhood of a point on a manifold is isomorphic to one of the four series: Lie algebra of all vector fields, or its subalgebras of divergenceless vector fields, Hamiltonian vector fields, or contact vector fields. It turned out that in the case of supermanifolds the classification is much richer: it consists of ten series and five excepional Lie superalgebras of vector fields. In my first lecture I will explain this classification. In the second lecture I will dwell on representation theory of these superalgebras. It turned out that, while representation theory of the Lie algebra of all vector fields is intimately related to the classical de Rham complex, representation theory of exceptional Lie superalgebras of vector fields gives rise to infinitely many complexes, called the exceptional de Rham complexes.