Quasi-Poisson superalgebras

Abstract: In 1985, Novikov and Balinskii introduced what became known as Novikov algebras in an attempt to construct generalizations of Witt Lie algebra. To their disappointment, Zelmanov showed that the only simple finite-dimensional Novikov algebra is one-dimensional (and corresponds to Witt algebra). The picture is much more interesting in the super case, where there are many more generalizations of Witt algebra, called superconformal Lie algebras. In 1988 Kac and Van de Leur gave a conjectural list of simple superconformal Lie algebras. Their list was amended with a Cheng-Kac superalgebra, which was constructed several years later. However, Novikov superalgebras are not flexible enough to describe all simple superconformal Lie algebras. In this talk, we shall present the class of quasi-Poisson algebras. Quasi-Poisson algebras have two products: it is a commutative associative (super)algebra, a Lie (super)algebra, and has an additional unary operation, subject to certain axioms. All known simple superconformal Lie algebras arise from finite-dimensional simple quasi-Poisson superalgebras. In this talk, we shall present basic constructions, describe the examples of quasi-Poisson superalgebras, and mention some results about their representations.

quantum algebrarings and algebras

Audience: researchers in the topic

European Non-Associative Algebra Seminar