Tag - Novikov algebra

Yuly Billig: Quasi-Poisson superalgebras

19th August, 2024

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 1-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.

Nurlan Ismailov: On the variety of right-symmetric algebras

1st July, 2024

The problem of the existence of a finite basis of identities for a variety of associative algebras over a field of characteristic zero was formulated by Specht in 1950. We say that a variety of algebras has the Specht property if any of its subvariety has a finite basis of identities. In 1988, A. Kemer proved that the variety of associative algebras over a field of characteristic zero has the Specht property. Specht’s problem has been studied for many well-known varieties of algebras, such as Lie algebras, alternative algebras, right-alternative algebras, and Novikov algebras. An algebra is called right-symmetric if it satisfies the identity (a,b,c) = (a,c,b) where (a,b,c) = (ab)c − a(bc) is the associator of a, b, c. The talk is devoted to the Specht problem for the variety of right-symmetric algebras. It is proved that the variety of right-symmetric algebras over an arbitrary field does not satisfy the Specht property.

Yvain Bruned: Novikov algebras and multi-indices in regularity structures

27th May, 2024

In this talk, we will present multi-Novikov algebras, a generalization of Novikov algebras with several binary operations indexed by a given set, and show that the multi-indices recently introduced in the context of singular stochastic partial differential equations can be interpreted as free multi-Novikov algebras. This is parallel to the fact that decorated rooted trees arising in the context of regularity structures are related to free multi-pre-Lie algebras.

Yanyong Hong: Novikov bialgebras, infinite-dimensional Lie bialgebras, and Lie conformal bialgebras

29th January, 2024

In this talk, I will introduce a bialgebra theory for the Novikov algebra, namely the Novikov bialgebra, which is characterized by the fact that its affinization (by a quadratic right Novikov algebra) gives an infinite-dimensional Lie bialgebra. A Novikov bialgebra is also characterized as a Manin triple of Novikov algebras. The notion of Novikov Yang-Baxter equation is introduced, whose skew-symmetric solutions can be used to produce Novikov bialgebras and hence Lie bialgebras. These solutions also give rise to skewsymmetric solutions of the classical Yang-Baxter equation in the infinite-dimensional Lie algebras from the Novikov algebras. Moreover, a similar connection between Novikov bialgebras and Lie conformal bialgebras will be introduced.