Tag - Poisson algebras

David Fernández: Non-commutative Poisson geometry and pre-Calabi-Yau algebras

2nd December, 2024

In order to define suitable non-commutative Poisson structures, M. Van den Bergh introduced double Poisson algebras and double quasi-Poisson algebras. Furthermore, N. Iyudu and M. Kontsevich found an insightful correspondence between double Poisson algebras and pre-Calabi-Yau algebras; certain cyclic A∞-algebras which can be seen as non-commutative versions of shifted Poisson manifolds. In this talk, I will present an extension of the Iyudu-Kontsevich correspondence to the differential graded setting. I will also explain how double quasi-Poisson algebras give rise to pre-Calabi-Yau algebras.

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.

Jason Gaddis: Rigidity of quadratic Poisson algebras

18th December, 2023

The Shephard-Todd-Chevalley Theorem gives conditions for the invariant ring of a polynomial ring to again be polynomial. However, this behaviour is rarely observed for non-commutative algebras. For example, the invariant ring of the first Weyl algebra by a finite group is not isomorphic to the first Weyl algebra. In this talk, I will discuss this rigidity in the context of quadratic Poisson algebras. A key example will be those Poisson polynomial algebras with skew-symmetric structure.

Amir Fernández Ouaridi: On the simple transposed Poisson algebras and Jordan superalgebras

4th December, 2023

We prove that a transposed Poisson algebra is simple if and only if its associated Lie bracket is simple. Consequently, any simple finite-dimensional transposed Poisson algebra over an algebraically closed field of characteristic zero is trivial. Similar results are obtained for transposed Poisson superalgebras. An example of a non-trivial simple finite-dimensional transposed Poisson algebra is constructed by studying the transposed Poisson structures on the modular Witt algebra. Furthermore, we show that the Kantor double of a transposed Poisson algebra is a Jordan superalgebra, that is, we prove that transposed Poisson algebras are Jordan brackets. Additionally, a simplicity criterion for the Kantor double of a transposed Poisson algebra is obtained.

Mykola Khrypchenko: Transposed Poisson structures

4th September, 2023

A transposed Poisson algebra is a triple (L,⋅,[⋅,⋅]) consisting of a vector space L with two bilinear operations ⋅ and [⋅,⋅], such that (L,⋅) is a commutative associative algebra; (L,[⋅,⋅]) is a Lie algebra; and the 'transposed' Leibniz law holds: 2z⋅[x,y]=[z⋅x,y]+[x,z⋅y] for all x,y,z∈L. A transposed Poisson algebra structure on a Lie algebra (L,[⋅,⋅]) is a (commutative associative) multiplication ⋅ on L such that (L,⋅,[⋅,⋅]) is a transposed Poisson algebra. I will give an overview of my recent results in collaboration with Ivan Kaygorodov (Universidade da Beira Interior) on the classification of transposed Poisson structures on several classes of Lie algebras.

Anne Moreau: Functorial constructions of double Poisson vertex algebras

30th August, 2023

To any double Poisson algebra we produce a double Poisson vertex algebra using the jet algebra construction. We show that this construction is compatible with the representation functor which associates to any double Poisson (vertex) algebra and any positive integer a Poisson (vertex) algebra. We also consider related constructions, such as Poisson reductions and Hamiltonian reductions. This allows us to provide various interesting examples of double Poisson vertex algebras, in particular from double quivers.

Şehmus Fındık: Symmetric polynomials in some certain noncommutative algebras

31st July, 2023

Let F be a finitely generated free algebra in a variety of algebras over a field of characteristic zero. A polynomial in F is called symmetric, if it is preserved under any permutation of the generators. The set S(F) of symmetric polynomials is a subalgebra of F. In this talk, we examine the algebras S(F), where F is the free metabelian associative, Lie, Leibniz, Poisson algebra or the free algebra generated by generic traceless matrices or the free algebra in the variety generated by Grassmann algebras.

Marion Boucrot: The relation between A∞-morphisms and pre-Calabi-Yau morphisms

2nd May, 2023

Pre-Calabi-Yau algebras were introduced in the last decade by  M. Kontsevich, A. Takeda and Y. Vlassopoulos using the necklace bracket. This notion is equivalent to a cyclic A∞-algebra for the natural bilinear form in the finite-dimensional case. Moreover, W-K. Yeung showed that double Poisson DG structures provide an example of pre-Calabi-Yau structures. In 2020, D. Fernandez and E. Herscovich proved that given a morphism of double Poisson DG algebras from A to B, one can produce a cyclic A∞-algebra and A∞-morphisms between the latter and the cyclic A∞-algebras associated to A and B. I will explain how to generalize this result to pre-Calabi-Yau algebras by doing an explicit construction of a (cyclic) A∞-algebra and A∞-morphisms given a pre-Calabi-Yau morphism.

Vladimir Dotsenko: Operad filtrations and quantization

24th April, 2023

The celebrated problem of deformation quantization discusses deformations of Poisson algebras into associative algebras, a question that is, in the end, motivated by quantum mechanics. I shall discuss this question and some of its generalisations from the purely algebraic point of view using the theory of operads. In particular, I shall show how to prove that there are, in a strict mathematical sense, only two meaningful deformation problems for Poisson algebras, namely deforming them in the class of all Poisson algebras or all associative algebras, and there is only one meaningful deformation problem for the so called almost Poisson algebras (also sometimes known as generic Poisson algebras), namely deforming them in the class of all almost Poisson algebras. For instance, this explains the existing body of work in the mathematical physics literature asserting that some classes of non-associative star products cannot be alternative, are always flexible etc.

Maxime Fairon: Around Van den Bergh’s double brackets

10th April, 2023

The notion of a double Poisson bracket on an associative algebra was introduced by M. Van den Bergh in order to induce a (usual) Poisson bracket on the representation spaces of this algebra. I will start by reviewing the basics of this theory and its relation to other interesting operations, such as Leibniz brackets and H0-Poisson structures. I will then explain some recent results and generalisations related to double Poisson brackets.