Tag - Poisson algebras

Bojko Bakalov: An operadic approach to vertex algebras and Poisson vertex algebras

8th December, 2022

I will start by reviewing the notions of vertex algebra, Poisson vertex algebra, and Lie conformal algebra, and their relations to each other. Then I will present a unified approach to all these algebras as Lie algebras in certain pseudo-tensor categories, or equivalently, as morphisms from the Lie operad to certain operads. As an application, I will introduce a cohomology theory of vertex algebras similarly to Lie algebra cohomology, and will show how it relates to the cohomology of Poisson vertex algebras and of Lie conformal algebras.

Vladimir Sokolov: Non-Abelian Poisson brackets on projective spaces

9th December, 2021

We discuss non-abelian Poisson structures on affine and projective spaces over ℂ. We also construct a class of examples of non-abelian Poisson structures on ℂPn-1 for n ≥ 3. These non-abelian Poisson structures depend on a modular parameter τ ∈ ℂ and an additional discrete parameter k ∈ ℤ, where 1 ≤ k < n and k,n are coprime. The abelianization of these Poisson structures can be lifted to the quadratic elliptic Poisson algebras qn,k(τ).

Vladimir Bazhanov: Quantum geometry of 3-dimensional lattices

26th October, 2021

In this lecture I will explain a relationship between incidence theorems in elementary geometry and the theory of integrable systems, both classical and quantum. We will study geometric consistency relations between angles of 3-dimensional (3D) circular quadrilateral lattices, lattices whose faces are planar quadrilaterals inscribable into a circle. We show that these relations generate canonical transformations of a remarkable 'ultra-local' Poisson bracket algebra defined on discrete 2D surfaces consisting of circular quadrilaterals. Quantization of this structure allowed us to obtain new solutions of the tetrahedron equation (the 3D analogue of the Yang-Baxter equation) as well as reproduce all those that were previously known. These solutions generate an infinite number of non-trivial solutions of the Yang-Baxter equation and also define integrable 3D models of statistical mechanics and quantum field theory. The latter can be thought of as describing quantum fluctuations of lattice geometry.

Bojko Bakalov: On the Cohomology of Vertex Algebras and Poisson Vertex Algebras

17th October, 2021

Following Beilinson and Drinfeld, we describe vertex algebras as Lie algebras for a certain operad of n-ary chiral operations. This allows us to introduce the cohomology of a vertex algebra V as a
Lie algebra cohomology. When V is equipped with a good filtration, its associated graded is a Poisson vertex algebra. We relate the cohomology of V to the variational Poisson cohomology studied previously by De Sole and Kac.

Pavel Kolesnikov: Derived algebras and their identities

9th July, 2020

In this talk we will consider a "differential counterpart" of the dendriform splitting procedure for operads. This problem has a very natural interpretation in the language of non-associative algebras. It is well-known that a (non-associative, in general) algebra equipped with a Rota-Baxter operator (a formalization of integration) gives rise to a system in a class of splitting algebras. The latter include dendriform (pre-associative), pre-Lie (left-symmetric), pre-Poisson, Zinbiel (pre-commutative) algebras, etc. What happens if we replace a Rota-Baxter operator with a derivation? The answer is well known for associative commutative algebras: the resulting class of systems obtained in this way coincides with the variety Nov of Novikov algebras. We will show in general that for an arbitrary binary operad Var the variety of derived Var-algebras coincides with the Manin white product of operads Var and Nov. If we allow the initial multiplication(s) to leave in the language of a derived algebra then the same sort of description can be obtained just by replacement of Nov with GD!, the Koszul dual to the operad of Gelfand-Dorfman algebras. We will also discuss similar statements for the "integral" case of Rota-Baxter operators.

Yuly Billig: Towards Kac-van de Leur Conjecture

2nd July, 2020

Superconformal algebras are graded Lie superalgebras of growth 1, containing a Virasoro subalgebra. They play an important role in Conformal Field Theory. In 1988 Kac and van de Leur made a conjectural list of simple superconformal algebras, which since has been amended with an exceptional superalgebra CK(6). It has been proposed to use conformal superalgebras to attack this conjecture, and Fattori and Kac established a classification of finite simple conformal superalgebras. It still needs to be proved that one can associate a finite conformal superalgebra to each simple superconformal algebra. In this talk we will show how to use the results of Billig-Futorny to prove that every simple superconformal algebra is polynomial, which implies that one can attach to it an affine conformal superalgebra. We will discuss the difference between finite and affine conformal algebras. We also introduce quasi-Poisson algebras and show how to use them to construct known simple superconformal algebras. Quasi-Poisson algebras may be viewed as a refinement of the notion of Novikov algebras. Quasi-Poisson algebras may be used for computations of automorphisms and twisted forms of superconformal algebras.

João Schwarz: Poisson birational equivalence and Coloumb branches of 3d N=4 SUSY gauge theories

14th May, 2020

In this talk we discuss a notion of birational equivalence suitable for Poisson affine varieties: namely, that their function fields are isomorphic as Poisson fields. Some very interesting questions on non-commutative birational geometry, such as the Gelfand-Kirillov Conjecture, make perfect sense in the quasi-classical limit, and naturally leads one to consider the Poisson birational class of the algebras they quantize. In this setting, we study the behaviour of Poisson birational equivalence on the quasi-classical limit of rings of differential operators. With this idea we solve a Poisson analogue of Noether's Problem, introduced by Julie Baudry and François Dumas, in a constructive fashion, for essentially all finite symplectic reflection groups. As applications of our method, we show the Poisson rationality of the Generalized Calogero-Moser spaces, introduced by Etingof and Ginzburg in 2002, and surprisngly for this author, all Coloumb branches of 3d, N=4 SUSY gauge theories - an important object in mathematical physics recently given a rigorous formulation by Nakajima in 2015, and later Nakajima, Braverman, Finkelberg in 2016.

Ualbai Umirbaev: A Dixmier theorem for Poisson enveloping algebras

14th May, 2020

We consider a skew-symmetric n-ary bracket on the polynomial algebra K[x1, . . .,xn,xn+1] (n ≥ 2) over a field K of characteristic zero defined by {a1, . . .,an}=J(a1, . . .,an,C), where C is a fixed element of K[x1, . . .,xn,xn+1] and J is the Jacobian. If n = 2 then this bracket is a Poisson bracket and if n ≥ 3 then it is an n-Lie-Poisson bracket on K[x1, . . .,xn,xn+1]. We describe the centre of the corresponding n-Lie-Poisson algebra and show that the quotient algebra K[x1, . . .,xn,xn+1]/(C-λ), where (C-λ) is the ideal generated by (C-λ), 0 ≠ λ ∈ K, is a simple central n-Lie-Poisson algebra if C is a homogeneous polynomial that is not a proper power of any non-zero polynomial. This construction includes the quotients P(𝔰𝔩2(K))/(C-λ) of the Poisson enveloping algebra P(𝔰𝔩2(K)) of the simple Lie algebra 𝔰𝔩2(K), where C is the standard Casimir element of 𝔰𝔩2(K) in P(𝔰𝔩2(K)). It is also proven that the quotients P(𝕄)/(C-λ) of the Poisson enveloping algebra P(𝕄) of the exceptional simple 7-dimensional Malcev algebra 𝕄 are central simple.