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Efim Zelmanov: Automorphism groups and Lie algebras of vector fields on affine varieties

26th May, 2022

Let V be an affine algebraic variety over a commutative ring K and let A be the K-algebra of regular (polynomial) functions on V.

The group of automorphisms of V, namely Aut(A), is, generally speaking, not linear. We will discuss the following two questions: which properties of linear groups extend to Aut(A), and which properties of finite-dimensional Lie algebras extend to the Lie algebra Der(A) of vector fields on V?

In particular, we will focus on analogues of classical theorems of Selberg, Burnside, and Schur for Aut(A) and an analogue of the Engel theorem for Der(A). In order to achive natural degree of generality and to include some interesting non-commutative cases we prove the theorems for PI-algebras.

Jef Laga: Arithmetic statistics and graded Lie algebras

17th March, 2022

I will explain how various results in arithmetic statistics by Bhargava, Gross, Shankar and others on 2-Selmer groups of Jacobians of (hyper)elliptic curves can be organised and reproved using the theory of graded Lie algebras, following earlier work of Thorne. This gives a uniform proof of these results and yields new theorems for certain families of non-hyperelliptic curves.

Alexander Stolin: 40 years of Lie bialgebras: From definition to classification

17th March, 2022

The history of Lie bialgebras began with the paper where the Lie bialgebras were defined: V. G. Drinfeld, "Hamiltonian structures on Lie groups, Lie bialgebras and the geometric meaning of the classical Yang-Baxter equations".

The aim of my talk is to celebrate 40 years of Lie bialgebras in mathematics and to explain how these important algebraic structures can be classified. This classification goes "hand in hand" with the classification of the so-called Manin triples and Drinfeld doubles also introduced in Drinfeld's paper cited above.

The ingenious idea how to classify Drinfeld doubles associated with Lie algebras possessing a root system is due to F. Montaner and E. Zelmanov. In particular, using their approach the speaker classified Lie bialgeras, Manin triples and Drinfeld doubles associated with a simple finite-dimensional Lie algebra 𝔤 (the paper was based on a private communication by E. Zelmanov and it was published in Comm. Alg. in 1999).

Further, in 2010, F. Montaner, E. Zelmanov and the speaker published a paper in Selecta Math., where they classified Drinfeld doubles on the Lie algebra of the formal Taylor power series 𝔤[[u]] and all Lie bialgebra structures on the polynomial Lie algebra 𝔤[u].

Finally, in March 2022 S. Maximov, E. Zelmanov and the speaker published an arXiv preprint, where they made a crucial progress towards a complete classification of Manin triples and Lie bialgebra structures on 𝔤[[u]].

Of course, it is impossible to compress a 40 years history of the subject in one talk but the speaker will try his best to do this.

David Galban: Cohomology and Representation Theory for Lie Superalgebras

14th March, 2022

This talk will consist of two parts. In the first, I will describe the cohomology groups for the subalgebra 𝔫+ relative to the BBW parabolic subalgebras constructed by D. Grantcharov, N. Grantcharov, Nakano and Wu, essentially with these calculations essentially providing the first steps towards an analogue of Kostant’s theorem for Lie superalgebras. In the second part, based on joint work with Nakano, I will analyze the sheaf cohomology groups RI indBG L𝔣(λ), where L𝔣(λ) is an irreducible representation for the detecting subalgebra 𝔣, providing analogues for the BBW theorem and Kempf’s vanishing theorem for sufficiently large λ.

Michel Racine: Lie Algebras afforded by Jordan algebras

3rd March, 2022

Given a (quadratic) Jordan algebra J over a ring k, one obtains three Lie algebras, the derivation algebra, the structure algebra, and the Tits algebra. We are particularly interested in the case where J is an Albert algebra.

Robert Spencer: (Some) Gram Determinants for An nets

1st March, 2022

The nets giving a diagrammatic description of the category of (tensor products of) fundamental representations of 𝔰𝔩n form a cellular category. We can then ask about the natural inner form on certain cell modules. In this talk, we will calculate the determinant of some of these forms in terms of certain traces of clasps or magic weave elements (for which there is a conjectured formula due to Elias). The method appears moderately general and gives a result which is hopefully illuminating and applicable to other monoidal, cellular categories.

Plamen Koshlukov: Gradings on upper triangular matrices

22nd February, 2022

The upper triangular matrix algebras are important in Linear Algebra, and represent a powerful tool in Ring Theory. They also appear in the theory of PI algebras.

In addition to the usual associative product, one can consider the Lie bracket and also the symmetric (Jordan) product on the upper triangular matrices.

We discuss the group gradings on the upper triangular matrices viewed as an associative, Lie and Jordan algebra, respectively. Valenti and Zaicev proved that the associative gradings are, in a sense, given by gradings on the matrix units. Di Vincenzo, Valenti and Koshlukov classified such gradings. Later on, Yukihide and Koshlukov, described the Lie and the Jordan gradings. In this talk we recall some of these results as well as a new development in a rather general setting, obtained by Yukihide and Koshlukov.

Joanna Meinel: Decompositions of tensor products: Highest weight vectors from branching

14th December, 2021

We consider tensor powers of the natural 𝔰𝔩n-representation, and we look for descriptions of highest weight vectors therein: We discuss explicit formulas for n=2, a recursion for n=3, and for bigger n we demonstrate how Jucys-Murphy elements allow us to compute highest weight vectors (both in theory and in practice using Sage).

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).

Marcelo Aguiar: Lie theory relative to a hyperplane arrangement

3rd December, 2021

A result due to Joyal, Klyachko, and Stanley relates free Lie algebras to partition lattices. We will discuss the precise relationship and interpret the result in terms of the braid hyperplane arrangement. We will then extend this result to arbitrary (finite, real, and central) hyperplane arrangements, and do the same with several additional aspects of classical Hopf-Lie theory. The Tits monoid of an arrangement, and the notion of lune, play central roles in the discussion.

Lisa Carbone: A Lie Group Analogue for the Monster Lie Algebra

29th November, 2021

The Monster Lie algebra 𝔪 is an infinite-dimensional Lie algebra constructed by Borcherds as part of his programme to solve the Conway-Norton Monstrous Moonshine Conjecture. We describe how one may approach the problem of associating a Lie group analogue for 𝔪 and we outline some constructions.

Arturo Pianzola: Derivations of twisted forms of Lie algebras

25th November, 2021

The main purpose of this talk is to explain how the theory of torsors can be used to study problems in infinite dimensional Lie theory. I will not assume that the audience is familiar with torsors. Definitions and examples will be given. The main application in this case is to provide a general framework (relative sheaves of Lie algebras) that explains/justifies a known result about the derivations of multiloop algebras.

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.

Oksana Bezushchak: Locally matrix algebras and algebras of Mackey

21st October, 2021

n this talk we will discuss:

1. Tensor decompositions of locally matrix algebras and their parametrization by Steinitz numbers.

2. Automorphisms and derivations of locally matrix algebras.

3. Automorphisms and derivations of Mackey algebras and Mackey groups. In particular, we describe automorphisms of all infinite simple finitary torsion groups (in the classification of J.Hall) and derivations of all infinite-dimensional simple finitary Lie algebras (in the classification of A.Baranov and H.Strade).

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.

Slava Futorny: Free Field Constructions for Affine Kac-Moody Algebras

15th October, 2021

Classical free field realizations of affine Kac-Moody algebras (introduced by M.Wakimoto, B.Feigin and E.Frenkel) play an important role in quantum field theory. B.Cox initiated the study of free field realizations for the non-standard Borel subalgebras which led to an important class of intermediate (or parabolic) Wakimoto modules. A uniform construction of such realizations will be discussed based on a joint work with L.Krizka and P.Somberg.

Maria Ofelia Ronco: Generalization of dendriform algebras

9th September, 2021

In a joint work with D. López N. and L.-F. Préville-Ratelle in 2015 we introduce a family of non-symmetric operads Dyckm, which satisfies that:

1. Dyck0 is the operad of associative algebras,

2. Dyck1 is the operad Dend of dendriform algebras, introduced by J.-L. Loday,

3. the vector space spanned by the set of m-Dyck paths has a natural structure of free Dyckm algebra over one element,

4. for any k ≥ 1, there exist degeneracy operators si : Dyckm → Dyckm-1 and face operators dj: Dyckm → Dyckm+1, which defines a simplicial complex in the category of non-symmetric operads.

The main examples of Dyckm algebra are the vector spaces spanned by the m-simplices of certain combinatorial Hopf algebras, like the Malvenuto-Reutenauer algebras and the algebra of packed words.

A well-known result on associative algebras states that, as an 𝒮-module, the operad of Ass of associative algebras is the composition Ass = Com ∘ Lie, where Com is the operad of commutative algebras and Lie is the operad of Lie algebras. The version of this result for dendriform algebras is that Dend = Ass ∘ Brace, where Brace is the operad of brace algebras.

Our goal is to introduce the notion of m-brace algebra, for m ≥ 2, and prove that there exists a Poincaré-Birkoff-Witt Theorem in this context, stating that Dyckm = Ass ∘ m-Brace.

Dmitry Leites: Classifications of simple Lie (super)algebras and algebras ‘more interesting’ than simple

2nd September, 2021

I intend to overview classifications of simple Lie (super)algebras of finite dimension and of polynomial growth. Various properties of complex Lie superalgebras resemble same of modular Lie algebras. I will encourage to consider these classifications without fanaticism: certain non-simple Lie (super)algebras, "close" to simple ones, are often "better" for us than simple ones.

Interesting features of deformations: semi-trivial deformations and (in super setting) odd parameters.

I'll formulate classification of finite-dimensional simple complex Lie superalgebras, odd parameters including.

I'll formulate a definition of Lie superalgebra suitable for any characteristic and classification of simple (finite-dimensional) Lie superalgebras over algebraically closed fields of characteristic 2. With a catch: modulo (a) classification of simple (finite-dimensional) Lie superalgebras (over the same field) and (b) classification of their gradings modulo 2. I'll mention conjectures on classification of modular Lie algebras and superalgebras.

Is it feasible to classify simple filtered Lie (super)algebras of polynomial growth? Interesting examples: deforms of the Poisson Lie (super)algebras, Lie (super)algebras of "matrices of complex size", etc.

Examples. Double extensions of simple Lie (super)algebras are definitely "more interesting" than the simple objects they extend.

Sergey Malev and Alexei Kanel-Belov: Evaluations of nonassociative polynomials on finite-dimensional algebras

5th August, 2021

Let p be a polynomial in several non-commuting variables with coefficients in an algebraically closed field K of arbitrary characteristic. It has been conjectured that for any n, for p multilinear, the image of p evaluated on the set Mn(K) of n by n matrices is either zero, or the set of scalar matrices, or the set sln(K) of matrices of trace 0, or all of Mn(K). In this talk we will discuss the generalization of this result for non-associative algebras such as Cayley-Dickson algebra (i.e. algebra of octonions), pure (scalar free) octonion Malcev algebra and basic low rank Jordan algebras.