Tag - Lie algebras

Dietrich Burde: Pre-Lie algebra structures on reductive Lie algebras and étale affine representations

27th February, 2023

Étale affine representations of Lie algebras and algebraic groups arise in the context of affine geometry on Lie groups, operad theory, deformation theory and Yang-Baxter equations. For reductive groups, every étale affine representation is equivalent to a linear representation and we obtain a special case of a prehomogeneous representation. Such representations have been classified by Sato and Kimura in some cases. The induced representation on the Lie algebra level gives rise to a pre-Lie algebra structure on the Lie algebra 𝔤 of G. For a Lie group G, a pre-Lie algebra structure on 𝔤 corresponds to a left-invariant affine structure on G. This refers to a well-known question by John Milnor from 1977 on the existence of complete left-invariant affine structures on solvable Lie groups. We present results on the existence of étale affine representations of reductive groups and Lie algebras and discuss a related conjecture of V. Popov concerning flattenable groups and linearizable subgroups of the affine Cremona group.

Kenji Iohara: On Elliptic Root Systems

20th February, 2023

In 1985, K. Saito introduced elliptic root systems as root systems belonging to a real vector space F equipped with a symmetric bilinear form I with signature (l, 2, 0). Such root systems are studied in view of simply elliptic singularities which are surface singularities with a regular elliptic curve in its resolution. K. Saito had classified elliptic root systems R with its one-dimensional subspace G of the radical of I, in the case when R/G ⊂ F/G is a reduced affine root system. In our joint work with A. Fialowski and Y. Saito, we have completed its classification; we classified the pair (R,G) whose quotient R/G ⊂ F/G is a non-reduced affine root system. In this talk, we give an overview of elliptic root systems and describe some of the new root systems we have found.

Karel Dekimpe: Di-semisimple Lie algebras and applications in post-Lie algebra structures

23rd January, 2023

We call a Lie algebra 𝔤 di-semisimple if it can be written as a vector space sum 𝔤 = 𝔰1 + 𝔰2, where 𝔰1 and 𝔰2 are semisimple subalgebras of 𝔤 and we say that 𝔤 is strongly di-semisimple if g can be written as a direct vector space sum of semisimple subalgebras. We will show that complex strongly di-semisimple Lie algebras have to be semisimple themselves. We will then use this result to show that if a pair of complex Lie algebras (𝔤,𝔫) with 𝔤 semisimple admits a so called post-Lie algebra structure, then 𝔫 must be isomorphic to 𝔤.

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.

Tekin Karadağ: Lie Structure on Hochschild and Hopf Algebra Cohomologies II

26th September, 2022

Murray Gerstenhaber constructed a graded Lie structure (Gerstenhaber bracket) on Hochschild cohomology, which makes Hochschild cohomology a Lie algebra. However, it is not easy to calculate bracket structure with the original definition. There is an alternative technique to compute Gerstenhaber bracket on Hochschild cohomology, introduced by Chris Negron and Sarah Witherspoon. It is also known that Hopf algebra cohomology has a bracket and the bracket is trivial when a Hopf algebra is quasi-triangular. We use a similar technique to the technique given by Negron and Witherspoon to calculate the Lie structure on Hochschild cohomology of the Taft algebra Tp for any integer p>2 which is a nonquasi-triangular Hopf algebra. Then, we find the corresponding bracket on Hopf algebra cohomology of Tp. We show that the bracket is indeed zero on Hopf algebra cohomology of Tp, as in all known quasi-triangular Hopf algebras. This example is the first known bracket computation for a nonquasi triangular algebra.

Tekin Karadağ: Lie Structure on Hochschild and Hopf Algebra Cohomologies I

19th September, 2022

Murray Gerstenhaber constructed Lie structure (Gerstenhaber bracket) on Hochschild cohomology, which makes Hochschild cohomology a graded Lie algebra. Later, it is shown that Hopf algebra cohomology also has a Lie structure. We will introduce a general formula for the bracket on Hopf algebra cohomology of any Hopf algebra with bijective antipode on the bar resolution that is reminiscent of Gerstenhaber’s original formula for Hochschild cohomology.

Victor Petrogradsky: Growth in Lie algebras

26th August, 2022

Different versions of the Burnside Problem ask what one can say about finitely generated periodic groups under additional assumptions. For associative algebras, Kurosh type problems ask similar questions about properties of finitely generated nil (more generally, algebraic) algebras. Similarly, one considers finitely generated restricted Lie algebras with a nil p-mapping. Now we study an oscillating intermediate growth in nil restricted Lie algebras.

Namely, for any field of positive characteristic, we construct a family of 3-generated restricted Lie algebras of intermediate oscillating growth. We call them Phoenix algebras, because of the following. a) For infinitely many periods of time the algebra is 'almost dying' by having a quasi-linear growth, namely the lower Gelfand-Kirillov dimension is 1, more precisely, the growth is of type n (ln ⋯ ln n)κ (ln q times), where q ∈ ℕ, κ > 0 are constants. b) On the other hand, for infinitely many n the growth function has a rather fast intermediate behaviour of type exp(n/(ln n)λ), λ being a constant determined by characteristic, for such periods the algebra is 'resuscitating'. c) Moreover, the growth function is bounded and oscillating between these two types of behaviour. d) These restricted Lie algebras have a nil p-mapping.

We also construct nil Lie superalgebras and nil Jordan superalgebras of similar oscillating intermediary growth over arbitrary field.

Jason Bell: Recent results on the Dixmier-Moeglin equivalence

12th August, 2022

Dixmier and Moeglin showed that if L is a finite-dimensional complex Lie algebra then the primitive ideals of the enveloping algebra U(L) are the prime ideals of Spec(U(L)) that are locally closed in the Zariski topology. In addition, they proved that a prime ideal P of U(L) is primitive if and only if the Goldie ring of quotients of U(L)/P has the property that its centre is just the base field of the complex numbers. Algebras that share this characterization of primitive ideals are said to satisfy the Dixmier-Moeglin equivalence. We give an overview of this property and mention some recent work on proving this equivalence holds for certain classes of twisted homogenous coordinate rings and classes of Hopf algebras of small Gelfand-Kirillov dimension.

Rob Muth: Superalgebra deformations of web categories

5th July, 2022

For a superalgebra A, and even subalgebra a, one may define an associated diagrammatic monoidal supercategory Web(A,a), which generalizes a number of symmetric web category constructions. In this talk, I will define and discuss Web(A,a)), focusing on two interesting applications: Firstly, Web(A,a) is equipped with an asymptotically faithful functor to the category of 𝔤𝔩n(A)-modules generated by symmetric powers of the natural module, and may be used to establish Howe dualities between 𝔤𝔩n(A) and 𝔤𝔩m(A) in some cases. Secondly, Web(A,a) yields a diagrammatic presentation for the ‘Schurification' TAa(n,d). For various choices of A/a, these Schurifications have proven connections to RoCK blocks of Hecke algebras, and conjectural connections to RoCK blocks of Schur algebras and Sergeev superalgebras.

Ting Xue: Introduction to linear algebraic groups

20th June, 2022

Algebraic groups are fundamental objects in representation theory and number theory. The course will discuss the structure theory of linear algebraic groups over algebraically closed fields. Topics include tori, parabolic subgroups and Borel subgroups, Lie algebras, root data, Weyl group, and classification of simple algebraic groups. If time permits, we will briefly discuss relation between compact Lie groups and algebraic groups.