Tag - Lie algebras

Vera Serganova: The celebrated Jacobson-Morozov theorem for Lie superalgebras via semisimplification functor for tensor categories

11th June, 2020

The famous Jacobson-Morozov theorem claims that every nilpotent element of a semisimple Lie algebra 𝔤 can be embedded into an 𝔰𝔩2-triple inside 𝔤. Let 𝔤 be a Lie superalgebra with reductive even part and x be an odd element of 𝔤 with non-zero nilpotent [x,x]. We give necessary and sufficient condition when x can be embedded in 𝔬𝔰𝔭(1|2) inside 𝔤. The proof follows the approach of Etingof and Ostrik and involves semisimplification functor for tensor categories. Next, we will show that for every odd x in 𝔤 we can construct a symmetric monoidal functor between categories of representations of certain superalgebras. We discuss some properties of these functors and applications of them to representation theory of superalgebras with reductive even part. (Joint work with Inna Entova-Aizenbud).

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.

Markus Linckelmann: Hochschild cohomology and modular representation theory

10th August, 2016

Modular representation theory of finite groups seeks to understand, and possibly classify, the algebras - called block algebras of finite groups - which arise as indecomposable direct factors of finite group algebras over a complete local principal ideal domain with residue field of prime characteristic p. The expectation is that 'few' algebras should arise in this way, and that this should in turn lead to significant structural connections between finite groups and their block algebras.

The key feature of block algebras of finite groups is the dichotomy of invariants attached to these algebras.

On the one hand, they have all the typical algebra-theoretic invariants - module categories, their derived categories and stable categories, as well as numerical invariants such as the numbers of isomorphism classes of simple modules, and cohomologivcal invariants such as their Hochschild cohomology.

On the other hand, they have p-local invariants, due to their provenance from group algebras - reminiscent of the local structure of a finite group which includes its Sylow p-subgroups and its associated fusion systems.

Essentially all prominent conjectures which drive modular representation theory revolve around the interplay between these two types of invariants. We describe this interplay with a focus on Hochschild cohomology and analogous cohomology rings which are defined p-locally. This involves a variety of angles - Hochschild cohomology is graded commutative, hence methods and notions from commutative algebra will play a role. Hochschild cohomology in positive degree is also a Lie algebra. We will investigate connections between the algebra structure of block algebras and the Lie algebra structure of its first Hochschild cohomology space.

Andrei Okounkov: Weyl groups and their generalizations in enumerative geometry

15th March, 2016

These lectures will be about enumerative K-theory of curves (and more general 1-dimensional sheaves) in algebraic threefolds. In the first lecture, we will set up the enumerative problem and survey what we know and what we conjecture about it. In particular, we will meet the fundamental building blocks of the theory: threefolds fibered in ADE surfaces. In the second lecture, we will learn what geometric representation theory says about these building blocks, and, in particular, meet the present day incarnation of the Weyl group, which is really a fundamental groupoid of a certain periodic hyperplane arrangement, associated to a certain geometrically defined infinite-dimensional Lie algebra. This Weyl group completely determines the curve counts, and so seems like a very fitting topic for Hermann Weyl lectures. In the third lecture, I plan to introduce some of the geometric ideas that go into the actual technical construction of the theory.