Tag - Non-associative rings

Andy Jenkins: The nilpotent cone for Lie superalgebras

19th October, 2020

Many aspects of the representation theory of a Lie algebra and its associated algebraic group are governed by the geometry of their nilpotent cone. In this talk, we will introduce an analogue of the nilpotent cone N for Lie superalgebras and show that for a simple classical Lie superalgebra the number of nilpotent orbits is finite. We will also show that the commuting variety X described by Duflo and Serganova, which has applications in the study of the finite-dimensional representation theory of Lie superalgebras, is contained in N. Consequently, the finiteness result on N generalizes and extends the work on the commuting variety.

Beth Romano: Vinberg theory and related invariant theory

28th August, 2020

An LMS online lecture course in Vinberg theory.

In recent years, Vinberg theory of graded Lie algebras has become relevant in many areas of number theory, from arithmetic statistics (e.g., in the work of Romano-Thorne) to the local Langlands correspondence (e.g., in the work of Reeder-Yu). These lectures will provide the algebraic background for number theory students to engage with research involving graded Lie algebras. We'll start by discussing some of the relevant aspects of the invariant theory of Lie algebras, including the Chevalley restriction theorem and the pioneering work of Kostant on invariant rings. We'll then define graded Lie algebras and look at the graded analogues of these theorems, based on work of Vinberg. Time permitting, we'll look at Slodowy slices and applications to families of algebraic curves. These lectures should give number theory students sufficient background to read, for example, Thorne's paper Vinberg's representations and arithmetic invariant theory and other related papers. But the lectures will also be a useful introduction to some beautiful aspects of Lie theory for students in algebra and representation theory. I'll assume students have some knowledge of Lie algebras, but I will review relevant background and provide examples throughout the lectures.

Albert Schwarz: Some questions on Jordan algebras inspired by quantum theory

27th August, 2020

One can formulate quantum theory taking as a starting point a convex set (the set of states) or a convex cone (the set of non-normalized states.) Jordan algebras are closely related to homogeneous cones, therefore they appear naturally in this formulation. There exists a conjecture that superstring can be formulated in terms of exceptional Jordan algebras. In my purely mathematical talk I'll formulate some results and conjectures on Jordan algebras coming from these ideas.

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.

Tomoyuki Arakawa: Urod algebra and translation for W-algebras

9th July, 2020

In 2016 Bershtein, Feigin and Litvinov introduced the Urod algebra, which gives a representation-theoretic interpretation of the celebrated Nakajima-Yoshioka blowup equations in the case that the sheaves are of rank two. In this talk we will introduce higher-rank Urod algebras. This is done by constructing translation functors for affine W-algebras.

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.

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.