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Brian Boe: Complexity and Support Varieties for Type P Lie Superalgebras

5th April, 2021

We compute the complexity, z-complexity, and support varieties of the (thick) Kac modules for the Lie superalgebras of type P. We also show the complexity and the z-complexity have geometric interpretations in terms of support and associated varieties; these results are in agreement with formulas previously discovered for other classes of Lie superalgebras. Our main technical tool is a recursive algorithm for constructing projective resolutions for the Kac modules. The indecomposable projective summands which appear in a given degree of the resolution are explicitly described using the combinatorics of weight diagrams. Surprisingly, the number of indecomposable summands in each degree can be computed exactly: we give an explicit formula for the corresponding generating function. I wrote an iOS app to implement the combinatorics quickly and graphically, and I’ll be demoing live some of the interesting features of these resolutions.

Daniel Nakano: A new Lie theory for simple classical Lie superalgebras

15th March, 2021

In 1977, Kac classified simple Lie superalgebras over ℂ and showed they play an analogous role to simple Lie algebras over the complex numbers. For simple algebraic groups and their Lie algebras, the notions of a maximal torus, Borel subgroups and the Weyl groups provide a uniform method to treat the structure and representation theory for these groups and Lie algebras. Historically, much of the work for simple Lie superalgebras has involved dealing with these objects using a case by case analysis.

Fifteen years ago, Boe, Kujawa and the speaker introduced the important concept of detecting subalgebras for classical Lie superalgebras. These algebras were constructed by using ideas from geometric invariant theory. More recently, D. Grantcharov, N. Grantcharov, Wu and the speaker introduced the BBW parabolic subalgebras. Given a Lie superalgebra 𝔤, one has a triangular decomposition 𝔤=𝔫- ⨁ 𝔣 ⨁ 𝔫+ with 𝔟=𝔣 ⨁ 𝔫- where 𝔣 is a detecting subalgebra and 𝔟 is a BBW parabolic subalgebra. This holds for all classical 'simple' Lie superalgebras, and one can view 𝔣 as an analogue of the maximal torus, and 𝔟 like a Borel subalgebra. This setting also provide a useful method to define semisimple elements and nilpotent elements, and to compute various sheaf cohomology groups R• indBG (-).

The goal of my talk is to provide a survey of the main ideas of this new theory and to give indications of the interconnections within the various parts of this topic. I will also indicate how our ideas can further unify the study of the representation theory of classical Lie superalgebras.

David Galban: First and Second Cohomology Groups for BBW Parabolics for Lie Superalgebras

1st March, 2021

For semisimple Lie algebras, a well-known theorem of Kostant computes the cohomology groups of parabolic subalgebras, but it is unknown whether an analogue of Kostant’s theorem exists for Lie superalgebras. Seeking to provide the first calculations in this direction, in this talk, 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. These classical Lie superalgebras have a triangular decomposition 𝔤 = 𝔫- + 𝔣 + 𝔫+, where 𝔣 is a detecting subalgebra as introduced by Boe, Kujawa and Nakano. I will show that there exists a Hochschild-Serre spectral sequence that collapses for all infinite families of classical simple Lie superalgebras. Using this, I will provide examples of computation of the first and second cohomologies for various 𝔫+.

Nick Davidson: Type P Webs and Howe Duality

16th February, 2021

Webs are combinatorially defined diagrams which encode homomorphisms between tensor products of certain representations of Lie (super)algebras. I will describe some recent work with Jon Kujawa and Rob Muth which defines webs for the type P Lie superalgebra, and then uses these webs to deduce an analogue of Howe duality for this Lie superalgebra.

Daniel Nakano: A new Lie theory for classical Lie superalgebras

4th December, 2020

In 1977, Kac classified simple Lie superalgebras over ℂ and showed they play an analogous role to simple Lie algebras over the complex numbers. For simple algebraic groups and their Lie algebras, the notions of a maximal torus, Borel subgroups and the Weyl groups provide a uniform method to treat the structure and representation theory for these groups and Lie algebras. Historically, much of the work for simple Lie superalgebras has involved dealing with these objects using a case by case analysis. Fifteen years ago, Boe, Kujawa and the speaker introduced the concept of detecting subalgebras for classical Lie superalgebras. These algebras were constructed by using ideas from geometric invariant theory. More recently, D. Grantcharov, N. Grantcharov, Wu and the speaker introduced the concept of a BBW parabolic subalgebra.

Given a Lie superalgebra 𝔤, one has a triangular decomposition 𝔤=𝔫– ⨁ 𝔣 ⨁ 𝔫+ with 𝔟 = 𝔣 ⨁ 𝔫– where 𝔣 is a detecting subalgebra and 𝔟 is a BBW parabolic subalgebra. This holds for all classical ‘simple’ Lie superalgebras, and one can view 𝔣 as an analogue of the maximal torus, and 𝔟 like a Borel subalgebra. This setting also provide a useful method to define semisimple elements and nilpotent elements, and to compute various sheaf cohomology groups R ∙ indBG (-). The goal of my talk is to provide a survey of the main ideas of this new theory and to give indications of the interconnections within the various parts of this topic. I will also indicate how this treatment can further unify the study of the representation theory of classical Lie superalgebras.

This video was produced by the Universidade de São Paulo, as part of the LieJor Online Seminar: Algebras, Representations, and Applications.

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.

Arun Ram: Integrable modules for affine Lie algebras

5th October, 2020

These modules naturally divide themselves into three categories: positive level, negative level and level 0. The positive level modules are highest weight, the negative level ones are lowest weight, and the level 0 ones are neither. But all three classes of modules have some nice character formulas, a good crystal theory in the sense of Kashiwara-Lusztig-Littelmann, and Borel-Weil-Bott type geometric constructions. The geometric constructions use, respectively, the thin affine flag variety (for positive level), the thick affine flag variety (for negative level), and the semi-infinite flag variety (for level 0).

Alexander Molev: Symmetrization map, Casimir elements and Sugawara operators

2nd October, 2020

The canonical symmetrization map is a 𝔤-module isomorphism between the symmetric algebra S(𝔤) of a finite-dimensional Lie algebra 𝔤 and its universal enveloping algebra U(𝔤). This implies that the images of 𝔤-invariants in S(𝔤) are Casimir elements. For each simple Lie algebra 𝔤 of classical type we consider basic 𝔤-invariants arising from the characteristic polynomial of the matrix of generators. We calculate the Harish-Chandra images of the corresponding Casimir elements. By using counterparts of the symmetric algebra invariants for the associated affine Kac-Moody algebras we obtain new formulas for generators of the centres of the affine vertex algebras at the critical level. Their Harish-Chandra images are elements of classical W-algebras which we produce in an explicit form.

Alexander Premet: Modular representations of Lie algebras and Humphreys’s Conjecture

25th September, 2020

Let G be a connected reductive algebraic group defined over an algebraically closed field of positive characteristic p and suppose that the derived subgroup of G is simply connected, p is a good prime for the root system of G and the Lie algebra 𝔤=Lie(G) admits a non-degenerate Ad G-invariant symmetric bilinear form. If G is a simple algebraic group of type other than A, the above assumptions mean that p is a good prime for G, i.e. p ≥ 3 if G is of type B, C or D, p ≥ 5 if G is of type G2, F4, E6 or E7, and p ≥ 7 if G is of type E8. If all components of G have type A, B, C, D we set R=ℤ[1/2].  If G has a component of exceptional type but has no components of type E8 we set R=ℤ[1/6]. If G has a component of type E8 we set R=ℤ[1/30]. Given a linear function χ on 𝔤 we denote by Uχ(𝔤) the reduced enveloping algebra of 𝔤 associated with χ. By the Kac-Weisfeiler conjecture (now a theorem), any Uχ(𝔤)-module has dimension divisible by pd(χ) where 2d(χ) is the dimension of the coadjoint G-orbit of χ. In my talk, based on a joint work with Lewis Topley, I'll discuss a natural question raised in the 1990s by Kac, Humphreys and myself and explain that for any χ ∈ 𝔤* the reduced enveloping algebra Uχ(𝔤) has an irreducible module of dimension pd(χ). Forms of finite W-algebras over the ring R and their reductions modulo good primes play a crucial role in our arguments. We also use some recent results on multiplicty-free primitive ideals of U(𝔤𝒞) associated with the rigid nilpotent orbits in complex simple Lie algebras 𝔤𝒞.

Dražen Adamović: On logarithmic and Whittaker modules for affine vertex algebras

6th August, 2020

Simple affine vertex algebras at admissible levels are semi-simple in the category O, but beyond the category O they contain interesting categories of representations with many new research challenges. We will first present our explicit lattice realizations of simple affine VOA Lk(𝔰𝔩2) at arbitrary admissible level k, and their modules in certain categories. Then we discuss the existence and explicit realization of logarithmic modules which appear as extensions of weight modules. The next natural task is to include Whittaker modules in the representation category. Although Whittaker modules are constructed using standard Lie-theoretic constructions, we will show that in order to understand the structure of affine Whittaker modules, one needs to apply vertex-algebraic techniques. We present explicit realization of Whittaker modules for some vertex algebras. We will discuss our recent efforts to generalize this realization in higher-rank cases.