Representations of Lie algebras – Page 2

Matthew Hamil: Homological residue fields and nilpotence theorems for Lie superalgebra representations

6th November, 2023

In the study of cohomology of finite group schemes it is well known that nilpotence theorems play a key role in determining the spectrum of the cohomology ring.

Balmer recently showed that there is a more general notion of a nilpotence theorem for tensor triangulated categories through the use of homological residue fields and the connection with the homological spectrum. The homological spectrum can be viewed as a topological space that realizes the Balmer spectrum in a concrete way.

Let 𝔤=𝔤₀ ⊕ 𝔤₁ be a Type I classical Lie superalgebra with an ample detecting subalgebra. In this talk, the speaker will consider the tensor triangular geometry for the stable category of finite-dimensional Lie superalgebra representations: stab(ℱ_{(𝔤,𝔤₀)}).

The localizing subcategories for the detecting subalgebra 𝔣 are classified which answers a question of Boe, Kujawa and Nakano. As a consequence of these results, the we prove a nilpotence theorem and determine the homological spectrum for the stable module category of ℱ_{(𝔣,𝔣₀)}.

The orbit structure of the reductive group G₀ on 𝔤₁ where Lie G₀=𝔤₀ along with the results for the detecting subalgebra is used to prove a nilpotence theorem for stab(ℱ_{(𝔤,𝔤₀)}) and to determine the homological spectrum in this case.

Thomas Creutzig: Tensor categories of modules of W-algebras

28th August, 2023

Let V be an affine vertex algebra of some simple Lie algebra 𝔤 and some level. Let KL be the category of V-modules whose conformal weight spaces are integrable 𝔤-modules. A famous result of Kazhdan and Lusztig tells us that for almost all levels KL is a braided tensor category and as such equivalent to a category of weight modules of the quantum group U_q(𝔤) of 𝔤 for suitable q.

It is desired to have similar results for suitable categories of W-algebras and superalgebras. In particular one wants to understand tensor structure and equivalences to quantum supergroups.

I will outline how to prove such statements and illustrate this in some examples.

Tiago Macedo: Finite-dimensional modules for map superalgebras

10th July, 2023

In this talk we will present recent results on the category of finite-dimensional modules for map superalgebras. Firstly, we will show a new description of certain irreducible modules. Secondly, we will use this new description to extract homological properties of the category of finite-dimensional modules for map superalgebras, most importantly, its block decomposition.

Kaiming Zhao: Simple smooth modules

29th May, 2023

Let L be a graded Lie algebra by integers with k-th homogenous space L_k where k are integers. An L-module V is called a smooth module if any vector in V can be annihilated by L_k for all sufficiently large k. Smooth modules for affine Kac-Moody algebras were introduced and studied by Kazhdan and Lusztig in 1993. I will show why this class of modules should be studied and what results are known now. An easy characterization for simple smooth modules for some Lie algebras will be provided.

Malihe Yousofzadeh: Finite Weight Modules over Affine Lie Superalgebras

22nd May, 2023

Nonzero real vectors of an affine Lie superalgebra act on a simple module either locally nilpotently or injectively. This helps us to divide simple finite weight modules over a twisted affine Lie superalgebra L into two subclasses called hybrid and tight. We will talk about the characterization as well as the classification problem of modules in each subclass. In this regard, the classification of bases of the root system of L is crucial. We will discuss how we can classify the bases and how we can use the obtained classification to study simple finite weight modules over L.

Lewis Topley: Modular representation theory and finite W-algebras

15th May, 2023

Finite W-algebras were introduced by Premet in full generality, and they quickly became quite famous for their many applications in the representation theory of complex semisimple Lie algebras, especially the classification of primitive ideals. However, these algebras first appeared in the representation theory of Lie algebras associated to reductive groups in positive characteristic. In this talk I will survey the history of finite W-algebras in modular representation theory, and explain some of the contributions I have made to the field. The main applications in this talk will be the construction and classification of 'small' modules of Lie algebras.

Nicholas Davidson: Superalgebra deformations of webs

13th May, 2023

Webs are certain diagrams used to represent homomorphisms between tensor products of representations for various Lie (super)algebras. These diagrams can be assembled into a monoidal 'web category', and typically there is a full functor from the web category into the category of representations of the associated Lie superalgebra. In this talk, I will discuss recent work that deforms web categories by decorating the diagrams with elements of some superalgebra A. These 'decorated' webs generalize several constructions that have previously appeared in the literature, including webs for the Lie algebra 𝔤𝔩_n(ℂ), the Lie superalgebra 𝔮(n). The webs also give a diagrammatic presentation for the so-called Schurification of A.

Matthew Hamil: Stratifying rings for the stable category of modules over detecting Lie subalgebras

12th May, 2023

Tensor triangulated categories arise across many areas in mathematics. Examples of tensor triangulated categories include the derived category of perfect complexes of a suitably nice scheme X, the stable category of kG-modules where G is a finite group and k is a field of characteristic dividing the order of G, and many more. In his 2005 paper titled The spectrum of prime ideals in tensor triangulated categories, Paul Balmer associates a topological space, now called the Balmer spectrum, to tensor triangulated categories in a manner analogous to the spectrum of a commutative ring.

In the early 2000s Boe, Kujawa, and Nakano published a series of papers studying Lie superalgebras. They show that for classical Lie superalgebras 𝔤 admitting stable and polar actions of the algebraic group G₀, there exist interesting subalgebras 𝔣 which 'detect' the cohomology of 𝔤 relative to its even part. They also consider the stable category of 𝔤 modules which are completely reducible over 𝔤₀. This is a TTC, and in special cases, including for the detecting subalgebras and the Lie superalgebra 𝔤𝔩(m|n), they give a concrete description of the Balmer spectrum using methods from geometric invariant theory. In our talk we discuss how to recover this result for the detecting subalgebras using a different approach. Namely, we will discuss some circumstances in which we have a stratifying ring.

Mátyás Domokos: An application of classical invariant theory to the study of identities and concomitants of irreducible representations of the simple 3-dimensional complex Lie algebra

8th May, 2023

To an n-dimensional representation of a finite dimensional Lie algebra one can naturally associate an algebra of equivariant polynomial maps from the space of m-tuples of elements of the Lie algebra into the space of n-by-n matrices. In the talk, we mainly deal with the special case of irreducible representations of the simple 3-dimensional complex Lie algebra, and discuss results on the generators of the corresponding associative algebra of concomitants as well as results on the quantitative behaviour of the identities of these representations.

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.

Tag - Representations of Lie algebras