Student Algebra Seminar 2020

Speaker: Joel Gibson

Kac-Moody algebras are Lie algebras defined by generators and relations depending on a (generalised) Cartan matrix. By varying the Cartan matrix, the Kac-Moody construction produces all of the semisimple finite-dimensional Lie algebras, their affinisations, and even more exotic algebras, in a consistent and uniform way. The primary use of these algebras, for me, was to understand language and results for dealing with affine Lie algebras. However, there are many reasons you might want to study these:

• The generators-and-relations approach gives a very concrete way of dealing with the classical semisimple Lie algebras (the finite case), as well as consistent language which can be extended to the affine case.
• The Kac-Moody language is used ubiquitously in categorification, and the definition of quantum groups.
• In fact, the definition of quantum groups exactly parallels that of Kac-Moody algebras.

All of my examples will be in the finite and affine cases. The rough plan for three parts is as follows:

• Part I: The Lie algebra. (Scanned lecture notes) The guiding idea. Generalised Cartan matrices and realisations of root data. Definition and overview of the construction. The invariant bilinear form and Casimir operator. The Weyl group. Running examples for affine sl₂.
• Part II: The affine case. Classification of finite type, affine type, and twisted affine type algebras. The canonical central and scaling elements. Connection to the loop construction. Affine root systems in general.
• Part III: Representation theory. Category O, integrable modules, simple modules, Verma modules, the Weyl-Kac character formula. Some applications of Kac-Moody algebras.

References:

• The canonical reference: Infinite Dimensional Lie Algebras by Victor Kac.
• For a shorter and more condensed tour of the Weyl group and the representation theory, Chapters I, II, and XIII of Kac-Moody Groups, their Flag Varieties and Representation Theory by Shrawan Kumar is good. If you are looking to learn about Kac-Moody groups, this is the canonical reference.
• If you are interested in taking care to define things integrally in order to create a Kac-Moody-Group-Scheme, or wondering about interesting modifications of the Cartan, then An Introduction to Kac-Moody Groups over Fields by Timothée Marquis has what you're looking for.

Speaker: Joseph Baine

In this talk, we introduce the anti-spherical module for affine Hecke algebras, the canonical basis, and the p-canonical basis. We will explicitly compute these bases in type Ã₁. These bases have deep connections to the theory of tilting modules for algebraic groups.

• Lecture notes

Speaker: Giulian Wiggins

(Scanned lecture notes) We introduce the Ext functor and compute some examples coming from group theory and sheaf theory. At the very least we will compute all the Exts between simple abelian groups, and we’ll do at least one example of group cohomology. Some good sources for self-learning this (and related) material are Hilton and Stammbach’s ‘A course in homological algebra’ and D.J. Benson’s ‘Representations and cohomology’ series.

Speaker: Anna Romanov

A few weeks ago, Joe told us how to construct a module for the affine Hecke algebra by inducing the sign representation of the finite Hecke algebra. This was an example of an anti-spherical module, and Joe explained to us that it has deep significance in modular representation theory. It turns out that we can do the same construction for any parabolic subgroup of any Coxeter group, and the resulting anti-spherical modules have significance all over representation theory. In the case of finite Weyl groups, they showed up in my thesis as a combinatorial description of Whittaker modules, which are a class of representations of a complex semisimple Lie algebra which generalize Verma modules. In this talk, I'll tell you a bit about this story.

Notes

References:

• Soergel, Kazhdan-Lusztig polynomials and a combinatoric for tilting modules
• Kostant, On Whittaker vectors in representation theory
• McDowell, On modules induced from Whittaker modules
• Milicic-Soergel, The composition series of modules induced from Whittaker modues
• Romanov, A Kazhdan-Lusztig algorithm for Whittaker modules

Speaker: Gaston Burrull

Part I: Chamber systems and the Coxeter complex. (Lecture Notes)

The theory of buildings originally arose to provide a systematic procedure for the geometric interpretation of semisimple Lie groups. This technique was developed—but not started—mainly by Tits and evolved a lot during the following years. In this fifty minute talk, we will introduce the notion of a chamber system and the construction of its geometric realisation. This leads to a particular case of the construction of the Coxeter complex associated with an arbitrary finite rank Coxeter system (W,S). The Coxeter complex turns out to be fundamental to visualize a bunch of non-trivial properties of a Coxeter system, therefore it is indispensable to understand it in the greatest possible detail. The advantage of this approach is that it is direct, hence there is no need to use so much machinery as Tits representation, root systems, or so much work with dual vector spaces. This gives a geometric insight into fundamental domains, galleries, walls, reduced expressions, roots, foldings, minimal coset representatives, etc.

References:

• Mark Ronan, Lectures on Buildings, 1989.
• Anne Thomas, Geometric and topological aspects of Coxeter groups and buildings, 2018.

Speaker: Giulian Wiggins

Given a G-space X, a G-equivariant sheaf on X could informally be described as a sheaf with an action of G that is compatible with the G-action on X. We make this definition more precise and explain the sheaf-theoretic analogues of basic G-module notions such as induction/restriction and free actions.

Speaker: Kane Townsend

Notes

We introduce Lehrer and Springer's theory on reflection subquotients of complex reflection groups with a focus on examples from Weyl groups. The theory states that the normaliser of a maximal eigenspace modulo its centraliser leads to a new reflection group on this maximal eigenspace. Furthermore we will then consider how regular elements apply to this theory. To finish we will state how these reflection subquotients appear in the study of finite groups of Lie type when considering Sylow Φ-tori.

Speaker: Joseph Baine

Notes

Quantum groups can be regarded as one parameter deformations of universal enveloping algebras of Lie algebras. When the parameter is specialised to an l-th root of unity, there are various notions of the quantum group at a root of unity. In this talk I will give a motivated introduction to quantum groups and Lusztig’s quantum group at a root of unity. I will try to keep things relatively explicit, and emphasise the philosophy that when l=p, the quantum group behaves like a 1-step algebraic group in characteristic p.

Speaker: Gastón Burrull

Notes I

In this series of talks, I will motivate and define the universal enveloping algebra U(g) of a Lie algebra g over a field, as a quotient of vector spaces with the additional structure of an unital associative algebra defined over k. I will begin with the universal property of U(g) and the Poincaré–Birkhoff–Witt theorem Theorem (this theorem is highly non-trivial, hence I will not provide details for this). Those combined statements show that the theory of representations of the Lie algebra g is the same as the theory of representations of U(g). I will define the notion of filtered and graded algebras. I will provide some examples of this, as U(g) or algebras of differential operators. In later talks, I will possibly define Verma modules as quotients of U(g), give a picture for sl2 and sl3 Lie algebras, and the Harish-Chandra isomorphism. I will also explore the relation between U(g)-modules and D-modules under the Beĭlinson-Bernstein localization theorem.