Representation Theory Seminar

A student-run seminar on representation theory, Lie theory, algebraic groups and quantum groups, tensor categories, and related structures.

About

The Representation Theory Seminar is an informal reading and discussion seminar devoted to modern representation theory and its surrounding geometry, algebra, and category theory.

The seminar is intended for students who want to build a working understanding of representation theory through talks, reading sessions, problem discussions, and expository notes.

Our emphasis is on:

• Lie algebras and algebraic groups;

• homological methods in representation theory;

• tensor categories and monoidal categories;

• quantum groups and tilting modules;

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Seminar Information

Item	Details
Format	Student talks, reading sessions, problem discussions
Frequency	Weekly
Location	UNSW / online / hybrid
Organiser	Yuze Zheng/Pengyu Jia
Audience	Abstract Math Lover?
Prerequisites	Some AG and Commutative Algebra/Module/Category Theory/Monoidal Category Theory

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Current Theme

2026 Term 2: Lie Algebras and Algebraic Groups and Some Common Sense in Representation Theory

The guiding question is:

What structures are preserved when we pass from algebraic objects to their categories of representations?

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Schedule

A First Glance at Lie Algebras and (Affine) Algebraic Groups by Yuze Zheng (Marco)

Time and Location: TBD

Abstract

This first seminar is intended as a conceptual opening rather than a technical lecture. The goal is to set up several viewpoints that will reappear throughout the seminar: algebraic theories, PROPs, functorial algebraic geometry, affine algebraic groups, and the Lie-theoretic idea of infinitesimal symmetry.

We begin with Lie algebras from the perspective of Lawvere theories and PROPs. Instead of treating a Lie algebra merely as a vector space equipped with a bracket, we regard it as a model of an algebraic theory. This perspective explains why many familiar constructions are not accidents. In particular, the classical adjunction

$$U:{\mathrm{Lie}}_k\rightleftarrows {\mathrm{Alg}}_k[-,-]:\mathrm{Lie}$$

between Lie algebras and associative algebras becomes conceptually natural. It comes from the morphism between Lawvere Theories.

We then turn to affine algebraic groups. Rather than starting from the classical picture of polynomial equations in affine space, we motivate the functor-of-points viewpoint. From this perspective, an affine algebraic group is a group-valued functor represented by an a commutative Hopf algebra. This language makes base change, families, infinitesimal points, and algebraic symmetries much more transparent than the purely classical viewpoint.

After introducing the basic examples of Lie algebras and algebraic groups, we discuss a more structural source of examples. If a finite-dimensional vector space carries the structure of a model of a linear PROP, then its automorphism functor is naturally an affine group scheme. Thus algebraic structures give rise to algebraic groups of symmetries.

Finally, we explain how the Lie algebra of this automorphism group gives a uniform definition of derivations of a PROP-model. In the familiar cases, this recovers the usual Leibniz rules for associative algebras, Lie algebras, and related structures. But the construction is more general: for any linear PROP-model, derivations arise as infinitesimal automorphisms. Consequently, their closure under commutators is not a separate calculation, but a formal consequence of the fact that they form the Lie algebra of an affine group scheme.

The seminar will therefore use Lie algebras and algebraic groups not only as objects of study, but also as a language for understanding algebraic structures, their symmetries, and their infinitesimal deformations.

Reading List

Primary References

1. Lie Algebras, Algebraic Groups, and Lie Groups by J.S. Milne

2. A Tour of Representation Theory by Martin Lorenz

3. A brief introduction to quantum groups by Pavel Etingof, Mykola Semenyakin

4. DIAGRAM CATEGORIES FOR Uq-TILTING MODULES AT ROOTS OF UNITY by HENNING HAAHR ANDERSEN AND DANIEL TUBBENHAUER

Supplementary References

1. Pavel Etingof et al., Tensor Categories.

2. Weibel, An Introduction to Homological Algebra.

3. Mac Lane, Categories for the Working Mathematician.

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Notes

Topics in HOMOLOGICAL ALGEBRA

Lie Algebra, Algebraic Group, Lie Group

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Contact

For questions, suggestions, or talk proposals, contact:

Yuze Zheng UNSW School of Mathematics and Statistics

Email: yuze.zheng@student.unsw.edu.au

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Archive

2026

Term	Theme	Page
Term 2	Lie Algebra and Algebraic Group	Current page
Term 3	TBA	TBA