Representation Theory Seminar: Hopf Algebra and Quantum Group

Representation Theory Seminar

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

2026 Term 3: Hopf Algebras and Representation Theory of Quantum Groups. See the weekly topics and full reading plan below.

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 reading, with discussion meetings after Weeks 2, 4, 6, 8, and 10
Term 3 Meeting Time	Please contact the organisers for meeting times.
Term 3 Location	Please contact the organisers for the meeting location.
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 3: Hopf Algebras and Representation Theory of Quantum Groups

This term, we study Hopf algebras and the representation theory of quantum groups through a ten-week reading programme. We develop the module and comodule viewpoints together, with particular attention to the tensor-categorical structures encoded by a Hopf algebra.

The first five weeks build the foundations: representation categories, tensor products and duality, fibre functors and reconstruction, classical examples, and highest-weight theory. The final five weeks focus on classical and quantum sl₂, quantum coordinate algebras, braiding and Drinfeld doubles, and representations at roots of unity, culminating in tilting modules, linkage, blocks, and translation functors.

The guiding question is:

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

Hopf Algebras, Tensor Categories, and Quantum Groups — Full Reading Plan (Google Docs)

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Term 3: Weekly Topics

The weekly topics below follow the ten-week reading plan. The plan includes five discussion meetings, after Weeks 2, 4, 6, 8, and 10.

Week	Topic
Week 1	Hopf Algebras and Their Rigid Representation Categories
Week 2	Representations of Hopf Algebras
Week 3	Fiber Functors, Reconstruction, and Affine Groups
Week 4	Fundamental Representation Categories and Dual Viewpoints
Week 5	Highest-Weight Phenomena through Examples
Week 6	Classical and Quantum sl₂ I: Algebras and Simple Modules
Week 7	Classical and Quantum sl₂ II: Tensor Products and the Coordinate Side
Week 8	Braiding, Drinfeld Doubles, and Centers
Week 9	Quantum Groups at Roots of Unity I: Integral Forms and Standard Modules
Week 10	Quantum Groups at Roots of Unity II: Tilting, Linkage, Blocks, and Translation

For detailed readings, references, guiding questions, and suggested exercises, see the Hopf Algebras, Tensor Categories, and Quantum Groups — Full Reading Plan (Google Docs).

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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

Hopf Algebras, Tensor Categories, and Quantum Groups — Full Reading Plan (Google Docs)

A First Glance at Lie Algebras and Affine Algebraic Groups (Lecture Notes PDF)

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 Algebras and Algebraic Groups	Past talks and notes
Term 3	Hopf Algebras and Representation Theory of Quantum Groups	Current theme

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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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Term 2 regular meetings: Wednesdays, 2:00 pm, Red Centre 3078.

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

Time and Location: Wednesday, 8 July 2026, 2:00 pm; Red Centre 3078

Abstract

This first talk is meant to set the conceptual language for the seminar. The guiding idea is that Lie algebras, affine algebraic groups, Lie groups, and derivations should not be introduced as isolated definitions, but as manifestations of algebraic theories, functorial geometry, and infinitesimal symmetry.

We begin with three Lawvere theories that will serve as basic examples throughout the talk: the Lawvere theory of Lie algebras, the Lawvere theory of rings, and the Lawvere theory of smooth rings. Lie algebras will be presented as models of an algebraic theory rather than merely as vector spaces equipped with brackets. The theory of rings, in particular associative algebras over a base field, then explains why the classical commutator construction is functorial. In this language, the familiar adjunction

$$ U:\mathrm{Lie}_k \rightleftarrows \mathrm{Alg}_k : (-)_{\mathrm{Lie}} $$

is a change-of-theory phenomenon: the right adjoint sends an associative algebra to its commutator Lie algebra, while the left adjoint is the universal enveloping algebra. The theory of smooth rings will later play the analogous role for smooth geometry.

We then motivate functorial algebraic geometry by asking a basic question: what is a polynomial function? Instead of regarding a polynomial only as a function on $k^n$, we regard it as something that can be evaluated naturally on $R^n$ for every commutative $k$-algebra $R$. This shifts the focus from a fixed set of classical points to a functor of points. From this viewpoint, affine schemes are representable functors on commutative $k$-algebras, and affine algebraic groups are representable group-valued functors, or equivalently commutative Hopf algebras.

The same idea has a smooth analogue. Using the Lawvere theory of smooth rings, we explain the viewpoint that smooth manifolds embed fully faithfully into a functor category. A manifold $M$ is sent to the functor

$$ M \longmapsto \left(R \longmapsto \operatorname{Hom}_{C^\infty\mathrm{Rng}} \bigl(C^\infty(M),R\bigr)\right), $$

where $R$ ranges over a suitable category of finitely generated smooth rings. Thus manifolds, like affine schemes, can be studied through their functors of points.

This functorial language lets us define the Lie algebra of a Lie group and the Lie algebra of an affine algebraic group in the same way. For a group-valued functor $G$, its Lie algebra is obtained from infinitesimal points at the identity. In the algebraic case this is expressed by

$$ \operatorname{Lie}(G) = \ker\bigl(G(k[\epsilon]/(\epsilon^2))\to G(k)\bigr), $$

and the smooth case has the analogous formula using the dual numbers as a smooth ring, or more generally as a Weil algebra. This recovers the usual tangent space at the identity of a Lie group and the usual Lie algebra of an affine algebraic group.

After this, we return to PROPs. If $\mathsf P$ is a linear PROP and $V$ is finite-dimensional, then the space of $\mathsf P$-model structures on $V$ is cut out by polynomial equations, hence is naturally an affine $k$-scheme. More generally, finite-dimensional models of $\mathsf P$ form a category enriched in affine $k$-schemes: for two models $A$ and $B$, the scheme $\underline{\operatorname{Hom}}_{\mathsf P}(A,B)$ represents linear maps $A\to B$ preserving all operations of the PROP. In particular, $\underline{\operatorname{Aut}}_{\mathsf P}(A)$ is an affine algebraic group.

Finally, we explain how this gives a uniform definition of derivations. An infinitesimal automorphism of a PROP-model has the form $\operatorname{id}+\epsilon D$. Requiring this map to preserve the PROP-structure linearizes the defining equations of the structure and produces the corresponding Leibniz rule. For associative algebras this is the ordinary derivation rule; for Lie algebras it is the Lie derivation rule; and the same mechanism applies to any finite-dimensional linear PROP-model. Thus derivations are not a family of unrelated definitions: they are Lie algebras of automorphism group schemes. Their closure under the commutator bracket is therefore a formal consequence of Lie theory.

The purpose of the talk is therefore to give the first map of the seminar: Lawvere theories explain algebraic syntax, functorial geometry explains spaces and groups of points, and affine group schemes explain infinitesimal symmetries and derivations.