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

The Infinitesimal Commutator: Why Derivations of PROP-Models Form a Lie Algebra

The Infinitesimal CommutatorWhy derivations of PROP-models form a Lie algebra1. PROP-models and automorphisms2. Derivations as tangent vectors3. Addition comes from first-order multiplication4. The infinitesimal square5. The two kernels6. The area layer is the tangent space again7. The commutator lands in the area layer

 

The Infinitesimal Commutator

Why derivations of PROP-models form a Lie algebra

The aim of this note is to prove the following fact in a way that avoids the usual calculation.

Let $\mathsf{P}$$\mathsf P$ be a $k$$k$-linear PROP, and let $A$$A$ be a model of $\mathsf{P}$$\mathsf P$. Then the derivations of $A$$A$ are closed under the commutator bracket

$$[D,E]=DE-ED.$$

Equivalently,

$${\mathrm{Der}}_{\mathsf{P}}(A)$$

is a Lie subalgebra of ${\mathrm{End}}_k(A)$$\operatorname{End}_k(A)$.

For associative algebras this is familiar. One checks directly that if $D$$D$ and $E$$E$ satisfy the Leibniz rule, then so does $DE-ED$$DE-ED$. But that calculation is not the reason. It is only the shadow of a more structural fact:

$$\text{the commutator of derivations is the area term of a group commutator.}$$

More precisely, derivations are tangent vectors to an automorphism group functor. Addition of derivations comes from first-order multiplication of infinitesimal automorphisms. The bracket of derivations comes from the group commutator of two infinitesimal automorphisms placed in two independent infinitesimal directions.

The proof is then almost formal.

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1. PROP-models and automorphisms

Let $k$$k$ be a commutative base ring, and let $\mathsf{P}$$\mathsf P$ be a one-coloured $k$$k$-linear PROP.

A model of $\mathsf{P}$$\mathsf P$ in $k$$k$-modules is a $k$$k$-module $A$$A$ together with structure maps

$$\rho (p):A^{\otimes m}\to A^{\otimes n}$$

for every operation

$$p:m\to n$$

in $\mathsf{P}$$\mathsf P$, compatible with identities, composition, tensor product, and the symmetric group actions.

Equivalently, $A$$A$ gives a strict symmetric monoidal $k$$k$-linear functor

$$\rho :\mathsf{P}\to {\mathrm{End}}_A,$$

where

$${\mathrm{End}}_A(m,n)={\mathrm{Hom}}_k(A^{\otimes m},A^{\otimes n}).$$

For every commutative $k$$k$-algebra $R$$R$, put

$$A_R=A{\otimes }_{k}R.$$

The structure maps extend $R$$R$-linearly to

$${\rho }_R(p):A_R^{{\otimes }_{R}m}\to A_R^{{\otimes }_{R}n}.$$

Define the automorphism functor of the PROP-model $A$$A$ by

$$G_A(R)={\mathrm{Aut}}_{\mathsf{P},R}(A_R).$$

Thus an element

$$g\in G_A(R)$$

is an $R$$R$-linear automorphism

$$g:A_R\to A_R$$

such that, for every operation $p:m\to n$$p:m\to n$,

$$g^{\otimes n}\circ {\rho }_R(p)={\rho }_R(p)\circ g^{\otimes m}.$$

Therefore

$$G_A:{\mathrm{CAlg}}_k\to \mathrm{Grp}$$

is a group-valued functor.

No representability is assumed. The argument only uses the functor of points.

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2. Derivations as tangent vectors

Consider the dual numbers

$$k[ϵ]/({ϵ}^2).$$

There is a quotient map

$$k[ϵ]/({ϵ}^2)\to k,ϵ\mapsto 0.$$

Applying $G_A$$G_A$ gives a group homomorphism

$$G_A(k[ϵ]/({ϵ}^2))\to G_A(k).$$

Define the tangent space at the identity by

$$\mathrm{Lie}(G_A)=\ker (G_A(k[ϵ]/({ϵ}^2))\to G_A(k)).$$

An element of this kernel is an infinitesimal automorphism reducing to the identity modulo $ϵ$$\epsilon$.

Since

$$A{\otimes }_{k}k[ϵ]/({ϵ}^2)\cong A\oplus ϵA,$$

such an automorphism is necessarily of the form

$$g=\mathrm{id}+ϵD$$

for some $k$$k$-linear endomorphism

$$D:A\to A.$$

Its inverse is

$$g^{-1}=\mathrm{id}-ϵD,$$

because

$${ϵ}^2=0.$$

Now impose the condition that $g$$g$ preserves the PROP-structure. For every operation $p:m\to n$$p:m\to n$, we require

$$g^{\otimes n}\circ \rho (p)=\rho (p)\circ g^{\otimes m}$$

over $k[ϵ]/({ϵ}^2)$$k[\epsilon]/(\epsilon^2)$.

Expand

$$g^{\otimes r}=(\mathrm{id}+ϵD{)}^{\otimes r}.$$

Since ${ϵ}^2=0$$\epsilon^2=0$, at most one copy of $D$$D$ can appear. Thus

$$g^{\otimes r}=\mathrm{id}+ϵD^{(r)},$$

where

$$D^{(r)}=\sum _{i=1}^r{\mathrm{id}}^{\otimes (i-1)}\otimes D\otimes {\mathrm{id}}^{\otimes (r-i)}.$$

For $r=0$$r=0$, set

$$D^{(0)}=0.$$

Taking the coefficient of $ϵ$$\epsilon$ gives

$$D^{(n)}\circ \rho (p)=\rho (p)\circ D^{(m)}.$$

This is the general Leibniz rule for a PROP-model.

So a derivation of $A$$A$ is a $k$$k$-linear endomorphism $D:A\to A$$D:A\to A$ such that, for every operation $p:m\to n$$p:m\to n$,

$$D^{(n)}\circ \rho (p)=\rho (p)\circ D^{(m)}.$$

Equivalently,

$${\mathrm{Der}}_{\mathsf{P}}(A)=\mathrm{Lie}(G_A).$$

Derivations are tangent vectors to the automorphism functor.

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3. Addition comes from first-order multiplication

Before proving closure under commutators, one should first see why derivations are closed under addition.

Let

$$D,E\in {\mathrm{Der}}_{\mathsf{P}}(A).$$

Then

$$\mathrm{id}+ϵD,\mathrm{id}+ϵE$$

are elements of

$$G_A(k[ϵ]/({ϵ}^2)).$$

Since $G_A(k[ϵ]/({ϵ}^2))$$G_A(k[\epsilon]/(\epsilon^2))$ is a group, their product is again an infinitesimal automorphism:

$$(\mathrm{id}+ϵD)(\mathrm{id}+ϵE)\in G_A(k[ϵ]/({ϵ}^2)).$$

But

$$(\mathrm{id}+ϵD)(\mathrm{id}+ϵE)=\mathrm{id}+ϵ(D+E)+{ϵ}^{2}DE.$$

Since ${ϵ}^2=0$$\epsilon^2=0$, this becomes

$$\mathrm{id}+ϵ(D+E).$$

Therefore

$$D+E\in {\mathrm{Der}}_{\mathsf{P}}(A).$$

Scalar closure is just as formal. For $a\in k$$a\in k$, the map

$$k[ϵ]/({ϵ}^2)\to k[ϵ]/({ϵ}^2),ϵ\mapsto aϵ$$

sends

$$\mathrm{id}+ϵD$$

to

$$\mathrm{id}+ϵ(aD).$$

Hence

$$aD\in {\mathrm{Der}}_{\mathsf{P}}(A).$$

So

$${\mathrm{Der}}_{\mathsf{P}}(A)$$

is a $k$$k$-submodule of ${\mathrm{End}}_k(A)$$\operatorname{End}_k(A)$.

At first order, group multiplication becomes addition.

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4. The infinitesimal square

To see the bracket, one needs two independent infinitesimal directions.

Let

$$R=k[{ϵ}_1,{ϵ}_2]/({ϵ}_1^2,{ϵ}_2^2).$$

As a $k$$k$-module,

$$R=k\oplus k{ϵ}_1\oplus k{ϵ}_2\oplus k{ϵ}_1{ϵ}_2.$$

The relations are

$${ϵ}_1^2=0,{ϵ}_2^2=0,$$

but

$${ϵ}_1{ϵ}_2\ne 0.$$

Thus $R$$R$ remembers two first-order directions and their mixed second-order area term.

Now define

$$R_0=R/({ϵ}_1{ϵ}_2).$$

Equivalently,

$$R_0=k[{ϵ}_1,{ϵ}_2]/({ϵ}_1^2,{ϵ}_2^2,{ϵ}_1{ϵ}_2).$$

There are quotient maps

$$R⟶R_0⟶k.$$

The first quotient kills the area term

$${ϵ}_1{ϵ}_2,$$

and the second quotient kills the two first-order directions

$${ϵ}_1,{ϵ}_2.$$

Geometrically, the arrows reverse:

$$\mathrm{Spec}k⟶\mathrm{Spec}{R}_0⟶\mathrm{Spec}R.$$

The picture is:

$$\text{origin}\subset \text{two infinitesimal axes}\subset \text{infinitesimal square}.$$

The bracket lives in the difference between the square and its two axes.

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5. The two kernels

Apply $G_A$$G_A$ to

$$R\to R_0\to k.$$

We get group homomorphisms

$$G_A(R)\to G_A(R_0)\to G_A(k).$$

Define

$$K_R=\ker (G_A(R)\to G_A(k)),$$

and

$$K_0=\ker (G_A(R_0)\to G_A(k)).$$

So $K_R$$K_R$ is the second-order infinitesimal unit neighbourhood, while $K_0$$K_0$ is its first-order quotient.

There is a natural homomorphism

$$K_R\to K_0.$$

The key point is that $K_0$$K_0$ is abelian.

Indeed, in $R_0$$R_0$ we have

$${ϵ}_1^2={ϵ}_2^2={ϵ}_1{ϵ}_2=0.$$

Thus every element of $K_0$$K_0$ has the form

$$\mathrm{id}+{ϵ}_{1}D+{ϵ}_{2}E$$

with $D,E\in {\mathrm{Der}}_{\mathsf{P}}(A)$$D,E\in \operatorname{Der}_{\mathsf P}(A)$.

The product is

$$\begin{array}{rl}& (\mathrm{id}+{ϵ}_{1}D+{ϵ}_{2}E)(\mathrm{id}+{ϵ}_1{D}^{\prime }+{ϵ}_2{E}^{\prime })\\ & =\mathrm{id}+{ϵ}_1(D+D^{\prime })+{ϵ}_2(E+E^{\prime }),\end{array}$$

because all products of first-order terms vanish in $R_0$$R_0$.

Therefore $K_0$$K_0$ is an additive group:

$$K_0\cong {\mathrm{Der}}_{\mathsf{P}}(A){ϵ}_1\oplus {\mathrm{Der}}_{\mathsf{P}}(A){ϵ}_2.$$

In particular, $K_0$$K_0$ is abelian.

Hence the homomorphism

$$K_R\to K_0$$

factors through the abelianization of $K_R$$K_R$:

$$K_R\to K_R^{\mathrm{ab}}\to K_0.$$

Therefore every group commutator in $K_R$$K_R$ maps to the identity in $K_0$$K_0$.

Equivalently,

$$[K_R,K_R]\subseteq \ker (K_R\to K_0).$$

But

$$\ker (K_R\to K_0)=\ker (G_A(R)\to G_A(R_0)).$$

This is the area layer.

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6. The area layer is the tangent space again

The kernel

$$\ker (G_A(R)\to G_A(R_0))$$

consists of automorphisms which become the identity after killing ${ϵ}_1{ϵ}_2$$\epsilon_1\epsilon_2$.

Therefore every element of this kernel has the form

$$\mathrm{id}+{ϵ}_1{ϵ}_{2}F$$

for some $k$$k$-linear endomorphism $F:A\to A$$F:A\to A$.

Because

$$({ϵ}_1{ϵ}_2{)}^2=0,$$

its inverse is

$$\mathrm{id}-{ϵ}_1{ϵ}_{2}F.$$

The condition that

$$\mathrm{id}+{ϵ}_1{ϵ}_{2}F$$

preserves the PROP-structure is precisely the first-order condition saying that $F$$F$ is a derivation.

Thus

$$\ker (G_A(R)\to G_A(R_0))\cong {\mathrm{Der}}_{\mathsf{P}}(A)\cdot {ϵ}_1{ϵ}_2.$$

Since

$${\mathrm{Der}}_{\mathsf{P}}(A)=\mathrm{Lie}(G_A),$$

we may also write

$$\ker (G_A(R)\to G_A(R_0))\cong \mathrm{Lie}(G_A)\cdot {ϵ}_1{ϵ}_2.$$

This is the tangent space labelled by the area element.

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7. The commutator lands in the area layer

Take

$$D,E\in {\mathrm{Der}}_{\mathsf{P}}(A),$$

and put

$$g_D=\mathrm{id}+{ϵ}_{1}D,g_E=\mathrm{id}+{ϵ}_{2}E$$

inside

$$K_R=\ker (G_A(R)\to G_A(k)).$$

Their group commutator

$$c=g_D{g}_E{g}_D^{-1}{g}_E^{-1}$$

also lies in $K_R$$K_R$.

Now reduce modulo ${ϵ}_1{ϵ}_2$$\epsilon_1\epsilon_2$. In

$$R_0=R/({ϵ}_1{ϵ}_2),$$

the group $K_0=\ker (G_A(R_0)\to G_A(k))$$K_0=\ker(G_A(R_0)\to G_A(k))$ is first-order and hence abelian. Therefore the image of the commutator $c$$c$ in $K_0$$K_0$ is the identity.

Thus

$$c\in \ker (G_A(R)\to G_A(R_0)).$$

But this kernel is the area-labelled tangent space

$${\mathrm{Der}}_{\mathsf{P}}(A)\cdot {ϵ}_1{ϵ}_2.$$

Hence

$$c=\mathrm{id}+{ϵ}_1{ϵ}_{2}F$$

 

$$F\in {\mathrm{Der}}_{\mathsf{P}}(A).$$

Easy to see that

$$F=DE-ED.$$

Since $F$$F$ is already known to be a derivation, we conclude that

$$DE-ED\in {\mathrm{Der}}_{\mathsf{P}}(A).$$

This proves that derivations of a PROP-model are closed under commutators.