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
Regular Time	Wednesdays, 2:00 pm
Location	Red Centre 3078
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: 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.

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

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 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.