Since I have a lot of free time this summer, I decided to run a seminar. Currently we are discussing the structure and representation theory of semisimple Lie algebras; later on we may talk about Hopf algebras and symmetric functions (and some places in enumerative combinatorics where these things crop up, and the connections to representation theory). There is pretty much no upper bound on how bizarre the talks could get. Anyway, I will always be available afterwards to talk about things if there are questions.

Also, I don't intend to spend much time covering material that you either have seen, or will likely end up seeing, in a course (for example PMATH 763 which is being taught here next Winter). I will define everything and state results, but I will omit laborious proofs if they are a bit tangential to the topic at hand (for example, the Baker-Campbell-Hausdorff formula for \( \log( e^X e^Y ) \), or certain properties of the exponential mapping and so on). Even if you have no idea what a Lie group or Lie algebra is, you can still participate as long as you are willing to accept things on faith or read about them on your own: this is not supposed to be an introduction to Lie theory.

I will make an effort to post very detailed slides of the material covered.

Some background notes aka "PMATH 365 in a box" - new multilinear algebra material added June 8

References

**Thu May 15**: Michael Baker, "Lie theory and its ubiquity"

Lie groups. Examples of matrix groups (classical groups). The exponential map. The adjoint representation. Lie algebras, and the Lie algebra associated to a Lie group. The Baker-Campbell-Hausdorff formula and the degree to which the Lie algebra controls the group structure. Differential/pushforward of a Lie group homomorphism. Basics of representation theory. Motivation for studying Lie theory; connections to other branches of mathematics. Overview of representation theory.**Thu May 22**: Michael Baker, "Representations of sl(2,C)"

First we look at an alternative construction of the Lie algebra $\mathfrak{g}$ of a Lie group $G$. We then develop the basic theory of representations of Lie algebras: semisimplicity, nilpotence, solvability, complete reducibility, Jordan-Chevalley decomposition, and then move on to the important special case of the Lie algebra $\mathfrak{sl}(2,\mathbf{C})$ of traceless complex matrices. This nicely illustrates the general semisimple strategy, and will moreover play a key role in what follows.**Thu May 29**: Michael Baker, "Representations of sl(3,C)" (slides not yet finished)

We introduce the Lie algebra $\mathfrak{sl}(3,\mathbf{C})$ of traceless complex matrices, and begin our classification of its representations. With some slight generalizations, we can apply the same strategy we used for $\mathfrak{sl}(2,\mathbf{C})$, although the analysis is now more involved.

**Thu Jun 5**: Michael Baker, "Representations of sl(3,C) again"

Continuing from last time, we describe the possible weights of a representation of $\mathfrak{sl}(3,\mathbf{C})$, give a complete classification of such representations, and in the process, work through a few illuminating examples.**Thu Jun 12**: Rui Philip Xiao, "Universal enveloping algebras" (notes are by Philip; abbreviated from these notes)

Any associative algebra can be made into a Lie algebra by introducing the commutator bracket $[a,b] = ab-ba$; a Lie algebra arising in this way is called a commutator algebra. To a Lie algebra $L$ we can construct a unital associative algebra $\mathcal{U}(L)$, called the universal enveloping algebra of $L$, such that every Lie algebra homomorphism from $L$ to some associative algebra factors through $\mathcal{U}(L)$ in a universal way. The PoincarĂ©-Birkhoff-Witt theorem gives a more concrete description of $\mathcal{U}(L)$ and shows that $L$ can be viewed as a Lie subalgebra of the commutator algebra of $\mathcal{U}(L)$. In this talk I will define the universal enveloping algebra of a Lie algebra and show its existence and uniqueness, then prove the PoincarĂ©-Birkhoff-Witt theorem. If time permits I will talk about more consequences of the PBW theorem.**Thu Jun 19**: Michael Baker, "~~Schur functors, Young tableaux, and representation theory~~Representation theory of finite groups"

Obligatory crash course on the representation theory of finite groups (aka "most of PMATH 745 in one hour"): Schur's lemma, new representations from old, modules and algebras, the group algebra of a finite group, character theory, induced representations and Frobenius reciprocity. Representations of the symmetric groups $\mathfrak{S}_d$,~~Schur-Weyl duality, Schur functors, Young tableaux and (some of) their close connections with representation theory.~~*Note: Unfortunately, after the abstract for the talk was submitted to the department, I realized I needed a significant amount of prerequisite material from the representation theory of finite groups. Therefore, I talked about this instead. The topics I originally had in mind will instead be presented on July 10.***Thu Jun 26**: Raymond Cheng, "Hopf algebras, symmetric functions and representation theory" (videos: part one, part two)

We introduce the notion of Hopf algebras, from which we will obtain a new perspective on the representation theory introduced thus far. The Hopf algebra of symmetric functions will then be introduced. Connections with the representation theory of the general linear group will also be mentioned.

**Thu Jul 3**: cancelled due to CUMC (I am giving a talk there on topological quantum field theories)**Thu Jul 10**: Michael Baker, "Young tableaux and representation theory"

Young tableaux. Representations of the symmetric groups $\mathfrak{S}_d$. Schur-Weyl duality for $\mathfrak{gl}(V)$.**Thu Jul 17**: cancelled (break week)**Thu Jul 24**: Yossef Musleh, "Clifford algebras and spin representations"

The (special) orthogonal/unitary groups and their geometry will be discussed. Possible topics include covering groups, Clifford algebras, spin representations (as time permits).**Thu Jul 31**: Michael Baker, "Clifford algebras and spin representations I" (video)

The Clifford algebra $\mathcal{C}l(V,q)$ associated to a quadratic vector space $(V,q)$ is a quantization of its exterior algebra whose multiplication carries very rich geometric information. Now an indispensable tool in geometry and mathematical physics, these algebras are also intimately related to projective representations of the orthogonal Lie groups. Continuing from last time, I will develop their basic theory: the universal property, the natural filtration, the groups Pin and Spin, the twisted adjoint representation, and so on. We will see a few examples in simple cases.

**Thu Aug 7**: Michael Baker, "Clifford algebras and spin representations II"

After briefly reviewing some of the objects introduced last time, I will prove that the twisted adjoint representation $\widetilde{\text{Ad}}$ exhibits $\text{Spin}(V,q)$ and $\text{Pin}(V,q)$ as double covers of $\text{SO}(V,q)$ and $\text{O}(V,q)$ respectively. We will then discuss some extra structures on Clifford algebras, and their classification (time permitting).**Thu Aug 14**: Michael Baker, "Bundles, spheres, and Clifford algebras"

After introducing the concept of a vector bundle, we will examine the problem of how many linearly independent vector fields can be found on the $n$-sphere $S^n$, and discuss how representations of Clifford algebras can be used to give a construction. Frank Adams later proved using deep methods of K-theory that this construction in fact gives the largest possible number of such vector fields.