## Student Representation Theory Seminar

### Structure

The Student Representation Theory Seminar (SRTS) is a weekly informal reading seminar at LSU that aims to introduce research-level topics to graduate students in Representation Theory. Each semester, we pick a topic and devout the semester to learning that topic. We start with the foundations (roughly the first half of the semester) before diving into modern research papers on the selected topic (roughly the second half of the semester). Our topic for the Fall 2024 semester is Quantum Groups.

Each week, a participant will give a talk lasting between one to one-and-a-half hours. The talks are not recorded, but notes will be uploaded to this website after each talk. All graduate students with an interest in Representation Theory are encouraged to attend. There is no prerequisites for attending the seminar; although, basic knowledge of representation theory of finite groups, Lie algebras, and algebraic geometry is certainly helpful.

### Logistics

Class Room: Lockett 381

Time: Thursday 1:30pm-3:00pm

Organizers: Colton Sandvik (csandv1@lsu.edu) and Gurleen Nanda (gnanda1@.lsu.edu)

Topic: Quantum Groups

### Resources

*P. Etingof, M. Semenyakin - A brief introduction to quantum groups. arXiv:2106.05252.

J. C. Jantzen - Lectures on Quantum Groups. Graduate Studies in Mathematics, Vol. 6.

C. Kassel - Quantum Groups. Graduate Texts in Mathematics, Vol. 155.

G. Lusztig -Introduction to Qunatum Groups. Modern Birkhäuser Classics.

* This will be our main reference for the first half of the semester

### Special Topics

For the last two months of the semester, we will have special topics talks. We encourage attendants to pick a reference and give a talk on it. These talks should be like a high level research talk (you can even pretend that it is your own research if you'd like). We have included some suggested topics and references here if you are having trouble finding something.

Applications of Quantum Groups to Knot Invariants

C. Kassel - Quantum Groups. Graduate Texts in Mathematics, Vol. 155.

Canonical Basis for Positive Half for Quantum Groups

(Chapter 10) P. Achar - Perverse Sheaves and Applications to Representation Theory. AMS Monographs, Vol. 258.

Affine Quantum Groups

(Section 5) P. Etingof, M. Semenyakin - A brief introduction to quantum groups. arXiv:2106.05252.

Soergel Character Formula for Tilting Modules

W. Soergel, Character formulas for tilting modules over quantum groups at roots of one, in Current developments in mathematics, 1997 (Cambridge, MA), 161–172, Int. Press, 1999.

W. Soergel, Character formulas for tilting modules over Kac–Moody algebras, Represent. Theory 2 (1998), 432–448.

Coherent Realizations of the Category of Representations of Quantum Groups

(Part 1) S. Arkhipov, R. Bezrukavnikov, and V. Ginzburg, Quantum groups, the loop Grassmannian, and the Springer resolution, J. Amer. Math. Soc. 17 (2004), 595–678.

Quantum Integrable Systems

M. Wadati, Quantum integrability and quantum groups, Physica D: Nonlinear Phenomena. 86, Issues 1-2 (1995), 19-26.

Elliptic Quantum Groups

H. Konno, Elliptic Quantum Groups. arxiv:2405.11193.