## **Seminar: quantum groups.**

**Time: Thursdays 4-6, location: zoom**
(password is available from Ivan, you can send an e-mail to ivan dot loseu at yale dot edu and, if you think
Ivan doesn't know you, tell some basics
about yourself).

Organized by Roman Bezrukavnikov
(IAS/MIT), Pavel Etingof (MIT)
Ivan Losev (IAS/Yale).

This picture illustrates some parts of the dynamics in the organizing committee.

**Preliminary program:**

1) Intro part: Chapters 4-8 in [J] (chapters 1-3 dealing with U_q(sl_2) are assumed as prerequisites).

2) Main part: may include a discussion of Lusztig's form at roots of unity and its representations following
[L] and/or original papers (references to be provided later) as well as a connection
of the categories of modules over quantum groups to an affine parabolic category O following
[KL] and/or [ABG]. Precise list of topics TBD. Prerequisites will become more definite when
a precise list of topics is selected but should, in any case, include basics on the BGG
category O.

**Schedule**:

Jan 21: Leonardo Maltoni, chapter 4 in [J]. video, notes.
Jan 28: Leonardo Maltoni, chapter 5 in [J].
video (just in case, the passcode is: 3p4F@&wx), notes.
Feb 4: Arun Kannan, chapter 6 in [J].
video (the passcode is: m%7gbR$5), notes.
Feb 11: Arun Kannan, continuation on chapter 6 in [J]. Video (passcode: !ZGD9*WV),
notes.
Feb 18, Elijah Bodish, chapter 7 in [J].
video (passcode: c#=hT%p5), Hand-written notes.
Feb 25, Elijah continues. Calder Morton-Ferguson starts on Chapter 8. video
(passcode: %0ih*S#+). Elijah's hand-written notes.
Elijah's typed notes
typed notes (for both lectures). Calder's notes.
March 4, Calder continues. video (passcode: a#T5CVTM),
notes.
March 11, Calder finishes. Jay Taylor starts on quantum groups at roots of unity. Video
(passcode: #LN.Q41U). Calder's complete notes. Jay's notes.
March 18, Jay continues. Update: Jay cannot make it to the talk, so it will be delivered by Ivan. We'll discuss the continuation
of the characteristic p story. video (passcode: 0Nf!%?JF),
Ivan's notes, Jay's notes.
March 25, Ivan finishes the story about algebraic groups in characteristic p. Jay talks about quantum
groups at roots of unity. Video (passcode: HQryB?05).
Ivan's notes on characteristic p, Jay's notes
on characteristic p, Jay's notes on roots of unity.
April 1, Jay continues on quantum groups at roots of unity -- and this is not a joke! video (passcode: 9tXkRm@X). Notes.
April 8, Jize Yu starts to speak on the representation theory of Lusztig's forms at roots of 1. As with the structure theory, he'll
start with the representation theory in characteristic p. Video. notes.
April 15, Jize speaks about representations of quantum groups at roots of unity. video,
Ivan's short note.
April 22, Jize finishes. Jin-Cheng Guu starts on the Kazhdan-Lusztig category of representations of affine Lie algebras.
**Preliminary list of references**

[ABG] S. Arkhipov, R. Bezrukavnikov, V. Ginzburg, *Quantum groups, the loop Grassmannian, and the Springer resolution.*
J. Amer. Math. Soc. 17 (2004), no. 3, 595–678.
[J] J.C. Jantzen, *Lectures on quantum groups*. Lectures on quantum groups. Graduate Studies in Mathematics, 6. American Mathematical Society, Providence, RI, 1996.
[KL] D. Kazhdan, G. Lusztig, *Tensor structures arising from affine Lie algebras.*
Parts I,II: J. Amer. Math. Soc. 6 (1993), no. 4, 905–947, 949–1011;
parts III,IV: J. Amer. Math. Soc. 7 (1994), no. 2, 335–381, 383–453.
[L] G. Lusztig, *Introduction to quantum groups*. Progress in Mathematics, 110. Birkhauser Boston,
Inc., Boston, MA, 1993.