**Record:** Yes

**Level:** Graduate

**Language:** English

**Prerequisite**

Basic Lie algebra background. An interest or experience in mathematical physics is helpful, as well as familiarity with representation theory of groups and associative algebras.

**Abstract**

Starting with an associative algebra, tensor products of modules do not necessarily have a natural module structure. For bialgebras (in particular, Hopf algebras), they do. Moreover, if the bialgebra is quasitriangular then the tensor product of modules and the tensor product of the same modules taken in the opposite order are isomorphic as modules. Quantum groups, studied since the 1980s, form a rich family of quasitriangular bialgebras which deform associative algebras naturally connected with certain (e.g. semisimple finite-dimensional) Lie algebras.

**Reference**

Title A Guide to Quantum Groups

Authors V. Chari and A. Pressley

Publisher Cambridge University Press, 1995

ISBN 0521558840, 9780521558846

Title Lectures on Quantum Groups

Author J. Jantzen

Publisher American Mathematical Society

ISBN 0821872346, 9780821872345

**Syllabus**

Tentatively, we will discuss the following topics:

1) Preliminaries: definition and basic properties of bi- and Hopf algebras; monoidal categories; cocommutative examples (group algebras, universal enveloping algebras)

2) Quasitriangularity; braided monoidal categories; diagrammatical calculus; basic example: Sweedler's Hopf algebra

3) Chevalley-Serre presentation of finite-dimensional semisimple Lie algebras, symmetrizable Kac-Moody algebras, BGG/Kac category O

4) Main example: Drinfeld-Jimbo quantum groups U_q(g) defined over C(q) (starting with quantum sl2)

5) Quantized universal enveloping algebras U_q(g) as topological Hopf algebras; precise relation to U(g).

6) Quasitriangular structure of Drinfeld-Jimbo quantum groups, completion via Tannakian approach (natural transformations of forgetful functor)

7) Pairings between Hopf algebras; bar involution, skew derivations, quasi-R-matrix (Lusztig approach to R); quantum double (Drinfeld approach to R)

8) Explicit formulas of R for quantum sl2; factorization of R-matrices via quantum Weyl group

9) Dual quantum group: quantization of the algebra of functions (RTT relation)

10) Affine quantum groups and applications to quantum integrability

**Lecturer Intro**

Dr. Bart Vlaar has joined BIMSA in September 2022 as an Associate Research Fellow. His research interests are in algebra and representation theory and applications in mathematical physics. He obtained a PhD in Mathematics from the University of Glasgow. Previously, he has held postdoctoral positions in Amsterdam, Nottingham, York and Heriot-Watt University. Before coming to BIMSA he visited the Max Planck Institute of Mathematics in Bonn.Dr. Bart Vlaar has joined BIMSA in September 2022 as an Associate Research Fellow. His research interests are in algebra and representation theory and applications in mathematical physics. He obtained a PhD in Mathematics from the University of Glasgow. Previously, he has held postdoctoral positions in Amsterdam, Nottingham, York and Heriot-Watt University. Before coming to BIMSA he visited the Max Planck Institute of Mathematics in Bonn.

**Lecturer Email:** B.Vlaar@hw.ac.uk

**TA:** Dr. Shuang Ming, sming@ucdavis.edu