Columbia University G6250
Hopf algebras, their representations, applications, and categorifications

Basic information

Semester: Fall 2008
Call number: 53048
Room/Time: MW 2:40pm--3:55pm, 307 Math
Instructor: Mikhail Khovanov
Office: 517 Math
Office Hours: Walk-in or by appointment
Discussions: TBA
TA: Daniel Krasner,


The first half of the course will center on Hopf algebras and their applications in low-dimensional topology. Topics will include various constructions of link and 3-manifold invariants from Hopf algebras and their representations, graphical calculi of representations of quantum groups, Drinfeld double. Along the way we'll review general methods and ideas of representation theory and homological algebra to prepare us for the remainder of the course--categorification of quantum groups, their representations, and quantum link invariants. We'll go over recent work of Aaron Lauda on categorification of quantum sl(2) and related work on categorification of more general quantum groups. If time allows, we'll do some of the following topics: sl(n) homology and tangle invariants, categorification of the Burau representation, category O and its role in categorification of the Hecke algebra and tensor products, categorified Schur-Weyl duality.


There is no textbook for the course. If you want to read one book on quantum groups and applications to 3D topology, I suggest Quantum groups and knot invariants, by C.Kassel, M.Rosso, and V.Turaev. Additionally, Quantum groups, by C.Kassel, will be placed on library reserve.

Online resources:

Hopf algebras and quantum groups

G.Kuperberg, Involutory Hopf algebras and 3-manifold invariants
G.Kuperberg, Non-involutory Hopf algebras and 3-manifold invariants
P.Cartier, A primer of Hopf algebras
B.Pareigis, Hopf algebras, algebraic, formal, and quantum groups. This is chapter 2 of his book Lectures on quantum groups and noncommutative geometry
J.Bernstein, Sackler lectures on quantum groups and TQFTs.
A.Ram, A survey of quantum groups (gunzipped ps file)
V.F.R.Jones, In and around the origin of quantum groups
O.Schiffmann, Lectures on Hall algebras

Hecke algebras

D.Goldschmidt, Group Characters, Symmetric Functions, and the Hecke Algebra

Categorification of Hopf algebras

L.Crane and I.B.Frenkel,Four dimensional topological quantum field theory, Hopf categories, and the canonical bases
J.Chuang and R.Rouquier Derived equivalences for symmetric groups and sl(2)-categorification
A.Lauda,A categorification of quantum sl(2)
A.Lauda and M.Khovanov,A diagrammatic approach to categorification of quantum groups I

Additional online resources

Homological algebra

C.Weibel,The Grothendieck group K_0 is chapter II of his K-book
C.Kassel, Homology and cohomology of associative algebras - A concise introduction to cyclic homology
D.Milicic, Lectures on derived categories

Representation theory

V.Serganova, Representation theory: representations of finite groups, symmetric groups, GL(n), quivers.
P.Etingof and students, Lectures and problems in representation theory covers representations of algebras, finite groups, and quivers.
H.Barcelo and A.Ram, Combinatorial representation theory
D.Gaitsgory, Geometric Representation theory Notes for a course on highest weight categories.

Hopf algebras

R.Street, Quantum groups:an entree to modern algebra

Quantum invariants of links and 3-manifolds

V.F.R.Jones The Jones polynomial
C.Blanchet, Introduction to quantum invariants of knots and links
K.Walker, On Witten's 3-manifold invariants
G.Kuperberg, Spiders for rank 2 Lie algebras
M.Freedman, C.Nayak, K.Walker, Z.Wang, On Picture (2+1)-TQFTs
C.Blanchet, Introduction to quantum invariants of 3-manifolds, topological quantum field theories, and modular categories