Introduction to categorical Kac-Moody actions
Ivan Losev (Northeastern University)
The goal of these lectures is to provide an elementary introduction to
categorical actions of Kac-Moody algebras from a representation
theoretic perspective.
In a naive way (that, of course, appeared first), a categorical KacMoody action is a collection of functors on a category that on the level
of Grothendieck groups give actions of Chevalley generators. Such
functors were first observed in the representation theory of
symmetric groups in positive characteristic and then for the BGG
category O of gln . Analyzing the examples, in 2004 Chuang and
Rouquier gave a formal definition of a categorical sl2 -action. Later
(about 2008) Rouquier and Khovanov-Lauda extended this definition
to arbitrary Kac-Moody Lie algebras.
Categorical Kac-Moody actions are very useful in Representation
theory and (potentially, at least) in Knot theory. Their usefulness in
Representation theory is three-fold. First, they allow to obtain
structural results about the categories of interest (branching rules for
the symmetric groups obtained by Kleshchev, or derived equivalences
between different blocks constructed by Chuang and Rouquier in
order to prove the Broué abelian defect conjecture). Second,
categories with Kac-Moody actions are often uniquely determined by
the "type of an action", sometimes this gives character formulas. Third,
the categorification business gives rise to new interesting classes of
algebras that were not known before: the KLR (Khovanov-LaudaRouquier) algebras. Potential applications to Knot theory include
categorical (hence stronger) versions of quantum knot invariants, this
area is very much still in development.
I will start from scratch and try to keep the exposition elementary, in
particular I will only consider Kac-Moody algebras of type A, i.e., sl n
∧
and sln . The most essential prerequisite is a good understanding of
the categorical language (e.g., functor morphisms). Familiarity with
classical representation theoretic objects such as affine Hecke
algebras or BGG categories O is also useful although these will be
recalled.
A preliminary plan is as follows:
1)
2)
3)
4)
5)
6)
7)
8)
9)
Introduction.
Examples: symmetric groups/type A Hecke algebras.
Formal definition of a categorical action.
Consequences of the definition: divided powers,
categorifications of reflections, categorical Serre relations,
crystals.
More examples: cyclotomic Hecke algebras.
Structural results: uniqueness minimal categorifications,
filtrations.
Yet more examples: BGG categories O.
Categorical tensor products and their uniqueness.
Time permitting - and yet more examples: categories O for
cyclotomic rational Cherednik algebras, computation of
multiplicities.
Here are some important topics related to categorical Kac-Moody
actions that will not be discussed:
a) Categorical actions in other types and those of quantum groups.
b) Categorification of the algebras U(n),U(g), etc.
c) Connections to categorical knot invariants.
a) is described in reviews http://arxiv.org/abs/1301.5868 by
Brundan and (a more advanced text) http://arxiv.org/abs/1112.3619
by Rouquier. The latter also deals with b). A more basic review for b)
is http://arxiv.org/abs/1112.3619 by Lauda dealing with the sl2 case
and also introducing diagrammatic calculus. I'm not aware of any
reviews on c), a connection to Reshetikhin-Turaev invariants was
established
in
full
generality
by
Webster
in
http://arxiv.org/abs/1309.3796.