Seminar: Gruber -- Tensor products and highest weight structures

Abstract: In this series of talks, I will discuss connections and interactions between
two types of structures that are ubiquitous in Lie theory and representation theory.
The first is the formalism of highest weight categories, which provides an axiomatic
framework for studying categories with "highest weight modules" (or "standard
modules").  The second is the theory of monoidal categories, i.e.  of categories
equipped with a tensor product bifunctor.  The primordial example of a highest weight
category with a monoidal structure is the category of rational representations of a
reductive algebraic group, and here the highest weight structure interacts with the
monoidal structure by way of the fact that tensor products of standard modules admit
filtrations whose successive subquotients are standard modules.  This kind of
interaction can surprisingly be observed in many other examples, and I will give an
explanation for this phenomenon via a monoidal enhancement of Brundan-Stroppel’s semi
infinite Ringel duality.  As applications, I will present solutions to two open
problems: One concerns the existence of monoidal structures on categories of
representations of affine Lie algebras at positive levels; the other concerns the
existence of highest weight structures on monoidal abelian envelopes of certain
"interpolation tensor categories".  All of this is based on joint work with Johannes
Flake.  

The program for the lecture series is as follows: 

In the first talk, I will mostly discuss the motivating examples (algebraic groups,
affine Lie algebras, interpolation categories), explain how they fit into the framework
of "lower finite" or "upper finite" highest weight categories, and state the two
aforementioned open problems.  

In the second talk, I will explain how lower finite and upper finite highest weight
categories are related via Ringel duality and how a monoidal structure on a lower finite
highest weight category gives rise to a monoidal structure on the Ringel dual upper
finite highest weight category.  This allows us to solve our first open problem: We
define a canonical monoidal structure on a parabolic version of the BGG category O for
an affine Lie algebra at positive level, and we construct a monoidal functor (a
"Kazhdan-Lusztig correspondence") to a category of representations of a quantum group at
a root of unity.  

The third and fourth lecture will be devoted to explaining the converse, that is, how a
monoidal structure on an upper finite highest weight category gives rise to a monoidal
structure on the Ringel dual lower finite highest weight category.  As an application,
we show that a large class of interpolation tensor categories (defined by Knop,
generalizing a construction of Deligne) can be embedded as categories of tilting objects
in monoidal lower finite highest weight categories.  This gives a uniform explanation
for the previously mysterious observation that many "abelian envelopes" of these
interpolation categories are highest weight categories.