3:30 pm, Seminar Hall
One motivation for categorical representation theory (Third Lecture - Colloquium)
University of Aarhus, Denmark.
First we review the standard approach to Categorification (following Frenkel, Khovanov, Rouquier): ideologically, in classical algebraic constructions vector spaces are replaced by categories (Abelian, additive or triangulated), groups and algebras are replaced by monoidal categories, actions become monoidal actions of monoidal categories. Standard examples include categorical Hecke algebras and categorifications of quantum groups. We discuss the de-categorification procedure and its benifits (canonical bases for usual (non-categorical) representations).
Next we discuss a different area or Representation Theory where categorification is required naturally. We define a canonical central extension of the group of automorphisms of an infinite-dimensional k-vector space k((t)) (known as Jacobian matrices). We provide a simple construction for the central extension using a purely categorical approach to Sato Grassmannian (due to Kapranov).
Then we outline a higher analog of this construction: k((t)) is replaced by k((t))((s)), the corresponding group is called the group of generalized Jacobian matrices, Sato Grassmannian is replaced by double Sato Grassmannian, and a canonical 3-cocycle for the group of generalized Jacobian matrices arises.
Yet no representations in vector spaces "feel" this higher central extension, so there is a call for categorical representation theory.