Kevin Coulembier (U Sydney)

A Borelic approach to cellular algebras.

Abstract:

Quasi-hereditary and cellular algebras appear naturally in the study of the fundamental questions in representation theory. Quasi-hereditary algebras, such as algebras describing category O, admit a natural class of modules known as 'standard modules'. Cellular algebras, such as Hecke algebras, admit 'cell modules'.

Hemmer and Nakano proved the remarkable result that the cell modules of the Hecke algebra behave as the standard modules of some bigger quasi-hereditary algebra, meaning they form a 'standard system'. This observation has been extended to a variety of other cellular algebras.

We introduce a theory of Borelic pairs (B,H) of arbitrary algebras A, inspired by König's notion of exact Borel subalgebras of quasi-hereditary algebras. We prove that A inherits structural properties from H. For instance, if H is semisimple, A is quasi-hereditary with exact Borel subalgebra B. Roughly speaking, if H is cellular, so is A, and then cell modules of A form a standard system if and only if the ones of H do so.

We apply this general theory to a variety of examples, including Brauer, Temperley-Lieb and Auslander algebras.