Tuesday, August 2, 15.15, 7.527
Kevin Coulembier (U Sydney)
A Borelic approach to cellular algebras.
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
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.