Elsevier

Advances in Mathematics

Volume 229, Issue 1, 15 January 2012, Pages 139-182
Advances in Mathematics

Affine cellular algebras

https://doi.org/10.1016/j.aim.2011.08.010Get rights and content
Under an Elsevier user license
open archive

Abstract

Graham and Lehrer have defined cellular algebras and developed a theory that allows in particular to classify simple representations of finite dimensional cellular algebras. Many classes of finite dimensional algebras, including various Hecke algebras and diagram algebras, have been shown to be cellular, and the theory due to Graham and Lehrer successfully has been applied to these algebras.

We will extend the framework of cellular algebras to algebras that need not be finite dimensional over a field. Affine Hecke algebras of type A and infinite dimensional diagram algebras like the affine Temperley–Lieb algebras are shown to be examples of our definition. The isomorphism classes of simple representations of affine cellular algebras are shown to be parameterised by the complement of finitely many subvarieties in a finite disjoint union of affine varieties. In this way, representation theory of non-commutative algebras is linked with commutative algebra. Moreover, conditions on the cell chain are identified that force the algebra to have finite global cohomological dimension and its derived category to admit a stratification; these conditions are shown to be satisfied for the affine Hecke algebra of type A if the quantum parameter is not a root of the Poincaré polynomial.

MSC

20C08
16G30
16G10
16P40
18G20
16E10
16D60
81R50

Keywords

Cellular algebra
Affine cellular algebra
Affine Hecke algebra
Simple module
Global dimension

Cited by (0)