BASED ALGEBRAS AND STANDARD BASES FOR QUASI-HEREDITARY ALGEBRAS

Quasi-hereditary algebras can be viewed as a Lie theory approach to th e theory of finite dimensional algebras. Motivated by the existence of certain nice bases for representations of semisimple Lie algebras and algebraic groups, we will construct in this paper nice bases for (spl it) quasi-hereditary algebras and characterize them using these bases. We first introduce the notion of a standardly based algebra, which is a generalized version of a cellular algebra introduced by Graham and Lehrer, and discuss their representation theory. The main result is th at an algebra over a commutative local noetherian ring with finite ran k is split quasi-hereditary if and only if it is standardly full-based . As an application, we will give an elementary proof of the fact that split symmetric algebras are not quasi-hereditary unless they are sem isimple. Finally, some relations between standardly based algebras and cellular algebras are also discussed.