Based algebras and standard bases for quasi-hereditary algebras

Abstract

Abstract. Quasi-hereditary algebras can be viewed as a Lie theory approach to the 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 (split) 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 that an algebra over a commutative local noetherian ring with finite rank is split quasi-hereditary if and only if it is standardly fullbased. As an application, we will give an elementary proof of the fact that split symmetric algebras are not quasi-hereditary unless they are semisimple. Finally, some relations between standardly based algebras and cellular algebras are also discussed.

Presenting cyclotomic q-Schur algebras

“…, and put L (λ) = Δ (λ)/ rad Δ (λ) We have the following theorem. This theorem is proven in a way similar to proofs in the general theory of standardly based algebras or cellular algebras (see [DR1], [GL], [M,Chapter 2]).…”

“…This definition is inspired by the presentation of generalized q-Schur algebras given in [Do]. Then we study the representation theory of S q which has properties similar to those of the theory of cellular algebras and of standardly based algebras introduced by [GL] and [DR1], respectively. The results in this section will be applied to obtain a presentation of cyclotomic q-Schur algebras in Section 7.…”

“…; see Appendix C.) And we do not know whether S q has such an involution. Nevertheless, we develop a certain representation theory of S q which is almost similar to the theory of standardly based algebras in the sense of [DR1], and also similar to the theory of cellular algebras (see, e.g., [GL], [M,Chapter 2]).…”

The Queer -Schur Superalgebra

“…(2) We remark that, for the q-Schur superalgebras S v (m|n, r) F of type M with m + n ≥ r, their irreducible representations at (odd) roots of unity have been classified in [11], while a nonconstructible classification of irreducible Q v (n, r) F -supermodules is obtained in [24,Theorem 6.32] by a generalised cellular structure [12,16]. We remark that Green's codeterminant basis was a first such basis for the Schur algebra.…”

Some properties for almost cellular algebras

“…There are several generalizations of cellular algebras. For example, affine cellular algebras if we extend the framework of cellular algebras to algebras that need not be finite dimensional over a field [13], relative cellular algebras if we allow different partial orderings relative to fixed idempotents [4], standardly based algebra by constructing a nice bases satisfy some conditions [3] and almost cellular algebras if we remove the compatible anti-involution from the definition of cellularity [7]. In this paper, we focus on the third generalization.…”