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.
Abstract. We use the monomial basis theory developed by Deng and Du to present an elementary algebraic construction of the canonical bases for both the Ringel-Hall algebra of a cyclic quiver and the positive part U + of the quantum affine sl n . This construction relies on analysis of quiver representations and the introduction of a new integral PBW-like basis for the Lusztig Z[v, v −1 ]-form of U + .
Let G be a nite group of Lie type and let k be a eld of characteristic distinct from the de ning characteristic of G. In studying the non-describing representation theory of G, the endomorphism algebra SG; k = End kG L J ind G P J k plays an increasingly important role. In type A, by work of Dipper and James, SG; k identi es with a q-Schur algebra and so serves as a link between the representation theories of the nite general linear groups and certain quantum groups. This paper presents the rst systematic study of the structure and homological algebra of these algebras for G of arbitrary type. Because SG; k has a reinterpretation as a Hecke endomorphism algebra, it may be analyzed using the theory of Hecke algebras. Its structure turns out to involve new applications of Kazhdan-Lusztig cell theory. In the course of this work, we prove two strati cation conjectures about Coxeter group representations made in CPS4 and we formulate a new conjecture about the structure of SG; k. We v erify this conjecture here in all rank 2 examples. Let G be a nite group of Lie type. Thus, G is the subgroup of xed points G for a rational endomorphism of a reductive algebraic group G de ned over an algebraically closed eld F of positive characteristic. For a eld k of characteristic distinct from that of F, consider the endomorphism algebra SG; k = End kG L PB ind G P k. Here ind G P k denotes the right permutation module over k de ned by the set fP g g g2G of right cosets of the parabolic subgroup P in G. Work of Dipper and James for G = GLn cf. below suggests that the algebras SG; k play an important role in the representation theory of kGfor all types. This paper presents the rst systematic study of these algebras. We establish new results concerning the structure and cohomology of SG; k v alid for all G. In the process, we also prove several conjectures made in CPS4; x6 for nite Coxeter groups, and we indicate new directions for further study.
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