Based algebras and standard bases

for quasi-hereditary algebras

Authors:
Jie Du and Hebing Rui

Journal:
Trans. Amer. Math. Soc. **350** (1998), 3207-3235

MSC (1991):
Primary 16G99, 17B10; Secondary 20C20, 20C30

DOI:
https://doi.org/10.1090/S0002-9947-98-02305-8

MathSciNet review:
1603902

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Abstract | References | Similar Articles | Additional Information

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 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 semisimple. Finally, some relations between standardly based algebras and cellular algebras are also discussed.

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Additional Information

**Jie Du**

Affiliation:
Schools of Mathematics, University of New South Wales, Sydney, 2052, Australia

Email:
jied@maths.unsw.edu.au

**Hebing Rui**

Affiliation:
Department of Mathematics, University of Shanghai for Science & Technology, Shanghai, 200093, People’s Republic of China

Email:
hbruik@online.sh.cn

DOI:
https://doi.org/10.1090/S0002-9947-98-02305-8

Received by editor(s):
September 13, 1996

Additional Notes:
Both authors gratefully acknowledge support received from Australian Research Council under Large ARC Grant A69530243. The second author is partially supported by the National Natural Science Foundation, grant no. 19501016, in P.R. China. He wishes to thank the University of New South Wales for its hospitality during his visit.

Article copyright:
© Copyright 1998
American Mathematical Society