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Transactions of the American Mathematical Society

Published by the American Mathematical Society, the Transactions of the American Mathematical Society (TRAN) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6850 (online) ISSN 0002-9947 (print)

The 2020 MCQ for Transactions of the American Mathematical Society is 1.43.

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Based algebras and standard bases for quasi-hereditary algebras
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by Jie Du and Hebing Rui PDF
Trans. Amer. Math. Soc. 350 (1998), 3207-3235 Request permission

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
  • MR Author ID: 242577
  • 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
  • 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.
  • © Copyright 1998 American Mathematical Society
  • 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