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Cellular algebras and quasi-hereditary algebras: a comparison


Authors: Steffen König and Changchang Xi
Journal: Electron. Res. Announc. Amer. Math. Soc. 5 (1999), 71-75
MSC (1991): Primary 16D80, 16G30, 20C30, 20G05; Secondary 16D25, 18G15, 20F36, 57M25, 81R05
DOI: https://doi.org/10.1090/S1079-6762-99-00063-3
Published electronically: June 24, 1999
MathSciNet review: 1696822
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Abstract | References | Similar Articles | Additional Information

Abstract: Cellular algebras have been defined in a computational way by the existence of a special kind of basis. We compare them with quasi-hereditary algebras, which are known to carry much homological and categorical structure. Among the properties to be discussed here are characterizations of quasi-hereditary algebras inside the class of cellular algebras in terms of vanishing of cohomology and in terms of positivity of the Cartan determinant.


References [Enhancements On Off] (What's this?)

  • 1. I. N. BERNSTEIN, I. M. GELFAND, AND S. I. GELFAND, A category of $\frak g$-modules. Funct. Anal. and Appl. 10, 87-92 (1976). MR 53:10880
  • 2. R. BRAUER, On algebras which are connected with the semisimple continous groups. Annals of Math. 38, 854-872 (1937).
  • 3. E. CLINE, B. PARSHALL, AND L. SCOTT, Finite dimensional algebras and highest weight categories. J. reine angew. Math., 391, 85-99 (1988). MR 90d:18005
  • 4. V. DLAB AND C. M. RINGEL, Quasi-hereditary algebras. Illinois J.Math. 33, 280-291 (1989). MR 90e:16023
  • 5. S. DONKIN, On Schur algebras and related algebras I. J. Alg. 104, 310-328 (1986). MR 89b:20084a
  • 6. J. J. GRAHAM AND G. I. LEHRER, Cellular algebras. Invent. Math. 123, 1-34 (1996). MR 97h:20016
  • 7. S. KÖNIG AND C. C. XI, On the structure of cellular algebras. Algebras and modules, II (Geiranger, 1996), 365-386, CMS Conf. Proc., 24, Amer. Math. Soc., Providence, RI, 1998. CMP 99:02
  • 8. S. KÖNIG AND C. C. XI, Cellular algebras: inflations and Morita equivalences. Preprint 97-078, Bielefeld, 1997. To appear in Journal of the London Math. Society.
  • 9. S. KÖNIG AND C. C. XI, On the number of cells of a cellular algebra. Preprint 97-126, Bielefeld, 1997. To appear in Comm. in Algebra.
  • 10. S. KÖNIG AND C. C. XI, A characteristic free approach to Brauer algebras. Preprint 98-005, Bielefeld, 1998.
  • 11. S. KÖNIG AND C. C. XI, When is a cellular algebra quasi-hereditary? Preprint 98-089, Bielefeld, 1998.
  • 12. S. KÖNIG AND C. C. XI, A self-injective cellular algebra is weakly symmetric. Preprint 99-017, Bielefeld, 1999.
  • 13. P. P. MARTIN, The structure of the partition algebras. J. Algebra 183, 319-358 (1996). MR 98g:05152
  • 14. B. PARSHALL AND L. SCOTT, Derived categories, quasi-hereditary algebras and algebraic groups. Proc. of the Ottawa-Moosonee Workshop in Algebra 1987, Math. Lecture Note Series, Carleton University and Université d'Ottawa (1988).
  • 15. H. WENZL, On the structure of Brauer's centralizer algebras. Annals of Math. 128, 173-193 (1988). MR 89h:20059
  • 16. C. C. XI, Partition algebras are cellular. To appear in Compos. Math.

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Additional Information

Steffen König
Affiliation: Fakultät für Mathematik, Universität Bielefeld, Postfach 100131, D-33501 Bielefeld, Germany
Email: koenig@mathematik.uni-bielefeld.de

Changchang Xi
Affiliation: Department of Mathematics, Beijing Normal University, 100875 Beijing, P. R. China
Email: xicc@bnu.edu.cn

DOI: https://doi.org/10.1090/S1079-6762-99-00063-3
Received by editor(s): March 15, 1999
Published electronically: June 24, 1999
Additional Notes: The research of C.C. Xi was partially supported by NSF of China (No. 19831070).
Communicated by: Dave Benson
Article copyright: © Copyright 1999 American Mathematical Society

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