Remote Access Electronic Research Announcements

Electronic Research Announcements

ISSN 1079-6762



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
Published electronically: June 24, 1999
MathSciNet review: 1696822
Full-text PDF Free Access

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?)

  • I. N. Bernšteĭn, I. M. Gel′fand, and S. I. Gel′fand, A certain category of ${\mathfrak g}$-modules, Funkcional. Anal. i Priložen. 10 (1976), no. 2, 1–8 (Russian). MR 0407097
  • R. Brauer, On algebras which are connected with the semisimple continous groups. Annals of Math. 38, 854–872 (1937).
  • E. Cline, B. Parshall, and L. Scott, Finite-dimensional algebras and highest weight categories, J. Reine Angew. Math. 391 (1988), 85–99. MR 961165
  • Vlastimil Dlab and Claus Michael Ringel, Quasi-hereditary algebras, Illinois J. Math. 33 (1989), no. 2, 280–291. MR 987824
  • S. Donkin, On Schur algebras and related algebras. I, J. Algebra 104 (1986), no. 2, 310–328. MR 866778, DOI
  • J. J. Graham and G. I. Lehrer, Cellular algebras, Invent. Math. 123 (1996), no. 1, 1–34. MR 1376244, DOI
  • 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.
  • 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.
  • 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.
  • S. König and C. C. Xi, A characteristic free approach to Brauer algebras. Preprint 98–005, Bielefeld, 1998.
  • S. König and C. C. Xi, When is a cellular algebra quasi-hereditary? Preprint 98–089, Bielefeld, 1998.
  • S. König and C. C. Xi, A self-injective cellular algebra is weakly symmetric. Preprint 99–017, Bielefeld, 1999.
  • Paul Martin, The structure of the partition algebras, J. Algebra 183 (1996), no. 2, 319–358. MR 1399030, DOI
  • 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).
  • Hans Wenzl, On the structure of Brauer’s centralizer algebras, Ann. of Math. (2) 128 (1988), no. 1, 173–193. MR 951511, DOI
  • C. C. Xi, Partition algebras are cellular. To appear in Compos. Math.

Similar Articles

Retrieve articles in Electronic Research Announcements of the American Mathematical Society with MSC (1991): 16D80, 16G30, 20C30, 20G05, 16D25, 18G15, 20F36, 57M25, 81R05

Retrieve articles in all journals with MSC (1991): 16D80, 16G30, 20C30, 20G05, 16D25, 18G15, 20F36, 57M25, 81R05

Additional Information

Steffen König
Affiliation: Fakultät für Mathematik, Universität Bielefeld, Postfach 100131, D-33501 Bielefeld, Germany
MR Author ID: 263193

Changchang Xi
Affiliation: Department of Mathematics, Beijing Normal University, 100875 Beijing, P. R. China

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