Download PDF - Matrices over upper triangular bimodules and Δ-filtered modules over quasi-hereditary algebrasAll rights reserved. Copyright exceptions and limitations apply

ArticleOriginal scientific text

Title

Matrices over upper triangular bimodules and Δ-filtered modules over quasi-hereditary algebras

Authors 1, 2

Affiliations

  1. Fakultät für Mathematik, Universität Bielefeld, P.O. Box 100 131, D-33501 Bielefeld, Germany
  2. Mathematisches Seminar, Universität Hamburg, Bundesstr. 55, D-20146 Hamburg, Germany

Abstract

Let Λ be a directed finite-dimensional algebra over a field k, and let B be an upper triangular bimodule over Λ. Then we show that the category of B-matrices mat B admits a projective generator P whose endomorphism algebra End P is quasi-hereditary. If A denotes the opposite algebra of End P, then the functor Hom(P,-) induces an equivalence between mat B and the category ℱ(Δ) of Δ-filtered A-modules. Moreover, any quasi-hereditary algebra whose category of Δ-filtered modules is equivalent to mat B is Morita equivalent to A.

Bibliography

  1. [Ba] M. Bauch, Subspace categories as categories of good modules over quasi-hereditary algebras, Arch. Math. (Basel) 62 (1994), 112-115.
  2. [BH1] T. Brüstle and L. Hille, Finite, tame and wild actions of parabolic subgroups in GL(V) on certain unipotent subgroups, J. Algebra 226 (2000), 347-360.
  3. [BH2] T. Brüstle and L. Hille, Actions of parabolic subgroups in GLn on unipotent normal subgroups and quasi-hereditary algebras, this issue, 281-294.
  4. [BHRZ] T. Brüstle, L. Hille, G. Röhrle and G. Zwara, The Bruhat-Chevalley order of parabolic group actions in general linear groups and degeneration for delta-filtered modules, Adv. Math. 148 (1999), 203-242.
  5. [DR] V. Dlab and C. M. Ringel, The module theoretical approach to quasi-hereditary algebras, in: Representations of Algebras and Related Topics (Kyoto, 1990), London Math. Soc. Lecture Note Ser. 168, Cambridge Univ. Press, Cambridge, 1992, 200-224.
  6. [D] Yu. A. Drozd, Matrix problems, small reduction and representations of a class of mixed Lie groups, ibid., 225-249.
  7. [GR] P. Gabriel and A. V. Roiter, Representations of Finite-Dimensional Algebras, Encyclopaedia Math. Sci. 73, Algebra VIII, Springer, 1992.
  8. [HR1] L. Hille and G. Röhrle, On parabolic subgroups of classical groups with a finite number of orbits on the unipotent radical, C. R. Acad. Sci. Paris Sér. I 325 (1997), 465-470.
  9. [HR2] L. Hille and G. Röhrle, A classification of parabolic subgroups in classical groups with a finite number of orbits on the unipotent radical, Transform. Groups 4 (1999), 35-52.
  10. [K] B. Keller, Chain complexes and stable categories, Manuscripta Math. 67 (1990), 379-417.
  11. [Q] D. Quillen, Higher algebraic K-theory I, in: Algebraic K-Theory I, Proc. Conf. Battelle Inst. 1972, Lecture Notes in Math. 341, Springer, Berlin, 1973, 85-147.
  12. [R] C. M. Ringel, Tame Algebras and Integral Quadratic Forms, Lecture Notes in Math. 1099, Springer, Berlin, 1984.
  13. [S] D. Simson, Linear Representations of Partially Ordered Sets and Vector Space Categories, Algebra Logic Appl. 4, Gordon and Breach, Montreux, 1992.
  14. [T] R. Tiefenbrunner, Darstellungen von Bimodulproblemen, Diss., FU Berlin, 1995.
Colloquium Mathematicum
2000/83/2
Export to PBN, Opens in new tab
Pages:
295-303
Main language of publication
English
Received
1999-08-02
Accepted
1999-11-22
Published
2000
Exact and natural sciences