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

Colloquium Mathematicae (2000)

- Volume: 83, Issue: 2, page 295-303
- ISSN: 0010-1354

## Access Full Article

top## Abstract

top## How to cite

topBrüstle, Thomas, and Hille, Lutz. "Matrices over upper triangular bimodules and Δ-filtered modules over quasi-hereditary algebras." Colloquium Mathematicae 83.2 (2000): 295-303. <http://eudml.org/doc/210788>.

@article{Brüstle2000,

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.},

author = {Brüstle, Thomas, Hille, Lutz},

journal = {Colloquium Mathematicae},

keywords = {categories of representations; quasi-hereditary algebras; orbits; Euler forms; directed algebras; exact categories; triangular matrix algebras; categories of prinjective modules; category equivalences},

language = {eng},

number = {2},

pages = {295-303},

title = {Matrices over upper triangular bimodules and Δ-filtered modules over quasi-hereditary algebras},

url = {http://eudml.org/doc/210788},

volume = {83},

year = {2000},

}

TY - JOUR

AU - Brüstle, Thomas

AU - Hille, Lutz

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

JO - Colloquium Mathematicae

PY - 2000

VL - 83

IS - 2

SP - 295

EP - 303

AB - 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.

LA - eng

KW - categories of representations; quasi-hereditary algebras; orbits; Euler forms; directed algebras; exact categories; triangular matrix algebras; categories of prinjective modules; category equivalences

UR - http://eudml.org/doc/210788

ER -

## References

top- [Ba] M. Bauch, Subspace categories as categories of good modules over quasi-hereditary algebras, Arch. Math. (Basel) 62 (1994), 112-115. Zbl0815.16005
- [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. Zbl0968.20023
- [BH2] T. Brüstle and L. Hille, Actions of parabolic subgroups in $G{L}_{n}$ on unipotent normal subgroups and quasi-hereditary algebras, this issue, 281-294. Zbl0978.16008
- [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. Zbl0953.20037
- [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.
- [D] Yu. A. Drozd, Matrix problems, small reduction and representations of a class of mixed Lie groups, ibid., 225-249. Zbl0829.16009
- [GR] P. Gabriel and A. V. Roiter, Representations of Finite-Dimensional Algebras, Encyclopaedia Math. Sci. 73, Algebra VIII, Springer, 1992.
- [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. Zbl0922.20049
- [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. Zbl0924.20035
- [K] B. Keller, Chain complexes and stable categories, Manuscripta Math. 67 (1990), 379-417. Zbl0753.18005
- [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. Zbl0292.18004
- [R] C. M. Ringel, Tame Algebras and Integral Quadratic Forms, Lecture Notes in Math. 1099, Springer, Berlin, 1984.
- [S] D. Simson, Linear Representations of Partially Ordered Sets and Vector Space Categories, Algebra Logic Appl. 4, Gordon and Breach, Montreux, 1992. Zbl0818.16009
- [T] R. Tiefenbrunner, Darstellungen von Bimodulproblemen, Diss., FU Berlin, 1995.

## Citations in EuDML Documents

top## NotesEmbed ?

topTo embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.