Actions of parabolic subgroups in GL_n on unipotent normal subgroups and quasi-hereditary algebras

Thomas Brüstle; Lutz Hille

Colloquium Mathematicae (2000)

  • Volume: 83, Issue: 2, page 281-294
  • ISSN: 0010-1354

Abstract

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Let R be a parabolic subgroup in G L n . It acts on its unipotent radical R u and on any unipotent normal subgroup U via conjugation. Let Λ be the path algebra k t of a directed Dynkin quiver of type with t vertices and B a subbimodule of the radical of Λ viewed as a Λ-bimodule. Each parabolic subgroup R is the group of automorphisms of an algebra Λ(d), which is Morita equivalent to Λ. The action of R on U can be described using matrices over the bimodule B. The advantage of this description is that each bimodule B gives rise to an infinite number of those actions simultaneously: for each d in ℕ t we obtain a parabolic group R(d), which is the group of invertible elements in Λ(d), together with a unipotent normal subgroup U(d) in R(d). All those bimodules B are upper triangular with respect to the natural order of Λ. Then, according to [BH2], Theorem 1.1, there exists a quasi-hereditary algebra A such that the orbits of R(d) on U(d) are in bijection to the isomorphism classes of Δ-filtered A-modules of dimension vector d. We compute the quiver and relations of the quasi-hereditary algebra A corresponding to the action of the parabolic group R(d) on U(d). Moreover, we show that the Lie algebra of R(d) can be identified with the algebra Λ(d), and the Lie algebra of U(d) is isomorphic to a bimodule B(d) over Λ(d).

How to cite

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Brüstle, Thomas, and Hille, Lutz. "Actions of parabolic subgroups in GL_n on unipotent normal subgroups and quasi-hereditary algebras." Colloquium Mathematicae 83.2 (2000): 281-294. <http://eudml.org/doc/210787>.

@article{Brüstle2000,
abstract = {Let R be a parabolic subgroup in $GL_n$. It acts on its unipotent radical $R_u$ and on any unipotent normal subgroup U via conjugation. Let Λ be the path algebra $k _t$ of a directed Dynkin quiver of type with t vertices and B a subbimodule of the radical of Λ viewed as a Λ-bimodule. Each parabolic subgroup R is the group of automorphisms of an algebra Λ(d), which is Morita equivalent to Λ. The action of R on U can be described using matrices over the bimodule B. The advantage of this description is that each bimodule B gives rise to an infinite number of those actions simultaneously: for each d in $ℕ^t$ we obtain a parabolic group R(d), which is the group of invertible elements in Λ(d), together with a unipotent normal subgroup U(d) in R(d). All those bimodules B are upper triangular with respect to the natural order of Λ. Then, according to [BH2], Theorem 1.1, there exists a quasi-hereditary algebra A such that the orbits of R(d) on U(d) are in bijection to the isomorphism classes of Δ-filtered A-modules of dimension vector d. We compute the quiver and relations of the quasi-hereditary algebra A corresponding to the action of the parabolic group R(d) on U(d). Moreover, we show that the Lie algebra of R(d) can be identified with the algebra Λ(d), and the Lie algebra of U(d) is isomorphic to a bimodule B(d) over Λ(d).},
author = {Brüstle, Thomas, Hille, Lutz},
journal = {Colloquium Mathematicae},
keywords = {Dynkin quivers; parabolic subgroups; unipotent radicals; categories of matrices; quasi-hereditary algebras; orbits; directed algebras; filtered modules},
language = {eng},
number = {2},
pages = {281-294},
title = {Actions of parabolic subgroups in GL\_n on unipotent normal subgroups and quasi-hereditary algebras},
url = {http://eudml.org/doc/210787},
volume = {83},
year = {2000},
}

TY - JOUR
AU - Brüstle, Thomas
AU - Hille, Lutz
TI - Actions of parabolic subgroups in GL_n on unipotent normal subgroups and quasi-hereditary algebras
JO - Colloquium Mathematicae
PY - 2000
VL - 83
IS - 2
SP - 281
EP - 294
AB - Let R be a parabolic subgroup in $GL_n$. It acts on its unipotent radical $R_u$ and on any unipotent normal subgroup U via conjugation. Let Λ be the path algebra $k _t$ of a directed Dynkin quiver of type with t vertices and B a subbimodule of the radical of Λ viewed as a Λ-bimodule. Each parabolic subgroup R is the group of automorphisms of an algebra Λ(d), which is Morita equivalent to Λ. The action of R on U can be described using matrices over the bimodule B. The advantage of this description is that each bimodule B gives rise to an infinite number of those actions simultaneously: for each d in $ℕ^t$ we obtain a parabolic group R(d), which is the group of invertible elements in Λ(d), together with a unipotent normal subgroup U(d) in R(d). All those bimodules B are upper triangular with respect to the natural order of Λ. Then, according to [BH2], Theorem 1.1, there exists a quasi-hereditary algebra A such that the orbits of R(d) on U(d) are in bijection to the isomorphism classes of Δ-filtered A-modules of dimension vector d. We compute the quiver and relations of the quasi-hereditary algebra A corresponding to the action of the parabolic group R(d) on U(d). Moreover, we show that the Lie algebra of R(d) can be identified with the algebra Λ(d), and the Lie algebra of U(d) is isomorphic to a bimodule B(d) over Λ(d).
LA - eng
KW - Dynkin quivers; parabolic subgroups; unipotent radicals; categories of matrices; quasi-hereditary algebras; orbits; directed algebras; filtered modules
UR - http://eudml.org/doc/210787
ER -

References

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  2. [BH1] T. Brüstle and L. Hille, z Finite, tame and wild actions of parabolic subgroups in GL(V) on certain unipotent subgroups, J. Algebra 226 (2000), 347-360. Zbl0968.20023
  3. [BH2] T. Brüstle and L. Hille, Matrices over upper triangular bimodules and Δ-filtered modules over quasi-hereditary algebras, this issue, 295-303. Zbl0978.16009
  4. [BHRR] T. Brüstle, L. Hille, C. M. Ringel and G. Röhrle, Modules without selfextensions over the Auslander algebra of k [ T ] / T n , Algebras Represent. Theory 2 (1999), 295-312. Zbl0971.16007
  5. [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
  6. [DR] V. Dlab and C. M. Ringel, The module theoretical approach to quasi-hereditary algebras, in: London Math. Soc. Lecture Note Ser. 168, Cambridge Univ. Press, Cambridge, 1992, 200-224. 
  7. [GR] P. Gabriel and A. V. Roiter, Representations of Finite-Dimensional Algebras, Encyclopaedia Math. Sci. 73, Algebra VIII, Springer, 1992. 
  8. [HR] 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
  9. [Ri] R. Richardson, Conjugacy classes in parabolic subgroups of semisimple algebraic groups, Bull. London Math. Soc. 6 (1974), 21-24. Zbl0287.20036
  10. [R] C. M. Ringel, Tame Algebras and Integral Quadratic Forms, Lecture Notes in Math. 1099, Springer, Berlin, 1984. 
  11. [S] T. A. Springer, Linear Algebraic Groups, Progr. Math. 9, Birkhäuser, 1981. Zbl0453.14022

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