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2000 | 83 | 2 | 281-294
Tytuł artykułu

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

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EN
Abstrakty
EN
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 $k 𝔸_t$ of a directed Dynkin quiver of type 𝔸 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).
Słowa kluczowe
Rocznik
Tom
83
Numer
2
Strony
281-294
Opis fizyczny
Daty
wydano
2000
otrzymano
1999-08-02
poprawiono
1999-11-22
Twórcy
  • Fakultät für Mathematik, Universität Bielefeld, P.O. Box 100 131, D-33501 Bielefeld, Germany
autor
  • Mathematisches Seminar, Universität Hamburg, Bundesstr. 55, D-20146 Hamburg, Germany
Bibliografia
  • [B] N. Bourbaki, Groupes et algèbres de Lie, Chaps. 4-6, Hermann, Paris, 1975.
  • [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.
  • [BH2] T. Brüstle and L. Hille, Matrices over upper triangular bimodules and Δ-filtered modules over quasi-hereditary algebras, this issue, 295-303.
  • [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.
  • [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
  • [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.
  • [GR] P. Gabriel and A. V. Roiter, Representations of Finite-Dimensional Algebras, Encyclopaedia Math. Sci. 73, Algebra VIII, Springer, 1992.
  • [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.
  • [Ri] R. Richardson, Conjugacy classes in parabolic subgroups of semisimple algebraic groups, Bull. London Math. Soc. 6 (1974), 21-24.
  • [R] C. M. Ringel, Tame Algebras and Integral Quadratic Forms, Lecture Notes in Math. 1099, Springer, Berlin, 1984.
  • [S] T. A. Springer, Linear Algebraic Groups, Progr. Math. 9, Birkhäuser, 1981.
Typ dokumentu
Bibliografia
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bwmeta1.element.bwnjournal-article-cmv83i2p281bwm
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