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

100%

Brüstle T., Hille L.

Colloquium Mathematicum

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2000

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tom 83

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nr 2

281-294

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).

2

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

100%

Brüstle T., Hille L.

Colloquium Mathematicum

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2000

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tom 83

|

nr 2

295-303

EN

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.

3

The quasi-hereditary algebra associated to the radical bimodule over a hereditary algebra

100%

Hille L., Vossieck D.

Colloquium Mathematicum

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2003

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tom 98

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nr 2

201-211

EN

Let Γ be a finite-dimensional hereditary basic algebra. We consider the radical rad Γ as a Γ-bimodule. It is known that there exists a quasi-hereditary algebra 𝓐 such that the category of matrices over rad Γ is equivalent to the category of Δ-filtered 𝓐-modules ℱ(𝓐,Δ). In this note we determine the quasi-hereditary algebra 𝓐 and prove certain properties of its module category.

Ograniczanie wyników

3 Colloquium Mathematicum

3 Hille L.

2 Brüstle T.

1 Vossieck D.

1 2003

2 2000

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

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

The quasi-hereditary algebra associated to the radical bimodule over a hereditary algebra