Complete reducibility of torsion free Cn-modules of finite degree

Britten, D. J.; Khomenko, Oleksandr; Lemire, Francine; Mazorchuk, Volodymyr

doi:10.1016/j.jalgebra.2004.02.021

Cited by 18 publications

(23 citation statements)

References 7 publications

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“…In Section 5 we present some consequences of our main result. In particular, we recover the main result of [Britten et al 2004] stated above.…”

Section: Introduction and Description Of The Resultssupporting

confidence: 85%

“…In particular,Ꮿ χ ,ξ is indecomposable, hence a block. From this and [Britten et al 2004] we thus get that every nontrivial block Ꮿ χ ,ξ is equivalent to the category of finite dimensional ‫-ރ‬modules. Our main result is the following: Theorem 1.…”

Section: Introduction and Description Of The Resultsmentioning

confidence: 87%

“…The first major step towards the solution of this problem was made in [Mathieu 2000], where all simple objects inᏯ were classified. Britten et al [2004] showed that the category Ꮿ is semisimple, hence completely understood. The aim of the present note is to describe the categoryᏯ.…”

Section: Introduction and Description Of The Resultsmentioning

confidence: 99%

“…If V is a pointed simple cuspidal weight g-module, then, obviously, all nontrivial weight spaces of V are one-dimensional, in which case one says that V is completely pointed (see [Britten et al 2004]). It is enough to consider blocks with completely pointed simple modules because of the following:…”

Section: Completely Pointed Simple Cuspidal Weight Modulesmentioning

confidence: 99%

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Blocks of the category of cuspidal sp_2n-modules

Mazorchuk¹,

Stroppel²

2011

Pacific J. Math.

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In this paper we show that every block of the category of cuspidal generalized weight modules with finite dimensional generalized weight spaces over the Lie algebra sp 2n ‫)ރ(‬ is equivalent to the category of finite dimensional ‫[[ރ‬t 1 , t 2 , . . . , t n ]]-modules. Introduction and description of the resultsFix the ground field to be the complex numbers. Fix n ∈ {2, 3, . . . } and consider the symplectic Lie algebra sp 2n =: g with a fixed Cartan subalgebra h and root space decompositionwhere denotes the corresponding root system. For a g-module V and λ ∈ h * setA g-module V is called• a weight module provided that V = λ∈h * V λ ;• a generalized weight module provided that V = λ∈h * V λ ;• a cuspidal module provided that for any α ∈ the action of any nonzero element from g α on V is bijective.If V is a generalized weight module, then the set {λ ∈ h * : V λ = 0} is called the support of V and is denoted by supp(V ).Denote byᏯ the full subcategory in g-mod that consists of all cuspidal generalized weight modules with finite dimensional generalized weight spaces, and by Ꮿ the full subcategory ofᏯ consisting of all weight modules. Understanding the categories Ꮿ andᏯ is a classical problem in the representation theory of Lie algebras. The first major step towards the solution of this problem was made in [Mathieu 2000], where all simple objects inᏯ were classified. Britten et al. [2004] MSC2000: 17B10.

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“…In Section 5 we present some consequences of our main result. In particular, we recover the main result of [Britten et al 2004] stated above.…”

Section: Introduction and Description Of The Resultssupporting

confidence: 85%

Section: Introduction and Description Of The Resultsmentioning

confidence: 87%

Section: Introduction and Description Of The Resultsmentioning

confidence: 99%

Section: Completely Pointed Simple Cuspidal Weight Modulesmentioning

confidence: 99%

See 2 more Smart Citations

Blocks of the category of cuspidal sp_2n-modules

Mazorchuk¹,

Stroppel²

2011

Pacific J. Math.

Self Cite

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show abstract

“…In fact, we prove the following: Remark that the result holds trivially when θ = (this is in fact a part of the definition of the category). Note that when θ = ∅ and g is of type C , Britten, Khomenko, Lemire, Mazorchuk have proved the semisimplicity of O ,θ in [7]. The result does not hold when θ = ∅ and g is of type A.…”

Section: The Category O S θmentioning

confidence: 98%