A Twisted Approach to Kostant's Problem

Mazorchuk, Volodymyr

doi:10.1017/s0017089505002776

Cited by 16 publications

(14 citation statements)

References 28 publications

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“…The core object ∆R w (e) of our study in Section 2 has an unexpected application to the so-called Kostant's problem from [10]; see also [9,Kapitel 6]. Although there are several classes of modules for which the answer is known to be positive (see [10], [17], [18] and references therein), a complete answer to this problem seems to be far away -the problem is not even solved for simple highest weight modules. In [10, 9.5] an example of a simple highest weight module in type B 2 , for which the answer is negative is mentioned (for details see [18, 11.5]).…”

Section: Vol 91 (2008)mentioning

confidence: 96%

Categorification of Wedderburn’s basis for $${\mathbb{C}}[S_{n}]$$

Mazorchuk

Stroppel

2008

Arch. Math.

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M. Neunhöffer studies in [21] a certain basis of C[Sn]with the origins in [14] and shows that this basis is in fact Wedderburn's basis, hence decomposes the right regular representation of Sn into a direct sum of irreducible representations (i.e. Specht or cell modules). In the present paper we rediscover essentially the same basis with a categorical origin coming from projective-injective modules in certain subcategories of the BGG-category O. Inside each of these categories, there is a dominant projective module which plays a crucial role in our arguments and will additionally be used to show that Kostant's problem ([10]) has a negative answer for some simple highest weight module over the Lie algebra sl4. This disproves the general belief that Kostant's problem should have a positive answer for all simple highest weight modules in type A.

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Section: Vol 91 (2008)mentioning

confidence: 96%

Categorification of Wedderburn’s basis for $${\mathbb{C}}[S_{n}]$$

Mazorchuk

Stroppel

2008

Arch. Math.

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“…( 2) Kostant's problem for M is to determine whether the latter map is surjective. The problem is very hard and the answer is not even known for the modules L(w), w ∈ W , in the general case, although many special cases are settled (see [Jo,Ma3,MS3,MS4,Ka,KM] are references therein). Taking into account the results of the previous subsections, the main result of [KM] can be formulated as follows: After Theorem 6 the above can be reformulated in terms of the socalled double-centralizer property (see [So1,KSX,MS5]).…”

Section: 3mentioning

confidence: 99%

Some homological properties of the category 𝒪, II

Mazorchuk

2010

Represent. Theory

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Abstract. We show, in full generality, that Lusztig's a-function describes the projective dimension of both indecomposable tilting modules and indecomposable injective modules in the regular block of the BGG category O, proving a conjecture from the first paper. On the way we show that the images of simple modules under projective functors can be represented in the derived category by linear complexes of tilting modules. These complexes, in turn, can be interpreted as the images of simple modules under projective functors in the Koszul dual of the category O. Finally, we describe the dominant projective modules and also the projective-injective modules in some subcategories of O and show how one can use categorification to decompose the regular representation of the Weyl group into a direct sum of cell modules, extending the results known for the symmetric group (type A).

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“…The aim of this paper is to show that for certain values of λ, the action mapǓ q g → End L(λ) fin is surjective. Here (End L(λ) fin stands for the locally finite part of End L(λ) with respect to the adjoint action ofǓ q g. For the Lie-algebraic case (q = 1), this problem is known as the classical Kostant's problem, see [7,8,21,22]. The complete answer to it is still unknown even in the q = 1 case.…”

Section: Introductionmentioning

confidence: 99%

Equivariant quantization of Poisson homogeneous spaces and Kostant's problem

2014

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We find a partial solution to the longstanding problem of Kostant concerning description of the so-called locally finite endomorphisms of highest weight irreducible modules. The solution is obtained by means of its reduction to a far-reaching extension of the quantization problem. While the classical quantization problem consists in finding ⋆-product deformations of the commutative algebras of functions, we consider the case when the initial object is already a noncommutative algebra, the algebra of functions within q-calculus. (2000). 17B37, 17B10, 53D17, 53D55. Mathematics Subject Classifications

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A Twisted Approach to Kostant's Problem

Cited by 16 publications

References 28 publications

Categorification of Wedderburn’s basis for $${\mathbb{C}}[S_{n}]$$

Categorification of Wedderburn’s basis for $${\mathbb{C}}[S_{n}]$$

Some homological properties of the category 𝒪, II

Equivariant quantization of Poisson homogeneous spaces and Kostant's problem

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