Parabolic category O for classical Lie superalgebras

Mazorchuk, Volodymyr

doi:10.1007/978-3-319-02952-8_9

Cited by 24 publications

(72 citation statements)

References 22 publications

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“…The second one (valid for arbitrary g) is as an abelian category generated by the modules induced from the underling Lie algebra. Then we determine the projective dimension of injective modules for gl(m|n) and use this to obtain the finitistic global homological dimension of the blocks, which builds on and extends some results of Mazorchuk in [CM2,Ma1,Ma2]. Concretely, we show that this global categorical invariant of the blocks is determined by the singularity of the core of the central character.…”

Section: Introductionmentioning

confidence: 72%

Homological invariants in category $\mathcal {O}$ for the general linear superalgebra

Coulembier

Serganova

2017

Trans. Amer. Math. Soc.

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We study three related homological properties of modules in the BGG category O for basic classical Lie superalgebras, with specific focus on the general linear superalgebra. These are the projective dimension, associated variety and complexity. We demonstrate connections between projective dimension and singularity of modules and blocks. Similarly we investigate the connection between complexity and atypicality. This creates concrete tools to describe singularity and atypicality as homological, and hence categorical, properties of a block. However, we also demonstrate how two integral blocks in category O with identical global categorical characteristics of singularity and atypicality will generally still be inequivalent. This principle implies that category O for gl(m|n) can contain infinitely many non-equivalent blocks, which we work out explicitly for gl(3|1). All of this is in sharp contrast with category O for Lie algebras, but also with the category of finite dimensional modules for superalgebras. Furthermore we characterise modules with finite projective dimension to be those with trivial associated variety. We also study the associated variety of Verma modules. To do this, we also classify the orbits in the cone of self-commuting odd elements under the action of an even Borel subgroup.MSC 2010 : 17B10, 16E30, 17B55

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Section: Introductionmentioning

confidence: 72%

Homological invariants in category $\mathcal {O}$ for the general linear superalgebra

Coulembier

Serganova

2017

Trans. Amer. Math. Soc.

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“…As shown in the proof of [Ma4,Theorem 3], the finitistic dimension of O λ is equal to the maximal projective dimension of an injective module in O λ and is subsequently always finite. Theorem 70 and Lemma 71 thus determine implicitly these finitistic dimensions of blocks.…”

Section: Lie Superalgebrasmentioning

confidence: 99%

Some homological properties of category 𝒪. IV

Coulembier

Mazorchuk

2016

Forum Mathematicum

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We study projective dimension and graded length of structural modules in parabolic-singular blocks of the BGG category O. Some of these are calculated explicitly, others are expressed in terms of two functions. We also obtain several partial results and estimates for these two functions and relate them to monotonicity properties for quasi-hereditary algebras. The results are then applied to study blocks of O in the context of Guichardet categories, in particular, we show that blocks of O are not always weakly Guichardet.MSC 2010 : 16E30, 17B10

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“…For a given λ ∈ Λ r,l k,ζ , we recall the definition and existence of tilting modules T r,l (λ) in O r,l k,ζ , provided by [Br3,Theorem 6.3] (also see [Mar,Section 4.3]). In the case r = (1, 1, .…”

Section: Tilting Modules In Parabolic Categoriesmentioning

confidence: 99%

Quantum group of type A and representations of queer Lie superalgebra

Chen

Cheng

2017

Journal of Algebra

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We establish a maximal parabolic version of the Kazhdan-Lusztig conjecture [CKW, Conjecture 5.10] for the BGG category O k,ζ of q(n)-modules of "±ζweights", where k ≤ n and ζ ∈ C \ 1 2 Z. As a consequence, the irreducible characters of these q(n)-modules in this maximal parabolic category are given by the Kazhdan-Lusztig polynomials of type A Lie algebras. As an application, closed character formulas for a class of q(n)-modules resembling polynomial and Kostant modules of the general linear Lie superalgebras are obtained.

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Parabolic category O for classical Lie superalgebras

Cited by 24 publications

References 22 publications

Homological invariants in category $\mathcal {O}$ for the general linear superalgebra

Homological invariants in category $\mathcal {O}$ for the general linear superalgebra

Some homological properties of category 𝒪. IV

Quantum group of type A and representations of queer Lie superalgebra

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