Super formal Darboux-Weinstein theorem and finite W-superalgebras

Shu, Bin; Xiao, Husileng

doi:10.1016/j.jalgebra.2020.01.007

Cited by 3 publications

(2 citation statements)

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“…As observed in [Zh,], the proof of Lemma 14 given in [Sk1] extends to basic classical Lie superalgebras; see also [ZS,Theorem 2.17], [SW,Theorem 4.1]. We formulate a version in Theorem 22 that is also applicable to the periplectic Lie superalgebras.…”

Section: Equivalence Of Categoriesmentioning

confidence: 90%

“…An analogue of Skryabin's equivalence also holds for basic classical and queer Lie superalgebras; see, e.g., [Zh,], [ZS,Theorem 2.17] and [SW,Theorem 4.1]. However, a precise connection between the category N (ζ) and the category of finite-dimensional modules over principal finite W -superalgebra for Lie superalgebras does not seem to have been described in the literature.…”

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confidence: 99%

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Whittaker Categories, Properly Stratified Categories and Fock Space Categorification for Lie Superalgebras

Chen

Cheng

Mazorchuk

2023

Commun. Math. Phys.

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We study the Whittaker category N (ζ) of the Lie superalgebra g for an arbitrary character ζ of the even subalgebra of the nilpotent radical associated with a triangular decomposition of g. We prove that the Backelin functor from either the integral subcategory or any strongly typical block of the BGG category to the Whittaker category sends irreducible modules to irreducible modules or zero. The category N (ζ) provides a suitable framework for studying finite W -superalgebras associated with an even principal nilpotent element. For the periplectic Lie superalgebras p(n), we formulate the principal finite W -superalgebras W ζ and establish a Skryabin-type equivalence. For a basic classical and a strange Lie superalgebras, we prove that the category of finite-dimensional modules over a given principal finite W -superalgebra W ζ is equivalent to N (ζ) under the Skryabin equivalence, for a non-singular character ζ. As a consequence, we give a super analogue of Soergel's Struktursatz for a certain Whittaker functor from the integral BGG category O to the category of finite-dimensional modules over W ζ . Contents 1. Introduction 2. Preliminaries 3. Blocks and simple objects in N (ζ) 4. Principal finite W -superalgebras for basic classical Lie superalgebras 5. Principal finite W -superalgebra of periplectic Lie superalgebra p(n) 6. Super Soergel Struktursatz for functor Wh ζ • Γ ζ (−) Appendix A. Whittaker categories of queer Lie superalgebras Appendix B. Equivalence of the categories N (ζ) References MSC 2010: 17B10 17B55

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Section: Equivalence Of Categoriesmentioning

confidence: 90%

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confidence: 99%