Finite W-superalgebras for basic Lie superalgebras

Zeng, Yang; Shu, Bin

doi:10.1016/j.jalgebra.2015.04.034

Cited by 22 publications

(59 citation statements)

References 42 publications

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“…By construction, π B is injective. By the same discussion as in [33,Proposition 4.9], one can verify that…”

Section: The Structure Theory Of Refined W -Superalgebrasmentioning

confidence: 74%

“…This work is a sequel to [33]- [35]. In [35], we asserted that the lower bounds of modular representation dimensions stated in the abstract are attainable under the assumption that the associated complex finite W -superalgebras admit one-or two-dimensional representations.…”

Section: Introductionmentioning

confidence: 92%

“…In this section, we will recall some knowledge on basic classical Lie superalgebras and finite W -(super)algebras for use in the sequel. We refer the readers to [4], [9] and [10] for Lie superalgebras, and [22], [23], [24], [30], [33] and [35] §2], we recall the list of basic classical Lie superalgebras over F for F = C or F = k. These Lie superalgebras, with even parts being Lie algebras of reductive algebraic groups, are simple over F (the general linear Lie superalgebras, though not simple, are also included), and they admit an even non-degenerate supersymmetric invariant bilinear form in the following sense.…”

Section: Preliminariesmentioning

confidence: 99%

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Minimal $W$-Superalgebras and the Modular Representations of Basic Lie Superalgebras

Zeng

Shu

2019

Publ. Res. Inst. Math. Sci.

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Let g = g0 + g1 be a basic Lie superalgebra over C, and e a minimal nilpotent element in g0. Set W ′ χ to be the refined W -superalgebra associated with the pair (g, e), which is called a minimal W -superalgebra. In this paper we present a set of explicit generators of minimal W -superalgebras and the commutators between them. In virtue of this, we show that over an algebraically closed field k of characteristic p ≫ 0, the lower bounds of dimensions in the modular representations of basic Lie superalgebras with minimal nilpotent pcharacters are attainable. Such lower bounds are indicated in [31] as the super Kac-Weisfeiler property.1.2. The theory of finite W -superalgebras was developed in the same time. In the work of Sole and Kac [26], finite W -superalgebras were defined in terms of BRST cohomology under the background of vertex algebras and quantum reduction. The theory of finite W -superalgebras for the queer Lie superalgebras over an algebraically closed field of characteristic p > 2 was first introduced and discussed by Wang and Zhao in [32], then studied by Zhao over the field of complex numbers in [36]. The topics on finite W -superalgebras attracted many researchers, and the structure theory of W -superalgebras is developed in various articles (cf. [2], [3], [17], [18], [19] and [20], etc.).In mathematical physics, W -(super)algebras are divided into four types: classical affine, classical finite, quantum affine, and quantum finite W -(super)algebras. These types of algebras are endowed with Poisson vertex algebras, Poisson algebras, vertex algebras, and associative algebras structures, respectively. In the present paper, finite W -superalgebras will be referred to the so-called quantum finite W -(super)algebras.

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“…By construction, π B is injective. By the same discussion as in [33,Proposition 4.9], one can verify that…”

Section: The Structure Theory Of Refined W -Superalgebrasmentioning

confidence: 74%

Section: Introductionmentioning

confidence: 92%

Section: Preliminariesmentioning

confidence: 99%

See 1 more Smart Citation

Minimal $W$-Superalgebras and the Modular Representations of Basic Lie Superalgebras

Zeng

Shu

2019

Publ. Res. Inst. Math. Sci.

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“…For Q χ = U/Um ′ , we have following functors between the category C of Whittaker U-modules and the category of U(g, e)modules. This theorem was first stated in [ZS1]. Following [Lo1] we give a proof to it by using the Poisson realization given above.…”

Section: 1mentioning

confidence: 91%

Super formal Darboux-Weinstein theorem and finite W-superalgebras

Shu

Xiao

2020

Journal of Algebra

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Let v = v0+v1 be a Z 2 -graded (super) vector space with an even C *action and χ ∈ v * 0 be a fixed point of the induced action. In this paper we prove an equivariant Darboux-Weinstein theorem for the formal polynomial algebraŝ A = S[v0] ∧χ ⊗ (v1). We also give a quantum version of it. Let g = g0 + g1 be a Lie superalgebra of type I and e ∈ g0 be a nilpotent element. We use the equivariant quantum Darboux-Weinstein theorem to give a Poisson geometric realization of the finite W -superalgebra U(g, e) in sense of Losev. An indirect relation between U(g, e) and the finite W-algebra U(g0, e) is presented. Finally we use such a realization to study the finite dimensional irreducible modules of U(g, e).

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“…The general definition of a finite W -algebra was given by A. Premet in [15] (see also [7]). In the case of Lie superalgebras, finite W -algebras have been extensively studied by mathematicians and physicists in [1,2,[10][11][12][13][14][18][19][20].…”

Section: Introductionmentioning

confidence: 99%