No abstract

In this paper we formulate a conjecture about the minimal dimensional representations of the finite W -superalgebra U (g C , e) over the field of complex numbers and demonstrate it with examples including all the cases of type A. Under the assumption of this conjecture, we show that the lower bounds of dimensions in the modular representations of basic Lie superalgebras are attainable. Such lower bounds, as a super-version of Kac-Weisfeiler conjecture, were formulated by Wang-Zhao in [35] for the modular representations of a basic Lie superalgebra g k over an algebraically closed field k of positive characteristic p.

Abstract. Let g := gl m|n be a general linear Lie superalgebra over an algebraically closed field k = F p of characteristic p > 2. A module of g is said to be of Kac-Weisfeiler if its dimension coincides with the dimensional lower bound in the super Kac-Weisfeiler property presented by Wang-Zhao in [10]. In this paper, we verify the existence of the Kac-Weisfeiler modules for gl m|n . We also establish the corresponding consequence for the special linear Lie superalgebras sl m|n with restrictions p > 2 and p ∤ (m − n).

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.

In this paper, we study the center Z of the finite W-algebra $${\mathcal{T}}({\mathfrak{g}},e)$$ associated with a semi-simple Lie algebra $$\mathfrak{g}$$ over an algebraically closed field $$\mathbb{k}$$ of characteristic p≫0, and an arbitrarily given nilpotent element $$e \in{\mathfrak{g}} $$ We obtain an analogue of Veldkamp’s theorem on the center. For the maximal spectrum Specm(Z), we show that its Azumaya locus coincides with its smooth locus of smooth points. The former locus reflects irreducible representations of maximal dimension for $${\mathcal{T}}({\mathfrak{g}},e)$$ .