Finite $W$-Superalgebras and Dimensional Lower Bounds for the Representations of Basic Lie Superalgebras

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.

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

“…In §2, some basics on Lie superalgebras and finite W -superalgebras are recalled. In §3, we first study the construction of refined reduced W -superalgebra (Q χ χ ) ad m ′ k over k, and then reformulate the PBW theorem for refined W -superalgebra W ′ χ over C. In the new setting-up of refined W -superalgebras, we refine the conjecture [35,Conjecture 1.3] in §4. We first introduce Conjecture 4.2, which is irrelevant to the judging parity we mentioned above.…”

“…For the case g is of type A(m, n), Conjecture 1.1 was confirmed in [35,Proposition 4.7], which was accomplished by conversion from the verification of the attainableness of lower-bounds of modular dimensions for basic Lie superalgebras of the same type by some direct computation; see [34] for more details. For the case g is of type B(0, n) with e being a regular nilpotent element in g, we certified Conjecture 1.1 in [35,Proposition 5.8]. In the present paper, we will certify this conjecture for minimal nilpotent elements.…”

“…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.…”

“…When we turn to finite-dimensional representations of finite W -superalgebras over complex numbers, their minimal dimensions will be crucial to small representations of modular Lie superalgebras. Under an assumption on the minimal dimensions of representations for complex finite W -superalgebrs, we proved in [35,Theorem 1.6] the accessibility of lower-bounds of dimensions for modular representations of basic Lie superalgebras (see the next subsection). Such an assumption is also predicted to be true, as a conjecture listed below (as an analogy of Premet's work on finite W -algebras).…”

Super formal Darboux-Weinstein theorem and finite W-superalgebras

“…Secondly we use it to study finite dimensional representation of U(g, e). In some special case, the finite dimensional representations of U(g, e) were studied in [BG], [PS1] and [ZS2]. Their result are relying on some explicit presentation of U(g, e) by generators and relations or Super-Yangian realization.…”