A family of transitive modular Lie superalgebras with depth one

Abstract

The embedding theorem is established for Z-graded transitive modular Lie superalgebras g = ⊕ −1 i r gi satisfying the conditions:(i) g0 p(g−1) and g0-module g−1 is isomorphic to the natural p(g−1)-module;(ii) dim g1 = 2 3 n(2n 2 + 1), where n = 1 2 dim g−1. In particular, it is proved that the finite-dimensional simple modular Lie superalgebras satisfying the conditions above are isomorphic to the odd Hamiltonian superalgebras. The restricted Lie superalgebras are also considered. Keywords: flag, divided power algebra, modular Lie superalgebra, embedding theorem MSC(2000): 17B50, 17B05 IntroductionWe know that the Z-graded transitive Lie (super) algebras with depth one play an important role in the classification problem (see [1][2][3][4] for examples). In this paper we are interested in describing a family of the transitive modular Lie superalgebras g = ⊕ −1 i r g i satisfying the conditions in the abstract. More precisely, we shall establish the embedding mappings from these transitive Lie superalgebras into the so-called odd Hamiltonian superalgebras over the fields of prime characteristic, which are analogous to the Leites superalgebras [5,6] or the odd Hamiltonian superalgebras [2] in characteristic zero; they are also called by physicists the Batalin-Vilkoviski algebras. In a modular case, following [2] we call these Lie superalgebras the odd Hamiltonian modular superalgebrs (see sec. 2 for a definition), denoted by H. We should mention that the finite-dimensional simple odd Hamiltonian modular superalgebras have no analogs in the Lie algebras or in the finite-dimensional simple Lie superalgebras in characteristic 0 (cf. [1,2,7]). However, the present paper is motivated by the results and the methods in the modular Lie algebra case and the authors benefit much from reading [3,8,9].As well known, the theory of the Lie superalgebras in characteristic zero has seen a remarkable evolution (see [1,2,10], for examples). But the results in the modular Lie superalgebra case seem not so plentiful. For example, the classification problem remains open for the finite-dimensional

Infinite-dimensional Hamiltonian Lie superalgebras

“…As in the situation of modular Lie algebras or non-modular Lie superalgebras, the Lie superalgebras of Cartan type are of great importance in the study of modular Lie superalgebras. Recent works on modular Lie superalgebras can be found in [3,11,12].It is well known that filtration structures play an important role in the classification of modular Lie algebras (see [1,15]) and non-modular Lie superalgebras (see [6,14]), respectively. We know that the Lie algebras and Lie superalgebras of Cartan type possess a natural filtration structure.…”

“…As in the situation of modular Lie algebras or non-modular Lie superalgebras, the Lie superalgebras of Cartan type are of great importance in the study of modular Lie superalgebras. Recent works on modular Lie superalgebras can be found in [3,11,12].…”