Infinite-dimensional Hamiltonian Lie superalgebras

Abstract

The natural filtration of the infinite-dimensional Hamiltonian Lie superalgebra over a field of positive characteristic is proved to be invariant under automorphisms by characterizing ad-nilpotent elements. We are thereby able to obtain an intrinsic characterization of the Hamiltonian Lie superalgebra and establish a property of the automorphisms of the Lie superalgebra.The theory of Lie superalgebras over fields of characteristic zero has seen a significant development; especially, Kac [6, 7] had classified the finite-dimensional simple Lie superalgebras and infinite-dimensional simple linearly compact Lie superalgebras. However, the classification problem is still open for the finitedimensional simple Lie superalgebras over fields of positive characteristic (see [2,19] for example). For the earliest papers on modular Lie superalgebras, the reader is refereed to [8,13]. 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. The natural filtrations of finite-dimensional modular Lie algebras of Cartan type were proved to be invariant in [5,9]. In the infinite-dimensional case, the same conclusion was proved in [4], by determining ad-nilpotent elements. In the finite-dimensional modular Lie superalgebras case, the invariance of the natural filtration was solved in [16,17]. In the case of infinite-dimensional modular Lie superalgebras of Cartan type, similar results for the generalized Witt superalgebra, the special superalgebra and the odd Hamiltonian superalgebra were obtained in [10,18], respectively.In the present paper, we consider the infinite-dimensional modular Hamiltonian superalgebra H(m, n), which is analogous to the one in the nonmodular situation (see [7]). We prove that the natural filtration of * Corresponding author