Derivations from the even parts into the odd parts for Hamiltonian superalgebras

TL;DR

This paper investigates the derivations from the even parts of Hamiltonian superalgebras to the odd parts of Witt superalgebras over fields of characteristic p>3, providing a detailed decomposition and classification of these derivations.

Contribution

It introduces a decomposition of the even and odd parts of specific Lie superalgebras and classifies the derivations between them, including outer derivations, over fields of characteristic p>3.

Findings

Decomposition of the torus of $H_{\overline{0}}$ and weight space of the special subalgebra of $W_{\overline{1}}$.

Complete determination of derivations from $H_{\overline{0}}$ to $W_{\overline{1}}$, including outer derivations.

Identification of the structure of derivations in the context of Hamiltonian and Witt superalgebras.

Abstract

Let $W_{\overline{1}}$ and $H_{\overline{0}}$ denote the odd parts of the general Witt modular Lie superalgebra $W$ and the even parts of the Hamiltonian Lie superalgebra $H$ over a field of characteristic $p>3$, respectively. We give a torus of $H_{\overline{0}}$ and the weight space decomposition of the special subalgebra of $W_{\overline{1}}$ with respect to the torus. By means of the derivations of the weight 0 and three series of outer derivations from $H_{\overline{0}}$ into $W_{\overline{1}}$, the derivations from the even parts of Hamiltonian superalgebra to the odd parts of Witt superalgebra are determined.