On the cohomology and extensions of first-class $n$-Lie superalgebras

TL;DR

This paper develops the cohomology theory and extension classification for first-class $n$-Lie superalgebras, introducing $T^*$-extensions and showing their role in describing finite-dimensional nilpotent metric cases.

Contribution

It introduces the cohomology and extension theory for first-class $n$-Lie superalgebras, including the concept of $T^*$-extensions, and characterizes nilpotent metric cases via isometry to $T^*$-extensions.

Findings

Established a relation between extensions and first cohomology groups.

Defined $T^*$-extensions for first-class $n$-Lie superalgebras.

Proved all finite-dimensional nilpotent metric cases are isometric to $T^*$-extensions.

Abstract

An $n$-Lie superalgebra of parity 0 is called a first-class $n$-Lie superalgebra. In this paper, we give the representation and cohomology for a first-class $n$-Lie superalgebra and obtain a relation between extensions of a first-class $n$-Lie superalgebra $\mathfrak{b}$ by an abelian one $\mathfrak{a}$ and $Z^1(\mathfrak{b}, \mathfrak{a})_{\bar{0}}$. We also introduce the notion of $T^*$-extensions of first-class $n$-Lie superalgebras and prove that every finite-dimensional nilpotent metric first-class $n$-Lie superalgebra $(\g,< ,>_{\g})$ over an algebraically closed field of characteristic not 2 is isometric to a suitable $T^*$-extension.