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

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 $T^{\ast }$-extensions of first-class $n$-Lie superalgebras and prove that every finite-dimensional nilpotent metric first-class $n$-Lie superalgebra $(\text{g},<,{>}_{\text{g}})$ over an algebraically closed field of characteristic not 2 is isometric to a suitable $T^{\ast }$-extension.