  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.

Representation Theory