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

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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 b by an abelian one a and Z¹(b, a)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 (,<,>) over an algebraically closed field of characteristic not 2 is isometric to a suitable T*-extension.