The automorphism group of a hermitian superalgebra

Abstract

A class of Lie superalgebras, which includes the conformal superalgebra of ordinary space-time was introduced in [Sternberg and Wolf]. As they depend on the choice of a hermitian structure on the even and odd parts of a supervector space we shall call them Hermitian superalgebras. The group of automorphisms of the conformal superalgebra was determined in [Sternberg]. It was shown to have two components. In this paper we shall describe all the automorphisms of any hermitian superalgebra. We shall also show that the conformal superalgebra is the only superalgebra g=g ₀⊕g ₁ with a real eight dimensional odd subspace (g ₁=C^2,2) on which the even subalgebra (g ₀=(u(2,2)/u(1))⊕u(1)) acts irreducibly via the defining representation of u(2,2) on C^2,2 shifted by the character that makes the supertrace equal to zero. Finally, we shall give a geometrical interpretation of the connected components of the automorphism group of this superalgebra by studying the Clifford algebras C(q+2,2) for q=0,1, and 2