Current superalgebras and unitary representations

In this paper we determine the projective unitary representations of finite dimensional Lie supergroups whose underlying Lie superalgebra is g=A⊗k, where k is a compact simple Lie superalgebra and A is a supercommutative associative (super)algebra; the crucial case is when A=Λs(R) is a Graßmann algebra. Since we are interested in projective representations, the first step consists in determining the cocycles defining the corresponding central extensions. Our second main result asserts that, if k is a simple compact Lie superalgebra with k1≠{0}, then each (projective) unitary representation of Λs(R)⊗k factors through a (projective) unitary representation of k itself, and these are known by Jakobsen's classification. If k1={0}, then we likewise reduce the classification problem to semidirect products of compact Lie groups K with a Clifford-Lie supergroup which has been studied by Carmeli, Cassinelli, Toigo and Varadarajan.