Current superalgebras and unitary representations

Abstract 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 = Λ s ( R ) is a Grasmann 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 k 1 ≠ { 0 } , then each (projective) unitary representation of Λ s ( R ) ⊗ k factors through a (projective) unitary representation of k itself, and these are known by Jakobsen's classification. If k 1 = { 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.