Current superalgebras and unitary representations

In this paper we determine the projective unitary representations of finite dimensional Lie supergroups whose underlying Lie superalgebra is $\frak{g} = A \otimes \frak{k}$,where$\frak{k}$isacompactsimpleLiesuperalgebraand$A$isasupercommutativeassociative(super)algebra;thecrucialcaseiswhen$A = \Lambda_s($\mathbb{R}$)$ is a Gra\ss{}mann algebra. Since we are interested in projective representations, the first step consists of determining the cocycles defining the corresponding central extensions. Our second main result asserts that, if $\mathfrak{k}$ is a simple compact Lie superalgebra with ${\mathfrak{k}}_1\mathrm{\ne }\{0\}$, then each (projective) unitary representation of ${\mathrm{\Lambda }}_s(\mathbb{R})\otimes \mathfrak{k}$ factors through a (projective) unitary representation of $\mathfrak{k}$ itself, and these are known by Jakobsen’s classification. If ${\mathfrak{k}}_1=\{0\}$, 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.