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 $\mathfrak{g}=A\otimes \mathfrak{k}$, where $\mathfrak{k}$ is a compact simple Lie superalgebra and $A$ is a supercommutative associative (super)algebra; the crucial case is when $A={\Lambda }_s(\mathbb{R})$ is a Graßmann 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 ${\mathfrak{k}}_1\ne \{0\}$, then each (projective) unitary representation of ${\Lambda }_s(\mathbb{R})\otimes \mathfrak{k}$ 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.