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 $\frak{g} = A \otimes \frak{k}$, where $\frak{k}$ 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 $\frak{k}$ is a simple compact Lie superalgebra with $\frak{k}_1\neq \{0\}$, then each (projective) unitary representation of $\Lambda_s(\mathbb{R})\otimes \frak{k}$ factors through a (projective) unitary representation of $\frak{k}$ itself, and these are known by Jakobsen's classification. If $\frak{k}_1 = \{0\}$, 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.
- Publication:
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arXiv e-prints
- Pub Date:
- July 2017
- DOI:
- 10.48550/arXiv.1707.00282
- arXiv:
- arXiv:1707.00282
- Bibcode:
- 2017arXiv170700282N
- Keywords:
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- Mathematics - Quantum Algebra