Current superalgebras and unitary representations

2017/07/02 by Neeb, Karl-Hermann, Yousofzadeh, Malihe
#FOS: Mathematics #Quantum Algebra (math.QA)

paper · doi:10.48550/arxiv.1707.00282

Abstract

In this paper we determine the projective unitary representations of finite dimensional Lie supergroups whose underlying Lie superalgebra is \frakg = A ⊗ \frakk, where \frakk 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 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 \frakk is a simple compact Lie superalgebra with \frakk₁≠ \0\, then each (projective) unitary representation of Λ_s(ℝ)⊗ \frakk factors through a (projective) unitary representation of \frakk itself, and these are known by Jakobsen's classification. If \frakk₁ = \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.

Related

• Finite dimensional compact and unitary Lie superalgebras
• A general approach to constructing minimal representations of Lie supergroups
• Classification of supersymmetries
• Categories of unitary representations of Banach-Lie supergroups and restriction functors
• Lie supergroups, unitary representations, and invariant cones