The Peter-Weyl Theorem for SU(1|1)

TL;DR

This paper extends the Peter-Weyl theorem to the supergroup SU(1|1), describing its irreducible representations, reducibility conditions, and explicit matrix elements, thereby generalizing classical harmonic analysis to a supergeometric setting.

Contribution

It provides a comprehensive analysis of SU(1|1) representations and establishes the Peter-Weyl theorem for the supercircle S^{1|2}, advancing harmonic analysis in supergeometry.

Findings

Classification of all finite-dimensional irreducible representations of SU(1|1)

Explicit computation of matrix elements for these representations

Proof of the Peter-Weyl theorem for the supercircle S^{1|2}

Abstract

We study a generalization of the results \in \cite{cfk} to the case of $SU(1|1)$ interpreted as the supercircle $S^{1|2}$. We describe all of its finite dimensional complex irreducible representations, we give a reducibility result for representations not containing the trivial character, and we compute explicitly the corresponding matrix elements. In the end we give the Peter-Weyl theorem for $S^{1|2}$.