SUSY structures, representations and Peter-Weyl theorem for $S^{1|1}$

TL;DR

This paper explores the supergroup $S^{1|1}$, analyzing its structure, representation theory, and establishing a Peter-Weyl theorem, revealing its unique properties and a large class of semisimple representations.

Contribution

It characterizes $S^{1|1}$ as the unique supergroup with specific properties and develops its representation theory, including a Peter-Weyl theorem for this supergroup.

Findings

$S^{1|1}$ is the only supergroup with these properties.

A large family of complex semisimple representations is described.

The Peter-Weyl theorem is proved for $S^{1|1}$.

Abstract

The real compact supergroup $S^{1|1}$ is analized from different perspectives and its representation theory is studied. We prove it is the only (up to isomorphism) supergroup, which is a real form of $({\mathbf C}^{1|1})^\times$ with reduced Lie group $S^1$, and a link with SUSY structures on ${\mathbf C}^{1|1}$ is established. We describe a large family of complex semisimple representations of $S^{1|1}$ and we show that any $S^{1|1}$-representation whose weights are all nonzero is a direct sum of members of our family. We also compute the matrix elements of the members of this family and we give a proof of the Peter-Weyl theorem for $S^{1|1}$.