SUSY structures, representations and Peter-Weyl theorem for S1|1

The real compact supergroup S1|1 is analysed 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 (C1|1)× with reduced Lie group S1, and a link with SUSY structures on C1|1 is established. We describe a large family of complex semisimple representations of S1|1 and we show that any S1|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 S1|1.