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

References in corpus (4)

• On classical finite and affine W-algebras
• Dirac operators and the Very Strange Formula for Lie superalgebras
• Algebraic supergroups with Lie superalgebras of classical type
• On the construction of Chevalley Supergroups

Cited by in corpus (2)

• Real forms of complex Lie superalgebras and supergroups
• Klein and Conformal Superspaces, Split Algebras and Spinor Orbits