Super Poisson-Lie structure on \(\text{SU}(mn)\) via \(\text{SL}(m|n, \mathbb C)^{\mathbb R}\)
A Poisson-Lie group \(G\) is a Lie group equipped with a Poisson bracket compatible with the group multiplication. These groups naturally come in dual pairs \((G, \tilde{G})\) where the Poisson-Lie bracket on \(G(\tilde{G})\) defines the multiplication on \(\tilde{G}(G)\).There may exist several Poisson-Lie structures on a given Lie group \(G\) and thus, there exist several dual pairs \((G, \tilde{G})\) where \(G\) is fixed and \(\tilde{G}\) varies. If \(G\) is compact, the canonical Poisson-Lie structure on \(G\) appears in many applications in mathematics and in mathematical physics. In this paper, the author wants to construct its superanalogue, that is, to consider a simple compact supergroup and to find out a Poisson bracket on it which would be compatible with the supergroup multiplication. Indeed, the paper deals with a super Poisson-Lie structure on the real Lie supergroup \(\text{SU}(m|n)\) via \(\text{SL}(m|n, \mathbb C)^{\mathbb R}\). He shows that the dual Poisson-Lie supergroup of \(\text{SU}(m|n)\) is \(s(AN)\), and conversely, he also obtains that \(s(AN)\) inherits a super Poisson-Lie structure from the realification of \(\text{SL}(m|n, \mathbb C)\) such that its dual Poisson-Lie supergroup is \(\text{SU}(m|n)\).
standard (graded) real Baxter-Lie superalgebra
standard (graded) real Poisson-Lie supergroup
scientific article; zbMATH DE number 5125064