Super Poisson-Lie structure on \(\text{SU}(mn)\) via \(\text{SL}(m|n, \mathbb C)^{\mathbb R}\)
type: Item
instance of: scholarly article
title: Super Poisson-Lie structure on \(\text{SU}(mn)\) via \(\text{SL}(m|n, \mathbb C)^{\mathbb R}\)
zbMATH Open document ID: 1169.17014
author: N. E. Zubov
published in: Journal of Geometry and Physics
publication date: 2007-02-13T00:00:00Z
review text: 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 \(\text{SU}(m|n)\).
Mathematics Subject Classification ID: 17B60
Mathematics Subject Classification ID: 53D17
Mathematics Subject Classification ID: 58A50
zbMATH DE Number: 5125064
zbMATH Keywords: standard (graded) real supergroup
zbMATH Keywords: standard (graded) real Baxter-Lie superalgebra
zbMATH Keywords: standard (graded) real Poisson-Lie supergroup
zbMATH Keywords: standard (graded) *-structure
reviewed by: Juan Núñez Valdés
MaRDI profile type: Publication
full work available at URL: j.geomphys.2006.04.002
OpenAlex ID: W98316489
cites work: On Poisson actions of compact Lie groups on symplectic manifolds
cites work: Lie superalgebras and Poisson-Lie supergroups
cites work: A convexity theorem for Poisson actions of compact Lie groups
cites work: Field theory on a supersymmetric lattice
cites work: QUASITRIANGULAR WZW MODEL
cites work: Algebras of functions on compact quantum groups, Schubert cells and quantum tori
cites work: Poisson Lie groups, dressing transformations, and Bruhat decompositions
cites work: Irreducible representations of the osp(2,1) and spl(2,1) graded Lie algebras
cites work: Graded Lie algebras: Generalization of Hermitian representations
cites work: IWASAWA DECOMPOSITION OF THE LIE SUPERGROUP SL(n,m,ℂ)
cites work: Q4834178
cites work: Dressing transformations and Poisson group actions
cites work: Classification of real simple Lie superalgebras and symmetric superspaces
cites work: Automorphism schemes and forms of Witt Lie algebras
DOI: 10.1016/J.GEOMPHYS.2006.04.002
description: scientific article; zbMATH DE number 5125064