On Poisson pairs associated to modified \(R\)-matrices
type: Item
instance of: scholarly article
title: On Poisson pairs associated to modified \(R\)-matrices
zbMATH Open document ID: 0824.58022
DOI: 10.1215/S0012-7094-94-07311-0
author: Dmitri I. Panyushev
author: Dimitri Gurevich
published in: Duke Mathematical Journal
publication date: 1995-11-09T00:00:00Z
review text: Let \(g\) be a finite dimensional semi-simple complex Lie algebra with adjoint group \(G\). Let \({\mathcal O} \subset g^*\) be a coadjoint orbit of \(G\) and \(r \in g \wedge g\) satisfy \[ [r_{12}, r_{13}] + [r_{12}, r_{23}] + [r_{13}, r_{23}] = z \phi,\quad z \in \mathbb{C}^*,\quad \phi \in (\wedge^ 3 g)^ G. \] The authors classify those orbits \(\mathcal O\) for which the skew biderivation of \(C^ \bullet ({\mathcal O})\) \[ \{f,g\}_ r = m(df \otimes dg (\rho \otimes \rho(r))),\quad f,g \in C^ \bullet ({\mathcal O}) \] satisfies the Jacobi identity. Here \(m\) is multiplication in \(C^ \bullet ({\mathcal O})\) and \(\rho : g \to \text{Vect} ({\mathcal O})\) is the infinitesimal left \(g\)-action on \(\mathcal O\). The compatibility of \(\{ , \}_ r\) with the Kostant-Kirillov-Souriau symplectic structure of \(\mathcal O\) is used to recover a result of \textit{S. Khoroshkin, A. Radul} and \textit{V. Rubtsov} [Commun. Math. Phys. 152, No. 2, 299-315 (1993; Zbl 0785.58024)] according to which for \(\mathcal O\) hermitian symmetric, the reduced Sklyanin bracket is compatible with the KKS structure.
Mathematics Subject Classification ID: 37J99
Mathematics Subject Classification ID: 17B37
Mathematics Subject Classification ID: 53C30
zbMATH DE Number: 572060
zbMATH Keywords: \(R\)-matrix
zbMATH Keywords: Poisson brackets
zbMATH Keywords: homogeneous spaces
reviewed by: Serge Parmentier
MaRDI profile type: Publication
cites work: Solutions of the classical Yang-Baxter equation for simple Lie algebras
cites work: Some Poisson structures associated to Drinfeld-Jimbo \(R\)-matrices and their quantization
cites work: Q5285803
cites work: Quantization of Poisson pairs: The \(R\)-matrix approach
cites work: A family of Poisson structures on Hermitian symmetric spaces
cites work: Some remarks on nilpotent orbits
cites work: Complexity and nilpotent orbits
cites work: Q5662587
full work available at URL: s0012-7094-94-07311-0
OpenAlex ID: W2047685394
Recommended article: Semisimple symplectic symmetric spaces
Recommended article: Algebraic deformation program on minimal nilpotent orbit
Recommended article: On extensions of canonical symplectic structure from coadjoint orbit of complex general linear group
Recommended article: Symplectic orbits in quantum state space
Recommended article: Q3319456
Recommended article: On the exactness of Kostant-Kirillov form and the second cohomology of nilpotent orbits
Recommended article: Weyl quantization for semidirect products
Recommended article: Semidirect products and the Pukanszky condition
Recommended article: Q4332797
Recommended article: Coadjoint orbits in duals of Lie algebras with admissible ideals
description: scientific article; zbMATH DE number 572060