Irreducible representations of Poisson-Lie superalgebras, and invariant differential operators
type: Item
instance of: scholarly article
title: Irreducible representations of Poisson-Lie superalgebras, and invariant differential operators
zbMATH Open document ID: 0541.58024
DOI: 10.1007/BF01083194
author: G. S. Shmelev
publication date: 1983-01-01T00:00:00Z
review text: Let the field k be either \({\mathbb{R}}\) or \({\mathbb{C}}\). We let \(\hat {\mathcal H}(M)\) denote the Lie superalgebra formed by the functions defined on a symplectic supermanifold \((M,\omega)\), dim M\(=(2n,m)\) relative to the Poisson bracket. Let the vector bundle \(E\to M\), having a finite- dimensional fiber V, be associated with the bundle of symplectic frames on M. Then the Hamiltonian vector fields on M act on the corresponding space of sections \(\Gamma\) (E). Now if \(L\to E\) is a linear bundle endowed with a connection \(\nabla\) whose curvature form is \(h\omega\), \(h\in k\), then \(\hat {\mathcal H}(M)\) acts in the space of sections of the bundle \(E\otimes L\) according to the rule \(f\to X_ f=L_ f\otimes id+id\otimes(\nabla +hf)\), where \(L_ f\) is the Hamiltonian vector field of the function f relative to the form \(\omega\). The aim of this note is to describe the differential operators c: \(\Gamma\) \((E{}_ 1\otimes L)\to \Gamma(E_ 2\otimes L)\) that commute with this action, under the assumption that the fibers of the bundles \(E_ 1\) and \(E_ 2\) are simple, finite-dimensional osp(m,2n)-modules.
Mathematics Subject Classification ID: 37J99
Mathematics Subject Classification ID: 58J70
Mathematics Subject Classification ID: 17A70
Mathematics Subject Classification ID: 53C05
zbMATH DE Number: 3860034
zbMATH Keywords: Lie superalgebra
zbMATH Keywords: symplectic supermanifold
zbMATH Keywords: Hamiltonian vector fields
zbMATH Keywords: differential operators
MaRDI profile type: Publication
cites work: Wave equation scattering in even-dimensional spaces
published in: Functional Analysis and Its Applications
description: scientific article; zbMATH DE number 3860034