Generalized transvectants-Rankin-Cohen brackets
type: Item
instance of: scholarly article
title: Generalized transvectants-Rankin-Cohen brackets
zbMATH Open document ID: 1046.53005
DOI: 10.1023/A:1022956710255
author: Valentin Ovsienko
author: P. Redou
published in: Letters in Mathematical Physics
publication date: 2003-05-11T00:00:00Z
review text: In this paper the authors introduce bilinear differential operators on the space of tensor densities on \(\mathbb{R}^n\). These operators generalize the \(\roman{sl}_2\)-invariant differential operators in one-dimensional case, called transvectants, discovered by P. Gordan (in 1887). The main result of the paper is the following theorem: Let \({\mathcal{F}}_\lambda\) be the space of tensor densities of degree \(\lambda\) on \(\mathbb{R}^n\). Then: (i) For every \(k=0, 1, 2, \dots ,\) there is a \(o(p+1,q+1)\)-invariant bilinear differential operator of order \(k\) \[ B_{2k}:{\mathcal F}_\lambda\otimes{\mathcal F}_\mu \longrightarrow {\mathcal F}_{\lambda+\mu+\frac{2k}{n}}, \] unique up to a constant provided neither of \(\lambda\) and \(\mu\) belongs to the set \[ \left\{0,-\frac 1n ,-\frac 2n , \dots , \frac{2-2k}{n} \right\} \cup \left\{\frac{n-2}{2n} , \frac{n-4}{2n} , \dots , \frac{n-2k}{2n} \right\}. \] (ii) There is no other bilinear \(o(p+1,q+1)\)-invariant differential operator on tensor densities, for generic values of \(\lambda\) or \(\mu\). (Here \(n=p+q\).)
reviewed by: Marian Ioan Munteanu
Mathematics Subject Classification ID: 53A55
Mathematics Subject Classification ID: 58J70
Mathematics Subject Classification ID: 53A30
zbMATH DE Number: 1908591
zbMATH Keywords: transvectants
zbMATH Keywords: conformal structures
zbMATH Keywords: invariant differential operators
MaRDI profile type: Publication
full work available at URL: a:1022956710255
OpenAlex ID: W2123864768
description: scientific article; zbMATH DE number 1908591