2-cocycles on the algebra of differential operators
type: Item
instance of: scholarly article
title: 2-cocycles on the algebra of differential operators
zbMATH Open document ID: 0671.17010
DOI: 10.1016/0021-8693(89)90237-8
author: Wanglai Li
published in: Journal of Algebra
publication date: 1989-01-01T00:00:00Z
review text: Let \({\mathbb{C}}[t,t^{-1}]\) be the Laurent polynomial ring over the complex field \({\mathbb{C}}\), and \({\mathfrak a}={\mathbb{C}}[t,t^{-1},D]\) the algebra of differential operators on \({\mathbb{C}}[t,t^{-1}],\) \({\mathfrak A}={\mathfrak a}\otimes gl_ n({\mathbb{C}}).\) We know that \[ \alpha (t^{\ell +m} D^{\ell}, t^{\ell '+m'} D^{\ell '})=\delta_{m,-m'} (-1)^{\ell} \ell ! \ell '! \left( \begin{matrix} m+\ell \\ \ell +\ell '\quad +1\end{matrix} \right) \] is a nontrivial 2-cocycle on \({\mathfrak a}\) and \[ \psi (t^{\ell +m} D^{\ell}\otimes A, t^{\ell '+m'} D^{\ell '}\otimes \quad A')=\delta_{m,-m'} (-1)^{\ell} \ell ! \ell '! \left( \begin{matrix} m+\ell \\ \ell +\ell '+1\end{matrix} \right)\quad tr AA' \] is a nontrivial 2-cocycle on \({\mathfrak A}\). In this paper it is proved that every 2-cocycle on \({\mathfrak a}\) is equivalent to a multiple of \(\alpha\) and every 2- cocycle on \({\mathfrak A}\) is equivalent to a multiple of \(\psi\), solving problems proposed by V. G. Kac.
Mathematics Subject Classification ID: 17B65
Mathematics Subject Classification ID: 17B56
zbMATH DE Number: 4097594
zbMATH Keywords: central extension
zbMATH Keywords: infinite dimensional Lie algebra
zbMATH Keywords: cohomology
zbMATH Keywords: Laurent polynomial ring
zbMATH Keywords: algebra of differential operators
zbMATH Keywords: 2-cocycle
MaRDI profile type: Publication
full work available at URL: 0021-8693(89)90237-8
OpenAlex ID: W1989199536
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description: scientific article; zbMATH DE number 4097594