On modules over matrix quantum pseudo-differential operators
type: Item
instance of: scholarly article
title: On modules over matrix quantum pseudo-differential operators
zbMATH Open document ID: 1009.17017
DOI: 10.1023/A:1015757612765
author: Carina Boyallian
author: Jose I. Liberati
published in: Letters in Mathematical Physics
publication date: 2002-08-15T00:00:00Z
review text: Consider the following (commutative) diagram of Lie algebras, where horizontal arrows denote passing to an algebra of matrices and vertical ones denote quantization: \[ \begin{matrix} \widehat D & \longrightarrow & \widehat {D^M}\\ \downarrow & & \downarrow\\ \widehat{S_q} & \longrightarrow & \widehat {S^M_q} \end{matrix} \] Here \(\widehat D\) is the central extension of differential operators on the circle, \(\widehat {S_q}\) is the central extension of quantum pseudo-differential operators on the circle, \(\widehat {D^M}\) is the central extension of \(M\times M\) matrix differential operators on the circle, and \(\widehat {S^M_q}\) is the central extension of \(M\times M\) matrix quantum pseudo-differential operators on the circle. The quasifinite highest-weight modules for these algebras were described by \textit{V. Kac} and \textit{A. Radul} in [Commun. Math. Phys. 157, 429-457 (1993; Zbl 0826.17027)] for the cases of \(\widehat D\) and \(\widehat{S_q}\) and by \textit{C. Boyallian}, \textit{V. Kac}, \textit{J. Liberati}, and \textit{C. Yan} in [J. Math. Phys. 39, 2910-2928 (1998; Zbl 0999.17032)] for the case of \(\widehat {D^M}\). Here the authors make a next logical step by doing the same for \(\widehat {S^M_q}\). Both the results and methods are very similar to those in the cited works.
reviewed by: Pasha Zusmanovich
Mathematics Subject Classification ID: 17B66
Mathematics Subject Classification ID: 17B10
Mathematics Subject Classification ID: 81R10
zbMATH DE Number: 1781861
zbMATH Keywords: quantum pseudo-differential operators
zbMATH Keywords: central extension
zbMATH Keywords: highest-weight modules
MaRDI profile type: Publication
full work available at URL: a:1015757612765
OpenAlex ID: W85889637
Recommended article: Quantizations of modules of differential operators
Recommended article: Lie subalgebras of the matrix quantum pseudodifferential operators
Recommended article: Differential calculus on compact matrix pseudogroups (quantum groups)
Recommended article: Quantization and injective submodules of differential operator modules
Recommended article: Q4217445
Recommended article: Modules over quantum polynomials
Recommended article: Classification of quasifinite modules over Lie algebras of matrix differential operators on the circle
Recommended article: Differential operators over quantum spaces
Recommended article: Q3719291
Recommended article: Modules over quantized coordinate algebras and PBW-bases
description: scientific article; zbMATH DE number 1781861