On modules over matrix quantum pseudo-differential operators
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.
scientific article; zbMATH DE number 1781861
scientific article