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Zbl 1009.17017
Boyallian, Carina; Liberati, Jose I.
On modules over matrix quantum pseudo-differential operators. (English)
[J] Lett. Math. Phys. 60, No.1, 73-85 (2002). ISSN 0377-9017; ISSN 1573-0530

Consider the following (commutative) diagram of Lie algebras, where horizontal arrows denote passing to an algebra of matrices and vertical ones denote quantization: $$\matrix \widehat D & \longrightarrow & \widehat {D^M}\\ \downarrow & & \downarrow\\ \widehat{S_q} & \longrightarrow & \widehat {S^M_q} \endmatrix$$ 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 {\it V. Kac} and {\it 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 {\it C. Boyallian}, {\it V. Kac}, {\it J. Liberati}, and {\it 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.
[Pasha Zusmanovich (Amsterdam)]

MSC 2000:: *17B66 Lie algebras of vector fields and related algebras
17B10 Representations of Lie algebras, algebraic theory
81R10 Repres. of infinite-dim. groups and algebras from quantum theory

Keywords: quantum pseudo-differential operators; central extension; highest-weight modules

Citations: Zbl 0826.17027; Zbl 0999.17032

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