Conference abstracts

Session S08 - Lie Groups and Representations

July 29, 18:30 ~ 18:50

## Lie subalgebras of the matrix quantum pseudo differential operators

### CIEM- Famaf, Argentina   -   khbatistelli@gmail.com

We give a complete description of the anti-involutions that preserve the principal gradation of the algebra $\mathscr{S}_{q,N}$ of $N \times N$ matrix quantum pseudodifferential operators and we describe the Lie subalgebras of its minus fixed points. We obtain, up to conjugation, two families of anti-involutions that show quite different results when $n=N$ and $n \begin{thebibliography} \bibitem[KR]{KR} {\sc V.~G.~Kac and A. Radul, \emph{Quasifinite highest weight modules over the Lie algebra of differential operators on the circle}, Comm. Math. Phys. \textbf{157} (1993), 429--457.} \bibitem[KWY]{KWY} {\sc V.~G.~Kac, W. Wang and C. Yan, {\em Quasifinite representations of classical Lie subalgebras of$W_{1+\infty}\$} Adv. Math. {\bf 139} (1998), 56--140.}

\bibitem[BKLY]{BKLY} {\sc C. Boyallian, V. Kac, J. Liberati and C. Yan, {\em Quasifinite highest weight modules over the Lie algebra of matrix differential operators on the circle}, Journal of Math. Phys. {\bf 39} (1998), 2910--2928.}

\bibitem[BL01]{BL01} {\sc C. Boyallian and J. Liberati {\em Classical Lie subalgebras of the Lie algebra of matrix differential operators on the circle}, Journal of Math. Phys. {\bf 42} (2001), 3735-3753.}

\end{thebibliography}

Joint work with Carina Boyallian (CIEM-Famaf).