The deformed Lie algebra of differential operators of order at most one

Abstract

In this paper, we study the algebraic properties of the deformed Lie algebra \documentclass[12pt]{minimal}\begin{document}${\mathcal L}$\end{document}L of differential operators of order at most one. We determine all the derivations, central extensions, and automorphism group of \documentclass[12pt]{minimal}\begin{document}${\mathcal L}$\end{document}L. Furthermore, the reducibility criterion for the Verma modules of the universal central extension of \documentclass[12pt]{minimal}\begin{document}${\mathcal L}$\end{document}L is achieved through a study of the Shapovalov form.