Quasi-periodic functions on the torus and sl(n)-elliptic Lie algebra

Abstract

We investigate properties of the sl(n) automorphic elliptic algebra \documentclass[12pt]{minimal}\begin{document}$\mathfrak {E}(sl(n))$\end{document}E(sl(n)). We prove it to be \documentclass[12pt]{minimal}\begin{document}$\mathbb {Z}$\end{document}Z quasi-graded Lie algebra which could be viewed as a deformation of a graded loop algebra. We show that it admits the decomposition into the direct sum of two subalgebras: \documentclass[12pt]{minimal}\begin{document}$\mathfrak {E}(sl(n))= \mathfrak {E}(sl(n))_+ + \mathfrak {E}(sl(n))_-$\end{document}E(sl(n))=E(sl(n))++E(sl(n))− consistent with the described quasi-grading. We prove that \documentclass[12pt]{minimal}\begin{document}$\mathfrak {E}(sl(n))^*_{\pm }=\mathfrak {E}(sl(n))_{\mp }$\end{document}E(sl(n))±*=E(sl(n))∓, i.e., Lie algebras \documentclass[12pt]{minimal}\begin{document}$\mathfrak {E}(sl(n)),$\end{document}E(sl(n)), \documentclass[12pt]{minimal}\begin{document}$\mathfrak {E}(sl(n))_+ ,$\end{document}E(sl(n))+, and \documentclass[12pt]{minimal}\begin{document}$\mathfrak {E}(sl(n))_-$\end{document}E(sl(n))− constitute the Manin triple. We explicitly construct a central extension of \documentclass[12pt]{minimal}\begin{document}$\mathfrak {E}(sl(n))$\end{document}E(sl(n)). We find its algebra of differentiations and its central extension which coincide with the quasi-graded deformation of the Virasoro algebra.