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

We investigate properties of the sl(n) automorphic elliptic algebra c{fraktur}(sl(n)). We prove it to be 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: c{fraktur}(sl(n)) = c{fraktur}(sl(n)) + + c{fraktur}(sl(n)) - consistent with the described quasi-grading. We prove that c{fraktur}(sl(n)) ± * = c{fraktur}(sl(n)) ±, i.e., Lie algebras c{fraktur}(sl(n)), c{fraktur}(sl(n)) + c{fraktur}(sl(n)) - and constitute the Manin triple. We explicitly construct a central extension of c{fraktur}(sl(n)). We find its algebra of differentiations and its central extension which coincide with the quasi-graded deformation of the Virasoro algebra. © 2012 American Institute of Physics.