CYCLIC HOMOLOGY OF DIFFERENTIAL-OPERATORS, THE VIRASORO ALGEBRA AND A Q-ANALOG

We show how methods from cyclic homology give easily an explicit 2-cocycle-phi on the Lie algebra of differential operators of the circle such that phi-restricts to the cocycle defining the Virasoro algebra. The same methods yield also a q-analogue of phi as well as an infinite family of linearly independent cocycles arising when the complex parameter q is a root of unity. We use an algebra of q-difference operators and q-analogues of Koszul and de Rham complexes to construct these "quantum" cocycles.