A Two Dimensional Analogue of the Virasoro Algebra

In this article, we study the cohomology of Lie algebras of vector fields of holomorphic type V ect1,0(M) on a complex manifold M . The main result is the introduction of a kind of order filtration on the continuous cochains on V ect1,0(M) and the calculation of the second term of the resulting spectral sequence. The filtration is very much in the spirit of the classical order filtration of Gelfand and Fuks, but we restrict ourselves to z-jets only for a local holomorphic coordinate z. This permits us to calculate all continuous cohomology (because of the collapse of our spectral sequence) of V ect0,1(Σ) for a compact Riemann surface Σ of genus g > 0. In a first section, we calculate the first three cohomology spaces of the Lie algebra W1 ⊗ C[[t]] which is regarded as the formal version of V ect1,0(Σ). In the last section, we recall why V ect0,1(Σ) can be regarded as the 2 dimensional analogue of the Witt algebra. Then, we define, following Etingof and Frenkel, a central extension which is consequently a 2 dimensional analogue of the Virasoro algebra our cohomology calculations showing that it is a universal central extension. Introduction The continuous cohomology of Lie algebras of C-vector fields [7], [8], [1], [6] has proven to be a subject of great geometrical interest: One of its most famous applications is the construction of the Virasoro algebra as the universal central extension of the Lie algebra of vector fields on the circle. Because of its interest in conformal field theory, it is tempting to generalize the Virasoro algebra to higher dimensions. In this perspective, the results of