A two-dimensional analogue of the Virasoro algebra

Abstract

In this article, we study the cohomology of Lie algebras of vector fields of holomorphic type Vect 1,0( M) on a complex manifold M. The main result is the introduction of a kind of order filtration on the continuous cochains on Vect 1,0( 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 the diagonal cohomology (because of the collapse of our spectral sequence) of Vect 0,1(Σ) for a compact Riemann surface Σ of genus g>0. In the second section, we calculate the first three cohomology spaces of the Lie algebra W 1⊗ C[[t]] which is regarded as the formal version of Vect 1,0(Σ). In the last section, we recall why Vect 0,1(Σ) can be regarded as the two-dimensional analogue of the Witt algebra. Then, we define, following Etingof and Frenkel, a central extension which is consequently a two-dimensional analogue of the Virasoro algebra — our cohomology calculations showing that it is a universal central extension.