The algebra of the projective line and cohomology of Diff(S1)

Abstract

Geometry and algebra can be hardly separated: “ L'algebre n'est qu'une geometrie ecrite; la geometrie n'est qu'une algebre figuree” . Geometric objects usually form algebras, such as Lie algebras of vector fields, associative algebras of differential operators, etc. In this chapter we consider the associative algebra of differential operators on the projective line. Projective geometry allows us to describe this complicated and interesting object in terms of tensor densities. The group Diff + ( S 1 ) and its cohomology play a prominent role, unifying different aspects of our study. The group Diff + ( S 1 ) of orientation-preserving diffeomorphisms of the circle is one of the most popular infinite-dimensional Lie groups connected to numerous topics in contemporary mathematics. The corresponding Lie algebra, Vect( S 1 ), also became one of the main characters in various areas of mathematical physics and differential geometry. Part of the interest in the cohomology of Vect( S 1 ) and Diff( S 1 ) is due to the existence of their non-trivial central extensions, the Virasoro algebra and the Virasoro group. We consider the first and the second cohomology spaces of Diff( S 1 ) and Vect( S 1 ) with some non-trivial coefficients and investigate their relations to projective differential geometry. Cohomology of Diff( S 1 ) and Vect( S 1 ) has been studied by many authors in many different settings; see [72] and [90] for comprehensive references. Why should we consider this cohomology here? The group Diff( S 1 ), the Lie algebra Vect( S 1 ) and the Virasoro algebra appear consistently throughout this book.