An Algebraic Analog of the Virasoro Group

The group of diffeomorphisms of a circle is not an infinite-dimensional algebraic group, though in many ways it behaves as if it were. Here we construct an algebraic model for this object, and discuss some of its representations, which appear in the Kontsevich-Witten theory of two-dimensional topological gravity through the homotopy theory of moduli spaces. [This is a version of a talk on 23 June 2001 at the Prague Conference on Quantum Groups and Integrable Systems.] 1. Some functors from commutative rings to groups 1.1 A formal diffeomorphism of the line, with coefficients in a commutative ring A, is an element g of the ring A[[x]] of formal power series with coefficients in A, such that g(0) = 0 and g(0) is a unit. More precisely, the group of formal diffeomorphisms of the line, defined over A, is the set G(A) = {g ∈ A[[x]] | g(x) = ∑ k≥0 gkx k+1 with g0 ∈ A } . Composition g0, g1 7→ (g0 ◦ g1)(x) = g0(g1(x)) of formal power series makes this set into a monoid with e(x) = x as identity element, and it is an exercise in induction to show that such an invertible power series [ie with leading coefficient a unit] possesses a composition inverse in G(A). Thus G defines a covariant functor from commutative rings to groups; in fact this functor is representable, in the sense that G(A) is naturally isomorphic to the set of ring homomorphisms from the polynomial algebra Z[gk | k ≥ 0][g 0 ] to A. Composition endows this representing algebra with the Hopf diagonal ∆g(x) = (g ⊗ 1)((1⊗ g)(x)) , making G into a (pro-)algebraic group [9]. The kernel of the homomorphism ǫ 7→ 0 : G(A[ǫ]/(ǫ)) → G(A) can be given the structure of a Lie algebra, which is naturally isomorphic to the Lie algebra over A spanned by the differentiation operators vk = x k+1 d dx , k ≥ 0 . satisfying [vk, vl] = (l − k)vk+l. 1.2 There is a closely related functor Ǧ from commutative rings to groups, which in some ways resembles the group of diffeomorphisms of the circle. This functor is Date: 15 June 2001. 1991 Mathematics Subject Classification. 81R10, 55S25. The author was supported in part by the NSF..