Results

**1 - 4**of**4**### Supported by the Austrian Federal Ministry of Education, Science and Culture

"... of univalent functions ..."

### On action of the Virasoro algebra on the space of univalent functions

, 704

"... We obtain explicit expressions for differential operators defining the action of the Virasoro algebra on the space of univalent functions. We also obtain an explicit Taylor decomposition for Schwarzian derivative and a formula for the Grunsky coefficients. 0.1. Virasoro algebra. Consider the Lie alg ..."

Abstract
- Add to MetaCart

We obtain explicit expressions for differential operators defining the action of the Virasoro algebra on the space of univalent functions. We also obtain an explicit Taylor decomposition for Schwarzian derivative and a formula for the Grunsky coefficients. 0.1. Virasoro algebra. Consider the Lie algebra Vect(S1) of vector fields on the circle |z | = 1, its basis Ln: = z is numerated by integers, and n+1 d dz [Ln, Lm] = (m − n)Ln+m Recall that the Virasoro algebra Vir is the extended Lie algebra Vect(S 1); its generators are Ln, where n ranges in Z, and ζ; the commutation relations are [Ln, Lm] = (m − n)Lm+n + 1 12 (n3 − n)δm+n,0ζ (0.1) [Ln, ζ] = 0 0.2. Space of univalent functions K. Denote by K the space of all the functions f(z) = z + c1z 2 + c2z 3 +... that are univalent in the disk |z | < 1. Recall that a function f is univalent, if z = u implies f(z) = f(u). The standard references are [10], [9]. 0.3. Action of Virasoro algebra on K. According to Kirillov2 [12], [13], also [14], the Lie algebra Vect(S1) acts on K via vector fields described in the following way. Let v(z) ∂ ∂z be a real analytic vector field on the circle. Then the corresponding tangent vector at a point f(z) ∈ K is f(z) 2