Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 5 (2009), 032, 11 pages      arXiv:0903.2647
Contribution to the Special Issue on Kac-Moody Algebras and Applications

Vector Fields on the Space of Functions Univalent Inside the Unit Disk via Faber Polynomials

Helene Airault
LAMFA CNRS UMR 6140, Insset, Université de Picardie Jules Verne, 48 rue Raspail, 02100 Saint-Quentin (Aisne), France

Received July 17, 2008, in final form March 07, 2009; Published online March 15, 2009

We obtain the Kirillov vector fields on the set of functions f univalent inside the unit disk, in terms of the Faber polynomials of 1/f(1/z). Our construction relies on the generating function for Faber polynomials.

Key words: vector fields; univalent functions; Faber polynomials.

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  1. Airault H., Malliavin P., Unitarizing probability measures for representations of Virasoro algebra, J. Math. Pures Appl. (9) 80 (2001), 627-667.
  2. Airault H., Ren J., An algebra of differential operators and generating functions on the set of univalent functions, Bull. Sci. Math. 126 (2002), 343-367.
  3. Airault H., Neretin Yu.A., On the action of Virasoro algebra on the space of univalent functions, Bull. Sci. Math. 132 (2008), 27-39, arXiv:0704.2149.
  4. Kirillov A.A., Geometric approach to discrete series of unirreps for Vir, J. Math. Pures Appl. (9) 77 (1998), 735-746.
  5. Kirillov A.A., Yur'ev D.V., Kähler geometry of the infinite-dimensional homogeneous space M = Diff+(S1)/Rot(S1), Funktsional. Anal. i Prilozhen. 21 (1987), no. 4, 35-46.
  6. Neretin Yu.A., Representations of Virasoro and affine Lie algebras, in Representation Theory and Noncommutative Harmonic Analysis, I, Encyclopaedia Math. Sci., Vol. 22, Springer, Berlin, 1994, 157-234.
  7. Schaeffer A.C., Spencer D.C., Coefficients regions for schlicht functions, American Mathematical Society Colloquium Publications, Vol. 35, New York, 1950.
  8. Schiffer M., Faber polynomials in the theory of univalent functions, Bull. Amer. Math. Soc. 54 (1948), 503-517.

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