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Conformally equivariant quantization : existence and uniqueness

Christian Duval;  Pierre Lecomte;  Valentin Ovsienko — 1999

Annales de l'institut Fourier

We prove the existence and the uniqueness of a conformally equivariant symbol calculus and quantization on any conformally flat pseudo-riemannian manifold ( M , g ) . In other words, we establish a canonical isomorphism between the spaces of polynomials on T * M and of differential operators on tensor densities over M , both viewed as modules over the Lie algebra o ( p + 1 , q + 1 ) where p + q = dim ( M ) . This quantization exists for generic values of the weights of the tensor densities and we compute the critical values of the weights yielding...

2-frieze patterns and the cluster structure of the space of polygons

Sophie Morier-Genoud;  Valentin Ovsienko;  Serge Tabachnikov — 2012

Annales de l’institut Fourier

We study 2-frieze patterns generalizing that of the classical Coxeter-Conway frieze patterns. The geometric realization of this space is the space of n -gons (in the projective plane and in 3-dimensional vector space) which is a close relative of the moduli space of genus 0 curves with n marked points. We show that the space of 2-frieze patterns is a cluster manifold and study its algebraic and arithmetic properties.

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