Projectively invariant symbol map and cohomology of vector fields Lie algebras intervening in quantization

1996/11/18 by Lecomte, P. B. A., Ovsienko, V. Yu.
#Differential Geometry (math.DG) #FOS: Mathematics

paper · doi:10.48550/arxiv.dg-ga/9611006

Abstract

We define the unique (up to normalization) symbol map from the space of linear differential operators on Rⁿ to the space of polynomial on fibers functions on T^* Rⁿ, equivariant with respect to the Lie algebra of projective transformations sl_n+1⊂\Vect(Rⁿ). We apply the constructed sl_n+1-invariant symbol to studying of the natural one-parameter family of \Vect(M)-modules on the space of linear differential operators on an arbitrary manifold M. Each of the \Vect(M)-action from this family can be interpreted as a deformation of the standard \Vect(M)-module S(M) of symmetric contravariant tensor fields on M. We define (and calculatein the case: M= Rⁿ) the corresponding cohomology of \Vect(M) related with this deformation. This cohomology realize the obstruction for existence of equivariant symbol and quantization maps. The projective Lie algebra sl_n+1 naturally appears as the algebra of symmetries on which the involved \Vect(M)-cohomology is trivial.

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