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

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dc.creatorLecomte, P. B. A.
dc.creatorOvsienko, V. Yu.
dc.date1996-11-18
dc.date.accessioned2026-07-07T09:12:54Z
dc.date.available2026-07-07T09:12:54Z
dc.descriptionWe define the unique (up to normalization) symbol map from the space of linear differential operators on $R^n$ to the space of polynomial on fibers functions on $T^* R^n$, equivariant with respect to the Lie algebra of projective transformations $sl_{n+1}\subset\Vect(R^n)$. 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: $M= R^n$) 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.
dc.description32 pages, LaTex
dc.identifierhttps://arxiv.org/abs/dg-ga/9611006
dc.identifierhttp://arxiv.org/abs/dg-ga/9611006
dc.identifier.urihttp://salesiana.dossiersoluciones.com/handle/123456789/152183
dc.subjectDifferential Geometry
dc.titleProjectively invariant symbol map and cohomology of vector fields Lie algebras intervening in quantization
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