Projectively equivariant symbol calculus

1998/09/11 by P. B. A. Lecomte, Pierre Lecomte, Lecomte, P. B. A. +3
Mathematics · Physics and Astronomy · #Advanced Topics in Algebra #Algebraic structures and combinatorial models #Differential Geometry (math.DG) #FOS: Mathematics #Nonlinear Waves and Solitons #Quantum Algebra (math.QA) #math.DG #math.QA

paper · pdf · doi:10.48550/arxiv.math/9809061

23 pages, LaTeX This article is a revised version of the electronic preprint dg-ga/9611006

arxiv created 1998/09/11 · openalex publication_date 1998/09/11 · arxiv updated 2009/11/30 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

The spaces of linear differential operators on ℝⁿ acting on tensor densities of degree λ and the space of functions on T^*ℝⁿ which are polynomial on the fibers are not isomorphic as modules over the Lie algebra \Vect(ℝⁿ) of vector fields on ℝⁿ. However, these modules are isomorphic as sl(n+1,ℝ)-modules where sl(n+1,ℝ)⊂ \Vect(ℝⁿ) is the Lie algebra of infinitesimal projective transformations. In addition, such an sl_n+1-equivariant bijection is unique (up to normalization). This leads to a notion of projectively equivariant quantization and symbol calculus for a manifold endowed with a (flat) projective structure. We apply the sl_n+1-equivariant symbol map to study the \Vect(M)-modules of linear differential operators acting on tensor densities, for an arbitrary manifold M.

Citations

• Conformally equivariant quantization

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