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Cartan connections and natural and projectively equivariant quantizations
Author(s) -
Mathonet P.,
Radoux F.
Publication year - 2007
Publication title -
journal of the london mathematical society
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.441
H-Index - 62
eISSN - 1469-7750
pISSN - 0024-6107
DOI - 10.1112/jlms/jdm030
Subject(s) - equivariant map , mathematics , invariant (physics) , symbol (formal) , pure mathematics , manifold (fluid mechanics) , natural (archaeology) , connection (principal bundle) , algebra over a field , calculus (dental) , geometry , linguistics , philosophy , mathematical physics , mechanical engineering , medicine , archaeology , dentistry , engineering , history
Abstract In this paper, the question of existence of a natural and projectively equivariant symbol calculus is analysed using the theory of projective Cartan connections. A close relationship is established between the existence of such a natural symbol calculus and the existence of an sl( m + 1, ℝ)‐equivariant calculus over ℝ m . Moreover, it is shown that the formulae that hold in the non‐critical situations over Rm for the sl( m + 1,ℝ)‐equivariant calculus can be directly generalized to an arbitrary manifold by simply replacing the partial derivatives by invariant differentiations with respect to a Cartan connection.