Weyl quantization of degree 2 symplectic graded manifolds

Melchior Grützmann, Jean Philippe Michel, Ping Xu

Research output: Contribution to journalArticlepeer-review

Abstract

Let S be a spinor bundle of a pseudo-Euclidean vector bundle (E,g) of even rank. We introduce a new filtration on the algebra D(M,S) of differential operators on S. As a main property, the associated graded algebra grD(M,S) is proved to be isomorphic to the algebra O(M) of smooth functions on M, where M is the degree 2 symplectic graded manifold canonically associated to (E,g). Accordingly, we establish the Weyl quantization of M as a map WQħ:O(M)→D(M,S), and prove that WQħ satisfies all the desired properties of quantizations. As an application, we obtain a bijection between Courant algebroid structures (E,g,ρ,〚⋅,⋅〛) equivalently characterized by Hamiltonian generating functions on M, and skew-symmetric Dirac generating operators D∈D(M,S). The operator D2 gives a new invariant of (E,g,ρ,〚⋅,⋅〛) which generalizes the square norm of the Cartan 3-form of a quadratic Lie algebra. We study this invariant in detail in the particular case of E being the double of a Lie bialgebroid (A,A).

Original languageEnglish (US)
Pages (from-to)67-107
Number of pages41
JournalJournal des Mathematiques Pures et Appliquees
Volume154
DOIs
StatePublished - Oct 2021

All Science Journal Classification (ASJC) codes

  • Mathematics(all)
  • Applied Mathematics

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