2014/10/13 by Grützmann, Melchior, Michel, Jean-Philippe, Xu, Ping · 1 citation
#15A66 #16W70 #17B63 #53B05 #53D17 #53D18 #53D55 #81S10 #Differential Geometry (math.DG) #FOS: Mathematics #FOS: Physical sciences #Mathematical Physics (math-ph) #Symplectic Geometry (math.SG)
paper · doi:10.48550/arxiv.1410.3346
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 main property, the associated graded algebra grD(M,S) is isomorphic to the algebra O(M) of functions on M, where M is the symplectic graded manifold of degree 2 canonically associated to (E,g). Accordingly, we define the Weyl quantization on M as a map WQ_ℏ:O(M)\toD(M,S), and prove that WQ_ℏ satisfies all desired usual properties. As an application, we obtain a bijection between Courant algebroid structures (E,g,ρ,[⋅,⋅]), that are encoded by Hamiltonian generating functions on M, and skew-symmetric Dirac generating operators D\inD(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 in detail the particular case of E being the double of a Lie bialgebroid (A,A^*).