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Graded Lie agebroids of Poisson almost commutative algebras. (English) Zbl 1385.53075

This paper introduces and studies the notion of abelian groups graded Lie algebroid structures on almost commutative algebras. In a previous work, the author introduced the notion of Poisson almost commutative algebra (PACA). In this work, the author first answers the natural question to find which classical geometric objects on Poisson manifold have their equivalents in the framework of PACA. Then the author recalls the notions of graded Schouten-Nijenhuis structure and graded Poisson bracket. He shows that any graded Poisson bracket induces a graded Lie algebroid. The corresponding Poisson cohomology is studied. Moreover, the author explicitly computes the Poisson cohomology of the quantum plane.

MSC:

53D17 Poisson manifolds; Poisson groupoids and algebroids
17B75 Color Lie (super)algebras
81R30 Coherent states
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Full Text: Euclid