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Noncommutative Poisson brackets on Loday algebras and related deformation quantization. (English) Zbl 1278.53084

This paper gives the construction of a natural dual-prePossion algebra \(\mathcal{A}^{\mathrm{poly}}({g^*})\) from a given Loday algebra \(g\), in analogue to the classical case when \(S(g)\) is a free Lie-Poisson algebra, if \(g\) is a Lie algebra. The author calls such a construction \(\mathcal{A}^{\mathrm{poly}}({g^*})\) the polynomial Loday-Poisson algebra and proves that it is the free dual-prePoisson algebra over \(g\). Moreover, the deformation quantization of \(\mathcal{A}^{\mathrm{poly}}({g^*})\) turns out to be an associative dialgebra.

MSC:

53D17 Poisson manifolds; Poisson groupoids and algebroids
46L87 Noncommutative differential geometry
58J42 Noncommutative global analysis, noncommutative residues
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