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arXiv · 1007.4501

Noncommutative Poisson brackets on Loday algebras and related deformation quantization

Kyousuke Uchino
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Abstract

Given a Lie algebra, there uniquely exists a Poisson algebra which is called a Lie-Poisson algebra over the Lie algebra. We will prove that given a Loday/Leibniz algebra there exists uniquely a noncommutative Poisson algebra over the Loday algebra. The noncommutative Poisson algebras are called the Loday-Poisson algebras. In the super/graded cases, the Loday-Poisson bracket is regarded as a noncommutative version of classical (linear) Schouten-Nijenhuis bracket. It will be shown that the Loday-Poisson algebras form a special subclass of Aguiar's dual-prePoisson algebras. We also study a problem of deformation quantization over the Loday-Poisson algebra. It will be shown that the polynomial Loday-Poisson algebra is deformation quantizable and that the associated quantum algebra is Loday's associative dialgebra.

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Kyousuke Uchino. 2010-07-26. Noncommutative Poisson brackets on Loday algebras and related deformation quantization. https://arxiv.org/abs/1007.4501

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