Journal of Symplectic Geometry

Volume 11 (2013)

Number 1

Noncommutative Poisson brackets on Loday algebras and related deformation quantization

Pages: 93 – 108

DOI: http://dx.doi.org/10.4310/JSG.2013.v11.n1.a5

Author

Kyousuke Uchino (Tokyo University of Science, Tokyo, Japan)

Abstract

Given a Lie algebra, there uniquely exists a Poisson algebra that 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 pre-Poisson 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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