Journal of Symplectic Geometry

Volume 11 (2013)

Number 1

Noncommutative Poisson brackets on Loday algebras and related deformation quantization

Pages: 93 – 108



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


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.

Published 1 February 2013