Open Access
March 2013 Noncommutative Poisson brackets on Loday algebras and related deformation quantization
Kyousuke Uchino
J. Symplectic Geom. 11(1): 93-108 (March 2013).


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-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. "Noncommutative Poisson brackets on Loday algebras and related deformation quantization." J. Symplectic Geom. 11 (1) 93 - 108, March 2013.


Published: March 2013
First available in Project Euclid: 1 March 2013

zbMATH: 1278.53084
MathSciNet: MR3022922

Rights: Copyright © 2013 International Press of Boston

Vol.11 • No. 1 • March 2013
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