Noncommutative Poisson brackets on Loday algebras and related deformation quantization

TL;DR

This paper introduces Loday-Poisson algebras as noncommutative analogs of Lie-Poisson algebras for Loday algebras, explores their structure as dual-prePoisson algebras, and demonstrates their deformation quantization leading to Loday's associative dialgebra.

Contribution

It establishes the existence and uniqueness of Loday-Poisson algebras over Loday algebras and investigates their deformation quantization.

Findings

Loday-Poisson algebras form a subclass of dual-prePoisson algebras.

Polynomial Loday-Poisson algebras are deformation quantizable.

Quantum algebras correspond to Loday's associative dialgebra.

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.