Noncommutative Poisson brackets on Loday algebras and related\n deformation quantization

2010/07/26 by Kyousuke Uchino, Uchino, Kyousuke
Mathematics · #Advanced Topics in Algebra #Algebraic structures and combinatorial models #FOS: Mathematics #Homotopy and Cohomology in Algebraic Topology #Quantum Algebra (math.QA) #Symplectic Geometry (math.SG)

paper · pdf · doi:10.48550/arxiv.1007.4501

openalex publication_date 2010/07/26 · openalex created_date 2022/10/06 · openalex updated_date 2026/07/28

Abstract

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

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