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Mathematics > Quantum Algebra

arXiv:1007.4501 (math)
[Submitted on 26 Jul 2010 (v1), last revised 11 Jun 2011 (this version, v3)]

Title:Noncommutative Poisson brackets on Loday algebras and related deformation quantization

Authors:Kyousuke Uchino
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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.
Comments: The final version
Subjects: Quantum Algebra (math.QA); Symplectic Geometry (math.SG)
Cite as: arXiv:1007.4501 [math.QA]
  (or arXiv:1007.4501v3 [math.QA] for this version)
  https://doi.org/10.48550/arXiv.1007.4501

Submission history

From: Kyousuke Uchino [view email]
[v1] Mon, 26 Jul 2010 16:39:21 UTC (12 KB)
[v2] Thu, 12 Aug 2010 02:14:38 UTC (12 KB)
[v3] Sat, 11 Jun 2011 15:42:43 UTC (12 KB)
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