Non-associative structures of commutative algebras related with quadratic Poisson brackets
type: Item
instance of: scholarly article
title: Non-associative structures of commutative algebras related with quadratic Poisson brackets
zbMATH Open document ID: 1457.16019
author: Orest D. Artemovych
author: Denis Blackmore
author: A. K. Prykarpats'kyi
published in: European Journal of Mathematics
publication date: 2020-04-15T00:00:00Z
review text: Let \(A\) be a finite-dimensional non-associative and non-commutative algebra over an algebraically closed field \(\mathbb{K}\). There is a general scheme of constructing non-trivial ultra-local and local quadratic Poisson structures on the loop space \(\widetilde{A}\) [\textit{A. A. Balinsky} and \textit{Yu. M. Burman}, J. Phys. A, Math. Gen. 27, No. 18, L693--L696 (1994; Zbl 0844.58024)]. The first result of the article under review is a necessary and sufficient condition for this pre-Poisson bracket to be compatible with the internal multiplication of the algebra \(A\). When \(A\) is associative, relationships are shown of such pre-Poisson bracket with the derivations of symmetric tensor algebras, and with Yang-Baxter structures on the adjacent Lie algebras. In case that \(A\) is an associative commutative algebra with a derivation \(\delta\) on \(A\), and \(\xi\) a fixed element of \(A\), the relationship between \(\delta\)-associated Balinsky-Novikov algebra \(A^{\delta,\xi}\) is shown, as well as that with the \(\delta\)-associated Lie algebra \(A^{\delta,L}\).
reviewed by: Wen-Fong Ke
Mathematics Subject Classification ID: 16N60
Mathematics Subject Classification ID: 13N15
Mathematics Subject Classification ID: 13P25
Mathematics Subject Classification ID: 16W10
Mathematics Subject Classification ID: 35A30
Mathematics Subject Classification ID: 35G25
Mathematics Subject Classification ID: 58J70
zbMATH DE Number: 7189622
zbMATH Keywords: Balinsky-Novikov algebra
zbMATH Keywords: pre-Poisson brackets
zbMATH Keywords: Lie-Poisson structure
zbMATH Keywords: local quadratic Poisson structures
MaRDI profile type: Publication
full work available at URL: s40879-020-00398-w
OpenAlex ID: W3004589026
cites work: Q4174811
cites work: THE EXPONENTIAL REPRESENTATION OF FLOWS AND THE CHRONOLOGICAL CALCULUS
cites work: Chronological algebras and nonstationary vector fields
cites work: Q4692311
cites work: On the realization of transitive Novikov algebras
cites work: The classification of Novikov algebras in low dimensions
cites work: Addendum: The classification of Novikov algebras in low dimensions: Invariant bilinear forms
cites work: Transitive Novikov algebras on four-dimensional nilpotent Lie algebras
cites work: On the algebraic structures connected with the linear Poisson brackets of hydrodynamics type
cites work: Quadratic Poisson brackets compatible with an algebra structure
cites work: Quadratic Poisson brackets and the Drinfeld theory for associative algebras
cites work: Q3744927
cites work: Solutions of the classical Yang-Baxter equation for simple Lie algebras
cites work: Q3057915
cites work: Q4209234
cites work: Novikov algebras and Novikov structures on Lie algebras
cites work: ZINBIEL ALGEBRAS AND COMMUTATIVE ALGEBRAS WITH DIVIDED POWERS
cites work: Q5748616
cites work: Q3346635
cites work: Q3719516
cites work: Nilpotency of Zinbiel algebras
cites work: Q3769641
cites work: A class of simple nonassociative algebras
cites work: Hamiltonian operators and algebraic structures related to them
cites work: Q5639810
cites work: Q5732712
cites work: Lie Rings of Derivations of Associative Rings
cites work: The Lie Structure of a Commutative Ring with a Derivation
cites work: Q4363041
cites work: Complete left-invariant affine structures on nilpotent Lie groups
cites work: CUP-Product for Leibnitz Cohomology and Dual Leibniz Algebras.
cites work: Dialgebras and related operads
cites work: Universal enveloping algebras of Leibniz algebras and (co)homology
cites work: The Lie structure of a commutative ring with a derivation
cites work: Q4273954
cites work: Q3129431
cites work: Differential-algebraic and bi-Hamiltonian integrability analysis of the Riemann hierarchy revisited
cites work: Q5658212
cites work: Q5641292
cites work: On computation of the logarithm of the Chen-Fliess series for nonlinear systems
cites work: What is a classical r-matrix?
cites work: A simple way of making a Hamiltonian system into a bi-Hamiltonian one
cites work: Novikov algebras and a classification of multicomponent Camassa-Holm equations
cites work: On simple Novikov algebras and their irreducible modules
cites work: Novikov-Poisson algebras
cites work: Classification of simple Novikov algebras and their irreducible modules of characteristic 0
cites work: Novikov-Poisson algebras and associative commutative derivation algebras
DOI: 10.1007/S40879-020-00398-W
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