Non-associative structures of commutative algebras related with quadratic Poisson brackets

There are studied algebraic properties of quadratic Poisson brackets on non-associative non-commutative algebras, compatible with their multiplicative structure. Their relations both with derivations of symmetric tensor algebras and Yang–Baxter structures on the adjacent Lie algebras are demonstrate...

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Bibliographic Details
Published in:European journal of mathematics 2020-03, Vol.6 (1), p.208-231
Main Authors: Artemovych, Orest D., Blackmore, Denis, Prykarpatski, Anatolij K.
Format: Article
Language:English
Subjects:
Algebra
Algebraic Geometry
Brackets
Lie groups
Mathematics
Mathematics and Statistics
Research Article
Tensors
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Summary:There are studied algebraic properties of quadratic Poisson brackets on non-associative non-commutative algebras, compatible with their multiplicative structure. Their relations both with derivations of symmetric tensor algebras and Yang–Baxter structures on the adjacent Lie algebras are demonstrated. Special attention is paid to quadratic Poisson brackets of Lie–Poisson type, examples of Balinsky–Novikov and Leibniz algebras are discussed. The non-associative structures of commutative algebras related with Balinsky–Novikov, Leibniz, Lie, and Zinbiel algebras are studied in detail.
ISSN:2199-675X
2199-6768
DOI:10.1007/s40879-020-00398-w