Non-associative structures of commutative algebras related with quadratic Poisson brackets
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 \(A^{\delta,L}\).
scientific article; zbMATH DE number 7189622
scientific article