Hamiltonian operators and related differential-algebraic Balinsky-Novikov, Riemann and Leibniz type structures on nonassociative noncommutative algebras
Authors:
- Orest D. Artemovych,
- Alexander A. Balinsky,
- Anatolij K. Prykarpatski
Abstract
We review main differential-algebraic structures lying in background of analytical constructing multi-component Hamiltonian operators as derivatives on suitably constructed loop Lie algebras, generated by nonassociative noncommutative algebras. The related Balinsky-Novikov and Leibniz type algebraic structures are derived, a new nonassociative "Riemann" algebra is constructed, deeply related with infinite multicomponent Riemann type integrable hierarchies. An approach, based on the classical Lie-Poisson structure on coadjoint orbits, closely related with those, analyzed in the present work and allowing effectively enough construction of Hamiltonian operators, is also briefly revisited. As the compatible Hamiltonian operators are constructed by means of suitable central extentions of the adjacent weak Lie algebras, generated by the right Leibniz and Riemann type nonassociative and noncommutative algebras, the problem of their description requires a detailed investigation both of their structural properties and finite-dimensional representations of the right Leibniz algebras defined by the corresponding structural constraints. Subject to these important aspects we stop in the work mostly on the structural properties of the right Leibniz algebras, especially on their derivation algebras and their generalizations. We have also added a short Supplement within which we revisited the classical Poisson manifold approach, closely related to our construction of Hamiltonian operators, generated by nonassociative and noncommutative algebras. In particular, we presented its natural and simple generalization allowing effectively to describe a wide class of Lax type integrable nonlinear Kontsevich type Hamiltonian systems on associative noncommutative algebras.
- Record ID
- CUT4e44693b83db4dd7bb4076d862bc5468
- Publication categories
- ;
- Author
- Journal series
- Proceedings of the International Geometry Center, ISSN 2072-9812, e-ISSN 2409-8906
- Issue year
- 2019
- Vol
- 12
- No
- 4
- Pages
- [1-30]
- Other elements of collation
- Bibliografia (na s.) - [27-30]; Bibliografia (liczba pozycji) - 101; Oznaczenie streszczenia - Abstr.; Numeracja w czasopiśmie - Vol. 12, No. 4
- Keywords in English
- Hamiltonian operators, Lie-Poisson structure, differenetial algebras, integrability, derivatives, loop-algebra, cocycles, Balinsky-Novikov algebra, right Leibniz algebra, Riemann algebra, group agebras, π-metrized Lie alegbras, pi--metrized Lie alegbras, Kontsevich type systems
- URL
- https://journals.onaft.edu.ua/index.php/geometry/article/view/1554 Opening in a new tab
- Related project
- Teoria równań i nierówności różniczkowych, układy dynamiczne na rozmaitościach, wybrane działy: analizy matematycznej topologii algebry i geometrii oraz probabilistyki teorii grafów i liczb. . Project leader at PK: , ,
Działalność statutowa - Language
- eng (en) English
- License
- Score (nominal)
- 20
- Uniform Resource Identifier
- https://cris.pk.edu.pl/info/article/CUT4e44693b83db4dd7bb4076d862bc5468/
- URN
urn:pkr-prod:CUT4e44693b83db4dd7bb4076d862bc5468
* presented citation count is obtained through Internet information analysis, and it is close to the number calculated by the Publish or PerishOpening in a new tab system.