Abstract
This work is intended as an attempt to extend the notion of bialgebra for Lie algebras to Leibniz algebras and also, the correspondence between the Leibniz bialgebras and its dual is investigated. Moreover, the coboundary Leibniz bialgebras, the classical r-matrices, and Yang–Baxter equations related to the Leibniz algebras are defined, and some examples are given. Finally, a method for the construction of a dynamical system on a Leibniz manifold via Leibniz bialgebra is presented.
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Notes
Note that, contrary to the Lie algebra the bracket of Leibniz algebra is not antisymmetric i.e. \([X,Y]\ne -[Y,X]\). This is the main difference between Leibniz algebra and Lie algebra.
Note that \(\mathfrak {g}\) acts on \(\mathfrak {g}\otimes \mathfrak {g}\) from left and right such that \(\mathfrak {g}\otimes \mathfrak {g}\) becomes a \(\mathfrak {g}\)-module.
Here we use the Einstein summation convention, i.e., we have the summation over the upper and lower indices.
In the following we use and generalize the similar method for Lie bialgebra as in [13] for the Leibniz bialgebra case.
Note that \([\xi ,\eta ]_{*}=\gamma ^{t}(\xi ,\eta )\), i.e., when \(\gamma =\delta ^{0}r\), we use \([\xi ,\eta ]^{r}\) instead of \([\xi ,\eta ]_{*}\).
Note that, our definitions for r-matrix and Yang–Baxter equations for the Leibniz algebra are different from the definition given in Ref. [9].
If \(B^{\mu \nu }\) be a combination of \(G^{\mu \nu }\) (the metric of the manifold M) and \(P^{\mu \nu }\) (the Poisson structure on M) then \(B^{\mu \nu }\) will be a metriplectic structure [10].
Note that for obtaining (6.10)–(6.13) one must use the relation \([X\otimes Y,Z\otimes W]=[X,Z]\otimes YW+XZ\otimes [Y,W]\,\forall X,Y,Z,W\in \mathfrak {g}\), but, this relation for the left (right) Leibniz algebra \(\mathfrak {g}\), is consistent with the left (right) fundamental identity if \(\mathrm {ad}^{(r)}\) and \(\mathrm {ad}^{(l)}\) be both derivation, i.e., if \(\mathfrak {g}\) is a left and right Leibniz algebra simultaneously. For this reason, relations (6.10)–(6.13) are only satisfied for especial Leibniz algebra, i.e., for (right and left) Leibniz algebras.
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Acknowledgements
We would like to express our deepest gratitude to M. Akbari-Moghanjoughi for carefully reading the manuscript and his useful comments. This research was supported by Azarbaijan Shahid Madani University (Research Fund No. 27.d.1518).
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Communicated by Michaela Vancliff.
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Rezaei-Aghdam, A., Sedghi-Ghadim, L. & Haghighatdoost, G. Leibniz Bialgebras, Classical Yang–Baxter Equations and Dynamical Systems. Adv. Appl. Clifford Algebras 31, 77 (2021). https://doi.org/10.1007/s00006-021-01177-w
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DOI: https://doi.org/10.1007/s00006-021-01177-w