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arXiv · 1811.00842

Weighted infinitesimal unitary bialgebras on matrix algebras and weighted associative Yang-Baxter equations

Yi Zhang·Xing Gao·Jia-wen Zheng
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Abstract

We equip a matrix algebra with a weighted infinitesimal unitary bialgebraic structure, via a construction of a suitable coproduct. Furthermore, an infinitesimal unitary Hopf algebra, under the view of Aguiar, is constructed on a matrix algebra. By exploring the relationship between weighted infinitesimal bialgebras and pre-Lie algebras, we construct a pre-Lie algebraic structure and then a new Lie algebraic structure on a matrix algebra. We also introduce the weighted associative Yang-Baxter equations (AYBEs) and obtain the relationship between solutions of weighted AYBEs and weighted infinitesimal unitary bialgebras. We give a bijection between the solutions of the associative Yang-Baxter equation of weight $\lambda$ and Rota-Baxter operators of weight $-\lambda$ on matrix algebras. As a consequence, weighted quasitriangular infinitesimal unitary bialgebras are constructed, which generalize the results studied by Aguiar. Finally, We show that any weighted quasitriangular infinitesimal unitary bialgebra can be made into a dendriform algebra.

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Yi Zhang, Xing Gao, Jia-wen Zheng. 2018-11-02. Weighted infinitesimal unitary bialgebras on matrix algebras and weighted associative Yang-Baxter equations. https://arxiv.org/abs/1811.00842

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