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arXiv · 2606.31735

Infinite-dimensional pre-Lie bialgebras induced from Leibniz-dendriform bialgebras and Zinbiel-dendriform bialgebras

Qinxiu Sun
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Abstract

In this paper, we establish a completed pre-Lie bialgebra structure on the tensor product of a Leibniz-dendriform bialgebra and a quadratic $\mathbb{Z}$-graded Zinbiel algebra. We also obtain such a structure on the tensor product of a Zinbiel-dendriform bialgebra and a quadratic $\mathbb{Z}$-graded Leibniz algebra. Moreover, a Zinbiel-dendriform bialgebra is precisely one whose affinization by a special quadratic $\mathbb{Z}$-graded Leibniz algebra is a completed pre-Lie bialgebra. Finally, using solutions of the ZD-YBE (resp.~LD-YBE) with invariant skew-symmetric parts in a Zinbiel-dendriform (resp.~ Leibniz-dendriform) algebra, we construct completed solutions possessing invariant symmetric parts of the $S$-equation in the induced pre-Lie algebra.

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Qinxiu Sun. 2026-06-30. Infinite-dimensional pre-Lie bialgebras induced from Leibniz-dendriform bialgebras and Zinbiel-dendriform bialgebras. https://arxiv.org/abs/2606.31735

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