Differential pre-Lie bialgebras and admissible $\mathcal{S}$-equations
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Abstract
The aim of this paper is to introduce the notion of differential pre-Lie bialgebra $(A,\delta,\zeta,\Phi,\Psi)$ and their admissibility conditions in terms of dual representations $(l_{\circ}^*-r_{\circ}^*,-r_{\circ}^*,\beta^*,V^*)$. Next, we show that differential pre-Lie bialgebras are characterized by matched pairs and Manin triples of differential pre-Lie algebras. Furthermore,
the coboundary case leads to the introduction of the admissible $\mathcal{S}$-equation in differential pre-Lie algebras, whose symmetric solutions are used to construct differential pre-Lie bialgebras.
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