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arXiv · 2401.13608

A bialgebra theory of Gel'fand-Dorfman algebras with applications to Lie conformal bialgebras

Yangyon Hong·Chengming Bai·Li Guo
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Abstract

Gel'fand-Dorfman algebras (GD algebras) give a natural construction of Lie conformal algebras and are in turn characterized by this construction. In this paper, we define the Gel'fand-Dorfman bialgebra (GD bialgebras) and enrich the above construction to a construction of Lie conformal bialgebras by GD bialgebras. As a special case, Novikov bialgebras yield Lie conformal bialgebras. We further introduce the notion of the Gel'fand-Dorfman Yang-Baxter equation (GDYBE), whose skew-symmetric solutions produce GD bialgebras. Moreover, the notions of $\mathcal{O}$-operators on GD algebras and pre-Gel'fand-Dorfman algebras (pre-GD algebras) are introduced to provide skew-symmetric solutions of the GDYBE. The relationships between these notions for GD algebras and the corresponding ones for Lie conformal algebras are given. In particular, there is a natural construction of Lie conformal bialgebras from pre-GD algebras. Finally, GD bialgebras are characterized by certain matched pairs and Manin triples of GD algebras.

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Yangyon Hong, Chengming Bai, Li Guo. 2024-01-24. A bialgebra theory of Gel'fand-Dorfman algebras with applications to Lie conformal bialgebras. https://arxiv.org/abs/2401.13608

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