Leibniz conformal bialgebras and the classical Leibniz conformal Yang-Baxter equation
Zhongyin Xu, Chengming Bai, Yanyong Hong
TL;DR
This work develops a conformal analogue of Leibniz bialgebras by introducing Leibniz conformal bialgebras and establishing their fundamental equivalences with matched pairs and conformal Manin triples. It then constructs a coboundary theory via the CLCYBE and connects symmetric solutions to triangular-like structures, using $\mathcal{O}$-operators and Leibniz-dendriform algebras to generate such solutions. A parallel theory for Novikov-type structures is developed through Novikov bi-dialgebras and the CDNYBE, linking them to a class of Leibniz conformal bialgebras and providing explicit examples and a unifying framework. The results expand the landscape of conformal algebraic bialgebras and offer tools for constructing and classifying such objects via operator-theoretic and dialgebraic methods.
Abstract
We introduce the notion of Leibniz conformal bialgebras, presenting a bialgebra theory for Leibniz conformal algebras as well as the conformal analogues of Leibniz bialgebras. They are equivalently characterized in terms of matched pairs and conformal Manin triples of Leibniz conformal algebras. In the coboundary case, the classical Leibniz conformal Yang-Baxter equation is introduced, whose symmetric solutions give Leibniz conformal bialgebras. Moreover, such solutions are constructed from $\mathcal{O}$-operators on Leibniz conformal algebras and Leibniz-dendriform conformal algebras. On the other hand, the notion of Novikov bi-dialgebras is introduced, which correspond to a class of Leibniz conformal bialgebras, lifting the correspondence between Novikov dialgebras and a class of Leibniz conformal algebras to the context of bialgebras. In addition, we introduce the notion of classical duplicate Novikov Yang-Baxter equation whose symmetric solutions produce Novikov bi-dialgebras and thus Leibniz conformal bialgebras.