On antisymmetric infinitesimal conformal bialgebras

TL;DR

This paper develops a new bialgebra theory for associative conformal algebras, introducing antisymmetric infinitesimal conformal bialgebras and linking them to Frobenius conformal algebras and the associative conformal Yang-Baxter equation.

Contribution

It introduces antisymmetric infinitesimal conformal bialgebras, connects them to Frobenius conformal algebras, and develops the associative conformal Yang-Baxter equation and its solutions.

Findings

Defined antisymmetric infinitesimal conformal bialgebras.

Established equivalence with double constructions of Frobenius conformal algebras.

Constructed solutions to the associative conformal Yang-Baxter equation.

Abstract

In this paper, we construct a bialgebra theory for associative conformal algebras, namely antisymmetric infinitesimal conformal bialgebras. On the one hand, it is an attempt to give conformal structures for antisymmetric infinitesimal bialgebras. On the other hand, under certain conditions, such structures are equivalent to double constructions of Frobenius conformal algebras, which are associative conformal algebras that are decomposed into the direct sum of another associative conformal algebra and its conformal dual as $\mathbb{C}[\partial]$-modules such that both of them are subalgebras and the natural conformal bilinear form is invariant. The coboundary case leads to the introduction of associative conformal Yang-Baxter equation whose antisymmetric solutions give antisymmetric infinitesimal conformal bialgebras. Moreover, the construction of antisymmetric solutions of associative conformal Yang-Baxter equation is given from $\mathcal{O}$-operators of associative conformal algebras as well as dendriform conformal algebras.