A further study on averaging commutative and cocommutative infinitesimal bialgebras and special apre-perm bialgebras
Quan Zhao, Guilai Liu
TL;DR
This work generalizes the averaging theory from commutative algebras to a bialgebra setting by developing quasi-triangular, triangular, and factorizable frameworks for averaging commutative and cocommutative infinitesimal bialgebras and their parallel special apre-perm bialgebras. It recasts the key equations (AAYBE and SAPP-YBE) in operator form via $O$-operators on admissible averaging algebras, establishing precise correspondences between solutions and associated bialgebra structures. The paper also reveals deep connections to symmetric averaging Rota-Baxter Frobenius algebras, double constructions, and Zinbiel-related representations, providing construction techniques and canonical decompositions (e.g., factorizable doubles) that unify the averaging and apre-perm perspectives. These results offer new tools for building and analyzing splitting structures in bialgebras and enrich the interplay between operads, representations, and duality in this area.
Abstract
In order to generalize the fact that an averaging commutative algebra gives rise to a perm algebra to the bialgebra level, the notion of a special apre-perm algebra was introduced as a new splitting of perm algebras, and it has been shown that an averaging commutative and cocommutative infinitesimal bialgebra gives rise to a special apre-perm bialgebra. In this paper, we give a further study on averaging commutative and cocommutative infinitesimal bialgebras and special apre-perm bialgebras. A solution of the averaging associative Yang-Baxter equation whose symmetric part is invariant gives rise to an averaging commutative and cocommutative infinitesimal bialgebra that is called quasi-triangular, and such solutions can be equivalently characterized as $\mathcal{O}$-operators of admissible averaging commutative algebras with weights. Moreover assuming the symmetric parts of such solutions to be zero or nondegenerate, we obtain typical subclasses of quasi-triangular averaging commutative and cocommutative infinitesimal bialgebras, namely the triangular and factorizable ones respectively. Both of them are shown to closely relate to symmetric averaging Rota-Baxter Frobenius commutative algebras. There is a parallel procedure developed for special apre-perm bialgebras. In particular, the fact that an averaging commutative and cocommutative infinitesimal bialgebra gives rise to a special apre-perm bialgebra is still available when these bialgebras are limited to the quasi-triangular cases.