Quasi-triangular averaging commutative and cocommutative infinitesimal bialgebras and special apre-perm bialgebras

Abstract

In order to generalize the fact that an averaging commutative algebra gives rise to a perm algebra at 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 present 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, by 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 remains valid when these bialgebras are limited to the quasi-triangular cases.