Monomial bialgebras

TL;DR

This paper constructs infinite families of solutions to the quantum and classical Yang-Baxter equations from a single initial solution, exploring their applications to Lie bialgebras and Hopf algebras.

Contribution

It introduces a method to generate infinite solutions of QYBE and CYBE parametrized by transitive arrays and signed permutations, with applications to algebraic structures.

Findings

Infinite families of solutions to QYBE and CYBE are constructed.

Applications to quasi-triangular structures on Lie bialgebras and Hopf algebras.

Analysis of associated Poisson-Lie and co-quasi-triangular structures.

Abstract

Starting from a single solution of QYBE (or CYBE) we produce an infinite family of solutions of QYBE (or CYBE) parametrized by transitive arrays and, in particular, by signed permutations. We are especially interested in cases when such solutions yield quasi-triangular structures on direct powers of Lie bialgebras and tensor powers of Hopf algebras. We obtain infinite families of such structures as well and study the corresponding Poisson-Lie structures and co-quasi-triangular algebras.