Monomial bialgebras

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.

💡 Research Summary

The paper “Monomial Bialgebras” develops a unified combinatorial framework for generating infinite families of solutions to the classical and quantum Yang‑Baxter equations (CYBE and QYBE) from a single seed solution. The key combinatorial object is a transitive array (or transitive matrix) – an (n\times n) matrix whose entries lie in a prescribed set (typically ({1,-1})) and satisfy the transitivity condition (a_{ik}\in{a_{ij},a_{jk}}) for all triples (i<j<k). When the matrix is almost skew‑symmetric, it corresponds uniquely to a signed permutation ((w,d)) with (w\in S_n) and (d\in{\pm1}^n).

Starting from a quasi‑triangular Lie bialgebra ((\mathfrak g,r)), the authors define for any transitive matrix (\varepsilon) a new element \