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Monomial bialgebras

Arkady Berenstein, Jacob Greenstein, Jian-Rong Li

TL;DR

This work develops a comprehensive framework for generating and manipulating solutions to the classical and quantum Yang-Baxter equations via transitive combinatorial data. It constructs monomial bialgebras and explicit twists that realize CYBE/QYBE on direct sums and tensor powers, including both classical and quantum twists and their relative variants. The paper ties these constructions to Poisson-Lie geometry and to dual (co-quasi-triangular) structures, deriving diagonal-embedding results and a broad array of R-matrix and twisted-multiplication formulas with concrete examples on matrix and quantum-matrix algebras. These results provide new families of (non)isomorphic quasi-triangular structures and illuminate the interplay between combinatorics, twists, and multi-factorbialgebra constructions, with potential implications for cluster-like structures and quantum matrix theory.

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.

Monomial bialgebras

TL;DR

This work develops a comprehensive framework for generating and manipulating solutions to the classical and quantum Yang-Baxter equations via transitive combinatorial data. It constructs monomial bialgebras and explicit twists that realize CYBE/QYBE on direct sums and tensor powers, including both classical and quantum twists and their relative variants. The paper ties these constructions to Poisson-Lie geometry and to dual (co-quasi-triangular) structures, deriving diagonal-embedding results and a broad array of R-matrix and twisted-multiplication formulas with concrete examples on matrix and quantum-matrix algebras. These results provide new families of (non)isomorphic quasi-triangular structures and illuminate the interplay between combinatorics, twists, and multi-factorbialgebra constructions, with potential implications for cluster-like structures and quantum matrix theory.

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.
Paper Structure (36 sections, 96 theorems, 298 equations, 1 table)

This paper contains 36 sections, 96 theorems, 298 equations, 1 table.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. General notation
  4. Completions
  5. Coalgebras
  6. Lie bialgebras
  7. Quasi-triangular Lie bialgebras
  8. Classical Drinfeld twists
  9. Relative classical twist
  10. The dual picture: Poisson algebras
  11. Quasi-triangular bialgebras
  12. Drinfeld twists
  13. Relative Drinfeld twists
  14. Duality in the quantum case
  15. Combinatorial background
  16. ...and 21 more sections

Key Result

Theorem 1.1

Let ${\mathfrak{g}}$ be a quasi-triangular Lie bialgebra with a classical r-matrix $r$ and let $n\ge 2$. Then for any $\boldsymbol d\in\{1,-1\}^n$, ${\mathfrak{g}}^{\oplus n}$ is a quasi-triangular Lie bialgebra with the classical r-matrix $\mathbf r^{(\boldsymbol\epsilon(\boldsymbol d))}$ where

Theorems & Definitions (189)

  • Theorem 1.1
  • Theorem 1.2
  • Conjecture 1.4
  • Theorem 1.5
  • Theorem 1.6
  • Corollary 1.7
  • Theorem 1.8
  • Corollary 1.9
  • Conjecture 1.12
  • Lemma 2.1
  • ...and 179 more