Yang–Baxter Equations, Hopf Algebras and Geometric Interpretations
- 1Simion Stoilow Institute of Mathematics of the Romanian Academy, 21 Calea Grivitei Street, 010702 Bucharest, Romania
Abstract
The Yang–Baxter equation is one of the most celebrated equations in science. There exist some correspondences between Yang–Baxter operators and Hopf algebras. For example, the quasitriangular Hopf algebras are used to define ribbon Hopf algebras and knot invariants. Finding new solutions for the Yang–Baxter equation and relating it to other algebraic structures is of great interest for scientists, as the classification of its solutions is out of reach for the moment. In the current paper, the new solutions arise from Boolean algebras and from racks. Our new solutions of the Yang–Baxter equation are generalizations of well-known solutions and they are related to the Euler formula. There exist self-dual Yang–Baxter operators as well as self-dual geometry theorems. There exists a partial dualization in the definition of entwining structures and there exists a partial dualization in our new theorem. Finding geometric representations for the phenomena from the Yang–Baxter theory is very useful at conferences and congresses. In geometry, the Morley’s trisector theorem was proved by the Fields medalist Alain Connes. We propose a similar theorem, which is a many circles version of a famous theorem. Connecting celebrated topics, the current paper also deals with an Euler formula proof for the original Johnson–Tzitzeica theorem.
Related
- A-Generalized Hessian pre-Lie algebras and A-Generalized Yang--Baxter Equations
- Yang-Baxter permutation group actions on distributive Yang-Baxter algebras
- Quantum wreath products and Yang-Baxter equations
- Symmetries of the Generalized Yang--Baxter Equations
- Matrix representations of BCn associative Yang--Baxter equations and related long-range spin chains
BibTeX
@article{nichita2026yangbaxter,
title = {{Yang–Baxter Equations, Hopf Algebras and Geometric Interpretations}},
author = {Florin Felix Nichita},
journal = {Axioms},
year = {2026},
month = nov,
doi = {10.3390/axioms15110829},
url = {https://www.mdpi.com/2075-1680/15/11/829},
note = {CC BY 4.0},
}
RIS
TY - JOUR
TI - Yang–Baxter Equations, Hopf Algebras and Geometric Interpretations
AU - Florin Felix Nichita
PY - 2026
DA - 2026/11/06
JO - Axioms
DO - 10.3390/axioms15110829
UR - https://www.mdpi.com/2075-1680/15/11/829
AB - The Yang–Baxter equation is one of the most celebrated equations in science. There exist some correspondences between Yang–Baxter operators and Hopf algebras. For example, the quasitriangular Hopf algebras are used to define ribbon Hopf algebras and knot invariants. Finding new solutions for the Yang–Baxter equation and relating it to other algebraic structures is of great interest for scientists, as the classification of its solutions is out of reach for the moment. In the current paper, the new solutions arise from Boolean algebras and from racks. Our new solutions of the Yang–Baxter equation are generalizations of well-known solutions and they are related to the Euler formula. There exist self-dual Yang–Baxter operators as well as self-dual geometry theorems. There exists a partial dualization in the definition of entwining structures and there exists a partial dualization in our new theorem. Finding geometric representations for the phenomena from the Yang–Baxter theory is very useful at conferences and congresses. In geometry, the Morley’s trisector theorem was proved by the Fields medalist Alain Connes. We propose a similar theorem, which is a many circles version of a famous theorem. Connecting celebrated topics, the current paper also deals with an Euler formula proof for the original Johnson–Tzitzeica theorem.
SN - 2075-1680
ER -