2022/06/17 by V. Bardakov, B. Chuzinov, Bardakov, V. +9
Chemistry · Mathematics · #08A02 #16T25 #17A42 #20F29 #20F36 #Advanced Differential Equations and Dynamical Systems #Advanced Topics in Algebra #Axial and Atropisomeric Chirality Synthesis #Exactly Solvable and Integrable Systems (nlin.SI) #FOS: Mathematics #FOS: Physical sciences #Group Theory (math.GR) #Mathematical Physics (math-ph)
paper · pdf · doi:10.48550/arxiv.2206.08906
openalex publication_date 2022/06/17 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
The n-simplex equation (n-SE) was introduced by A. B. Zamolodchikov as a generalization of the Yang--Baxter equation, which is the 2-simplex equation in these terms. In the present paper we suggest some general approaches to constructing solutions of n-simplex equations, describe some types of solutions, introduce an operation which under some conditions allows us to construct a solution of (n+m+k)-SE from solution of (n+k)-SE and (m+k)-SE. We consider the tropicalization of rational solutions and discuss a way to generalize it. We prove that if a group G is an extension of a group H by a group K, then we can find a solution of the n-SE on G from solutions of this equation on H and on K. Also, we find solutions of the parametric Yang-Baxter equation on H with parameters in K. For studying solutions of the 3-simplex equations we introduce algebraic systems with ternary operations and give examples of these systems which gives solutions of the 3-SE. We find all elementary verbal solutions of the 3-SE on free groups.