Solving the Yang-Baxter, tetrahedron and higher simplex equations using Clifford algebras

Abstract

Bethe Ansatz was discoverd in 1932. Half a century later its algebraic structure was unearthed: Yang-Baxter equation was discovered, as well as its multidimensional generalizations [tetrahedron equation and $d$-simplex equations]. Here we describe a universal method to solve these equations using Clifford algebras. The Yang-Baxter equation ($d=2$), Zamalodchikov's tetrahedron equation ($d=3$) and the Bazhanov-Stroganov equation ($d=4$) are special cases. Our solutions form a linear space. This helps us to include spectral parameters. Potential applications are discussed.

arXiv commentv2 70 pages, Includes missing solutions, all of which are succinctly captured in Theorems 1 and 2

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Cited by in corpus (8)

• Algebraic classification of Hietarinta's solutions of Yang-Baxter equations~:~invertible $4\times 4$ operators
• Integrable models on Rydberg atom chains
• Periodic Motzkin chain: Ground states and symmetries
• The Yang-Baxter integrability of the critical Ising chain
• Tetrahedron equation and Schur functions
• Majorana fermions solve the tetrahedron equations as well as higher simplex equations
• All 4 x 4 solutions of the quantum Yang-Baxter equation
• Almost local integrable models from supersymmetry algebras