Let's Baxterise

Abstract

We recall the concept of Baxterisation of an R-matrix, or of a monodromy matrix, which corresponds to build, from one point in the $R$-matrix parameter space, the algebraic variety where the spectral parameter(s) live. We show that the Baxterisation, which amounts to studying the iteration of a birational transformation, is a ``win-win'' strategy: it enables to discard efficiently the non-integrable situations, focusing directly on the two interesting cases where the algebraic varieties are of the so-called ``general type'' (finite order iteration) or are Abelian varieties (infinite order iteration). We emphasize the heuristic example of the sixteen vertex model and provide a complete description of the finite order iterations situations for the Baxter model. We show that the Baxterisation procedure can be introduced in much larger frameworks where the existence of some underlying Yang-Baxter structure is not used: we Baxterise L-operators, local quantum Lax matrices, and quantum Hamiltonians.

38 pages, no figure

References in corpus (4)

• Planar algebras, I
• Complexity spectrum of some discrete dynamical systems
• Random Matrix Theory and Classical Statistical Mechanics. I. Vertex Models
• Rational dynamical zeta functions for birational transformations

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• Representations of the quantum doubles of finite group algebras and solutions of the Yang--Baxter equation
• Integrable boundary conditions for multi-species ASEP
• Random Matrix Theory and higher genus integrability: the quantum chiral Potts model
• A new braid-like algebra for Baxterisation
• The 16-vertex model and its even and odd 8-vertex subcases on the square lattice
• Modular forms, Schwarzian conditions, and symmetries of differential equations in physics
• Baxterisation of the fused Hecke algebra and R-matrices with gl(N)-symmetry
• Stochastic Baxterisation of a fused Hecke algebra