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Let’s Baxterise. (English) Zbl 0990.82007
We recall the concept of Baxterisation of an $R$-matrix, or of a monodromy matrix, which corresponds to building, 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 nonintegrable 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.
##### MSC:
 82B23 Exactly solvable models; Bethe ansatz 81R12 Relations of groups and algebras in quantum theory with integrable systems