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<p>Given an arbitrary choice of two sets of nonzero Boltzmann weights for <i>n</i>-color lattice models, we provide explicit algebraic conditions on these Boltzmann weights which guarantee a solution (i.e., a third set of weights) to the Yang–Baxter equation. Furthermore, we provide an explicit one-dimensional parametrization of all solutions in this case. These <i>n</i>-color lattice models are so named because their admissible vertices have adjacent edges labeled by one of <i>n</i> colors with additional restrictions. The two-colored case specializes to the six-vertex model, in which case our results recover the familiar quadric condition of Baxter for solvability. The general <i>n</i>-color case includes important solutions to the Yang–Baxter equation like the evaluation modules for the quantum affine Lie algebra <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="26_2025_770_Article_IEq1.gif" Format="GIF" Height="23" Rendition="HTML" Resolution="72" Type="Linedraw" Width="51" /> </InlineMediaObject> <EquationSource Format="TEX">\(U_q(\hat{\mathfrak {sl}}_n)\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>U</mi> <mi>q</mi> </msub> <mrow> <mo stretchy="false">(</mo> <msub> <mover accent="true"> <mi mathvariant="fraktur">sl</mi> <mo stretchy="false">^</mo> </mover> <mi>n</mi> </msub> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation>. Finally, we demonstrate the invariance of this class of solutions under natural transformations, including those associated with Drinfeld twisting.</p>

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Solving the n-Color Ice Model

  • 期刊论文

  • 发表日期：

    2025年7月9日

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  • Patrick Addona,
  • Ethan Bockenhauer,
  • Ben Brubaker,
  • Michael Cauthorn,
  • Cianan Conefrey-Shinozaki,
  • David Donze,
  • William Dudarov,
  • Jessamyn Dukes,
  • Andrew Hardt,
  • Cindy Li,
  • Jigang Li,
  • Yanli Liu,
  • Neelima Puthanveetil,
  • Zain Qudsi,
  • Jordan Simons,
  • Joseph Sullivan,
  • Autumn Young

摘要

Given an arbitrary choice of two sets of nonzero Boltzmann weights for n-color lattice models, we provide explicit algebraic conditions on these Boltzmann weights which guarantee a solution (i.e., a third set of weights) to the Yang–Baxter equation. Furthermore, we provide an explicit one-dimensional parametrization of all solutions in this case. These n-color lattice models are so named because their admissible vertices have adjacent edges labeled by one of n colors with additional restrictions. The two-colored case specializes to the six-vertex model, in which case our results recover the familiar quadric condition of Baxter for solvability. The general n-color case includes important solutions to the Yang–Baxter equation like the evaluation modules for the quantum affine Lie algebra \(U_q(\hat{\mathfrak {sl}}_n)\) U q ( sl ^ n ) . Finally, we demonstrate the invariance of this class of solutions under natural transformations, including those associated with Drinfeld twisting.