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Axioms 2012, 1(2), 186-200; doi:10.3390/axioms1020186
Article

From Coalgebra to Bialgebra for the Six-Vertex Model: The Star-Triangle Relation as a Necessary Condition for Commuting Transfer Matrices

Departments of Physics and Mathematics, University of Wisconsin-Parkside 900 Wood Road, Kenosha, WI 53141, USA
Received: 2 July 2012 / Revised: 7 August 2012 / Accepted: 16 August 2012 / Published: 27 August 2012
(This article belongs to the Special Issue Hopf Algebras, Quantum Groups and Yang-Baxter Equations)
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Abstract

Using the most elementary methods and considerations, the solution of the star-triangle condition (a2+b2-c2)/2ab = ((a’)^2+(b’)^2-(c’))^2/2a’b’ is shown to be a necessary condition for the extension of the operator coalgebra of the six-vertex model to a bialgebra. A portion of the bialgebra acts as a spectrum-generating algebra for the algebraic Bethe ansatz, with which higher-dimensional representations of the bialgebra can be constructed. The star-triangle relation is proved to be necessary for the commutativity of the transfer matrices T(a, b, c) and T(a’, b’, c’).
Keywords: vertex model; bialgebra; coalgebra; Bethe ansatz vertex model; bialgebra; coalgebra; Bethe ansatz
This is an open access article distributed under the Creative Commons Attribution License (CC BY 3.0).
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Schmidt, J.R. From Coalgebra to Bialgebra for the Six-Vertex Model: The Star-Triangle Relation as a Necessary Condition for Commuting Transfer Matrices. Axioms 2012, 1, 186-200.

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