We present the general diagonal and, in some cases, non-diagonal solutions of the boundary Yang-Baxter equation for a number of related interaction-round-a-face models, including the standard and dilute A_L, D_L and E_{6,7,8} models.

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991 anJ 51 ]chem-tats.tam-dnoc[ 2v3121169/tam-dnco:viXraSolutionsoftheBoundaryYang-BaxterEquationfor

A–D–EModels

RogerE.Behrend1andPaulA.Pearce2

DepartmentofMathematics,UniversityofMelbourne,

Parkville,Victoria3052,Australia

Abstract

Wepresentthegeneraldiagonaland,insomecases,non-diagonalsolutionsoftheboundaryYang-Baxterequationforanumberofrelatedinteraction-round-a-facemod-els,includingthestandardanddiluteAL,DLandE6,7,8models.

1.Introduction

Atwo-dimensionallatticespinmodelinstatisticalmechanicscanbeconsideredassolvablewithperiodicboundaryconditionsifitsbulkBoltzmannweightssatisfytheYang-Baxterequation[1],andasadditionallysolvablewithcertainnon-periodicboundaryconditionsifitadmitsboundaryweightswhichsatisfytheboundaryYang-Baxterequation[2].

Manysuchmodelsarenowknown.Restrictingourattentiontointeraction-round-a-facemodels,thesearetheA∞oreight-vertexsolid-on-solidmodel[3],theA(1)

Lorcyclicsolid-on-solidmodels[4],theALorAndrews-Baxter-Forrestermodels[5,6,7],thedilute

ALmodels[8],andcertainhigher-rankmodelsassociatedwithA(1)(1)(1)(1)

(2)n,Bn,Cn,DnandAn[9].Here,wepresentgeneralformsoftheboundaryweightsforsomeofthesepreviously-consideredmodelsandforsomeadditional,relatedmodels.

Webegin,inSection2,byoutliningthestandardrelations,includingtheYang-BaxterequationandtheboundaryYang-Baxterequation,whichmaybesatis edbythebulkandboundaryweightsofaninteraction-round-a-facemodel,andwede netwoimportanttypesofboundaryweight,diagonalandnon-diagonal.InSection3,weconsidercertainintertwiningpropertieswhichmaybesatis edbythebulkandboundaryweightsoftwoappropriately-relatedinteraction-round-a-facemodels.InSections4–9,weobtainboundaryweights,mostlyofthediagonaltype,whichrepresentgeneralsolutionsoftheboundaryYang-Baxterequationforvariousmodels.Thesemodels,someoftheirrelationshipsandthesectionsinwhichtheyareconsideredareindicatedinFigure1.Weconclude,inSection10,withadiscussionofgeneraltechniquesforsolvingtheboundaryYang-Baxterequation.