On solutions to the YangBaxter equation related to sl(n)
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In this thesis the problem of constructing solutions to the YangBaxter equation is considered. Such solutions are known as Rmatrices and we study a certain class of these related to the quantum affine sl(n) algebra. Using a variety of unrelated methods the matrix elements for different representations of the quantum group are constructed. In the process the structure of the solutions and their symmetries are detailed including a realisation of the Rmatrix...[Show more]
dc.contributor.author  Bosnjak, Gary  

dc.date.accessioned  20171030T23:27:28Z  
dc.date.available  20171030T23:27:28Z  
dc.identifier.other  b47392824  
dc.identifier.uri  http://hdl.handle.net/1885/132692  
dc.description.abstract  In this thesis the problem of constructing solutions to the YangBaxter equation is considered. Such solutions are known as Rmatrices and we study a certain class of these related to the quantum affine sl(n) algebra. Using a variety of unrelated methods the matrix elements for different representations of the quantum group are constructed. In the process the structure of the solutions and their symmetries are detailed including a realisation of the Rmatrix as a "composite object". Among the new results obtained is a formula for the elements of the general quantum affine sl(n) Rmatrix for symmetric tensor representations with arbitrary weights in terms of multivariable qhypergeometric series. This formula is shown to be factorised by more elementary Rmatrices without the difference property. An explicit formula for the factors in terms of simple products is derived from the general formula by evaluating the Rmatrix at special values of the spectral parameter. Using this factorisation a simple proof that the newly obtained Rmatrix can be stochastic is given. Symmetries of the Rmatrix generate identities of hypergeometric series which may be unknown. This new factorised representation of the Rmatrix is compared with other constructions developed in the literature. It is shown that there is agreement up to simple transforms between all the Rmatrices considered, thereby linking different approaches to solving YangBaxter equation. In the process comparisons between different formulae for the matrix elements are made which reveal that the 3D approach based on a new solution to the tetrahedron equation is the most efficient construction for this class of Rmatrices. In some cases comparisons can only be made in the rational limit and using the newly obtained trigonometric Rmatrix a quantum deformation of their construction is given. These deformations are used to discover new structure of the trigonometric Rmatrix, such as a new Loperator factorisation in the rank 1 case as well some new formulae for the generating function of the operator action. Some progress is made towards a more general formula for matrix elements in the case of arbitrary highest weight representations of sl(n). Using a factorisation approach by Derkachov et al. explicit formulae for the elements of the factors in the case n=3 is presented. These factors are shown to be related to the new trigonometric factorisation presented in this thesis. Finally, the stochastic Rmatrix is linked to recent developments in nearequilibrium stochastic systems of interacting particles of KPZ universality class. The factorisation of the matrix is shown to be equivalent to a "convolution" of the probability function describing these models. A generalisation of this probability function in the case of sl(3) is proposed which contains an extra parameter and seems to satisfy the sumtounity rule.  
dc.language.iso  en  
dc.subject  YangBaxter equation  
dc.subject  Tetrahedron equation  
dc.subject  Kardar–Parisi–Zhang equation  
dc.subject  Integrable systems  
dc.subject  Quantum algebras  
dc.subject  Representation theory  
dc.subject  Hypergeometric series  
dc.subject  High energy physics  
dc.subject  Mathematical physics  
dc.subject  Statistical mechanics  
dc.subject  Simple exclusion process  
dc.subject  Orthogonal polynomials  
dc.title  On solutions to the YangBaxter equation related to sl(n)  
dc.type  Thesis (PhD)  
local.contributor.supervisor  Mangazeev, Vladimir  
local.contributor.supervisorcontact  vladimir.mangazeev@anu.edu.au  
dcterms.valid  2017  
local.description.notes  the author deposited 31/10/17  
local.type.degree  Doctor of Philosophy (PhD)  
dc.date.issued  2017  
local.contributor.affiliation  Department of Theoretical Physics, ANU College of Science, The Australian National University  
local.identifier.doi  10.25911/5d70f1fd50878  
local.mintdoi  mint  
Collections  Open Access Theses 
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