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On solutions to the Yang-Baxter equation related to sl(n)

Bosnjak, Gary

Description

In this thesis the problem of constructing solutions to the Yang-Baxter equation is considered. Such solutions are known as R-matrices 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 R-matrix...[Show more]

dc.contributor.authorBosnjak, Gary
dc.date.accessioned2017-10-30T23:27:28Z
dc.date.available2017-10-30T23:27:28Z
dc.identifier.urihttp://hdl.handle.net/1885/132692
dc.description.abstractIn this thesis the problem of constructing solutions to the Yang-Baxter equation is considered. Such solutions are known as R-matrices 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 R-matrix as a "composite object". Among the new results obtained is a formula for the elements of the general quantum affine sl(n) R-matrix for symmetric tensor representations with arbitrary weights in terms of multivariable q-hypergeometric series. This formula is shown to be factorised by more elementary R-matrices without the difference property. An explicit formula for the factors in terms of simple products is derived from the general formula by evaluating the R-matrix at special values of the spectral parameter. Using this factorisation a simple proof that the newly obtained R-matrix can be stochastic is given. Symmetries of the R-matrix generate identities of hypergeometric series which may be unknown. This new factorised representation of the R-matrix is compared with other constructions developed in the literature. It is shown that there is agreement up to simple transforms between all the R-matrices considered, thereby linking different approaches to solving Yang-Baxter 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 R-matrices. In some cases comparisons can only be made in the rational limit and using the newly obtained trigonometric R-matrix a quantum deformation of their construction is given. These deformations are used to discover new structure of the trigonometric R-matrix, such as a new L-operator 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 R-matrix is linked to recent developments in near-equilibrium 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 sum-to-unity rule.
dc.language.isoen
dc.subjectYang-Baxter equation
dc.subjectTetrahedron equation
dc.subjectKardar–Parisi–Zhang equation
dc.subjectIntegrable systems
dc.subjectQuantum algebras
dc.subjectRepresentation theory
dc.subjectHypergeometric series
dc.subjectHigh energy physics
dc.subjectMathematical physics
dc.subjectStatistical mechanics
dc.subjectSimple exclusion process
dc.subjectOrthogonal polynomials
dc.titleOn solutions to the Yang-Baxter equation related to sl(n)
dc.typeThesis (PhD)
local.contributor.supervisorMangazeev, Vladimir
local.contributor.supervisorcontactvladimir.mangazeev@anu.edu.au
dcterms.valid2017
local.description.notesthe author deposited 31/10/17
local.type.degreeDoctor of Philosophy (PhD)
dc.date.issued2017
local.contributor.affiliationDepartment of Theoretical Physics, ANU College of Science, The Australian National University
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