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

### 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.author | Bosnjak, Gary | |
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dc.date.accessioned | 2017-10-30T23:27:28Z | |

dc.date.available | 2017-10-30T23:27:28Z | |

dc.identifier.uri | http://hdl.handle.net/1885/132692 | |

dc.description.abstract | 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 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.iso | en | |

dc.subject | Yang-Baxter 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 Yang-Baxter 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 | |

Collections | Open Access Theses |

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