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

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. Show more