## 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] 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.

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