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Thesis (Ph.D.) - University of Queensland, 2005. / Includes bibliography.

In this dissertation, we study two areas of interest in knot theory: Random knots in the unit cube, and the Yang-Baxter solutions constructed from Frobenius algebras. The study of random knots can be thought of as a model of DNA strings situated in confinement. A random knot with n vertices is a polygonal loop formed by selecting n distinct points in the unit cube, for a positive integer n, and connecting these points by straight line segments successively, such that the last point selected is joined with the first one. We present a step by step description of our algorithm and Maple codes for generating random knots in the unit cube, with a given vertex number n. To detect non-trivial knots, we use a knot invariant called the determinant. We present an algorithm and its Maple code for computing the determinant for random knots. For each vertex number n, we generate large number of random knots and form data sets of values of the determinant. Then we analyze our data sets in various ways. For instance, for each vertex number n, we form data sets of the number of p-colorable random knots by finding the set of prime divisors of each determinant output. We define the stick number for p-colorability to be the minimum number of line segments required to form a p-colorable knot. We use our data sets to find upper bounds for stick numbers for p-colorability, for primes p _ 191. We also find distributions of p-colorable knots and small determinant values. The second topic on random knots is the linking number of random links. A random link is a collection of disjoint random knots produced simultaneously. We present descriptions of our algorithm and its Maple code for constructing random links of two components, and calculating their linking numbers in detail. By running the code for 1000 times, for the vertex number n less than or equal to 30, we obtain data sets of linking numbers for two-component random links such that each component is a random knot with n vertices. Then we find the distribution of linking numbers and calculate upper bounds for the stick number for the linking numbers ` _ 15. The second area we investigate is applications of Fobenius algebras to knot theory. Chain complexes and Yang-Baxter solutions (R-matrices) are constructed by the skein theoretic approach using Frobenius algebras, and deformed R-matrices are constructed by using 2-cocyles. We compute cohomology groups, Yang-Baxter solutions and their cocycle deformations for group algebras, polynomial algebras and complex numbers. We construct knot and link invariants using these R-matrices from Frobenius algebras via Turaev’s criteria. Then a series of skein relations of the invariant are introduced for oriented knot or link diagrams. We also present calculations of the Frobenius skein invariant for various knots and links.

A little over 20 years ago Drinfeld presented the quantum (or Drinfeld) double construction. This construction takes any Hopf algebra and embeds it in a larger quasi-triangular Hopf algebra, which contains an algebraic solution to the constant Yang–Baxter equation. One such class of algebras consists of the Drinfeld doubles of finite groups, which are currently of interest due to their connections with non-Abelian anyons. The smallest non-commutative Drinfeld double of a finite group algebra is the Drinfeld double of D3 , the dihedral group of order six, which was recently used to construct solutions to the Yang–Baxter equation cor- responding to 2-state and 3-state integrable spin chains with periodic boundary conditions. In this thesis we construct R-matrices from the Drinfeld double of dihedral group algebras, D(Dn) and consider their associated integrable systems. The 3-state spin chain from D(D3) is generalised to include open boundaries and it is also shown that there exists a more general R-matrix for this algebra. For general D(Dn) an R-matrix is constructed as a descendant of the zero-field six-vertex model.

A little over 20 years ago Drinfeld presented the quantum (or Drinfeld) double construction. This construction takes any Hopf algebra and embeds it in a larger quasi-triangular Hopf algebra, which contains an algebraic solution to the constant Yang–Baxter equation. One such class of algebras consists of the Drinfeld doubles of finite groups, which are currently of interest due to their connections with non-Abelian anyons. The smallest non-commutative Drinfeld double of a finite group algebra is the Drinfeld double of D3 , the dihedral group of order six, which was recently used to construct solutions to the Yang–Baxter equation cor- responding to 2-state and 3-state integrable spin chains with periodic boundary conditions. In this thesis we construct R-matrices from the Drinfeld double of dihedral group algebras, D(Dn) and consider their associated integrable systems. The 3-state spin chain from D(D3) is generalised to include open boundaries and it is also shown that there exists a more general R-matrix for this algebra. For general D(Dn) an R-matrix is constructed as a descendant of the zero-field six-vertex model.

In this thesis we review some aspects of Yang-Baxter deformations of semi-symmetric space sigma models. We start by giving a short review of the sigma model description of superstrings and then we offer a self contained introduction to the Yang-Baxter deformation technique. We then show how to obtain an integrable deformation of the hybrid sigma model. Also, we show that the gravity dual of beta-deformed ABJM theory can be obtained as a Yang-Baxter deformation. This is done by selecting a convenient combination of Cartan generators in order to construct an Abelian r-matrix satisfying the classical Yang-Baxter equation. / Nesta tese revisamos alguns aspectos das deformações de Yang-Baxter de modelos sigma em espaços semi-simétricos. Damos uma breve revisão do modelo sigma de supercordas e, em seguida, oferecemos uma introdução ao método de deformação de Yang-Baxter. Em seguida, mostramos como obter uma deformação integrável do modelo sigma híbrido. Além disso, mostramos que o dual gravitacional da teoria ABJM beta-deformada pode ser obtida como uma deformação de Yang-Baxter. Isso é feito selecionando-se uma combinação conveniente de geradores de Cartan para construir uma matriz r Abeliana satisfazendo a equação clássica de Yang-Baxter.

In this thesis we improve and extend an algebraic technique pioneered by M. Gaudin. The technique is based on an infinite dimensional Lie algebra and a related family of mutually commuting Hamiltonians. In order to find energy eigenvalues of such Hamiltonians one has to solve the equations of Bethe ansatz. However, in most cases analytical solutions are not available. In this study we examine a special case for which analytical solutions of Bethe ansatz equations are not needed. Instead, some special properties of these equations are utilized to evaluate the energy eigenvalues. We use this method to find exact expressions for the energy eigenvalues of a class of interacting boson models. In addition to that, we also introduce a q-deformation of the algebra of Gaudin. This deformation leads us to another family of mutually commuting Hamiltonians which we diagonalize using algebraic Bethe ansatz technique. The motivation for this deformation comes from a relationship between Gaudin algebra and a spin extension of the integrable model of F. Calogero. Observing this relation, we then consider a well known periodic version of Calogero&#039 / s model which is due to B. Sutherland. The search for a Gaudin-like algebraic structure which is in a similar relationship with the spin extension of Sutherland&#039 / s model naturally leads to the above mentioned q-deformation of Gaudin algebra. The deformation parameter q and the periodicity d of the Sutherland model are related by the formula q=i{pi}/d.

Made available in DSpace on 2016-06-02T20:16:47Z (GMT). No. of bitstreams: 1 3474.pdf: 452458 bytes, checksum: 7857dc28822e6d45234586ddd2b5e98e (MD5) Previous issue date: 2011-03-02 / Universidade Federal de Minas Gerais / In this work we study the solutions of the Yang-Baxter equation associated to nineteen vertex models invariant by the parity-time symmetry from the perspective of algebraic geometry. We determine the form of the algebraic curves constraining the respective Boltzmann weights and found that they possess a universal structure. This allows us to classify the integrable manifolds in four different families reproducing three known models besides uncovering a novel nineteen vertex model in a unified way. The introduction of the spectral parameter on the weights is made via the parameterization of the fundamental algebraic curve which is a conic. The diagonalization of the transfer matrix of the new vertex model and its thermodynamic limit properties are discussed. We point out a connection between the form of the main curve and the nature of the excitations of the corresponding spin-1 chains. / Nesta dissertação estudamos as possíveis soluções da equação de Yang-Baxter para modelos de dezenove vértices invariantes por simetria de paridade e reversão temporal do ponto de vista da geometria algébrica. Determinamos a forma das curvas algébricas que vinculam os respectivos pesos de Boltzmann e descobrimos que suas estruturas são universais. Com tal observação foi possível classificar, de uma maneira unificada, as variedades algébricas integráveis em quatro diferentes famílias, três delas já conhecidas e uma delas correspondendo a um novo modelo de dezenove vértices. A introdução de um parâmetro espectral nos pesos de Boltzmann é feita através da parametrização da curva algébrica fundamental, que é uma crônica. A diagonalização da matriz de transferência do novo modelo de vértices bem como suas propriedades no limite termodinâmico são discutidas. Mencionamos ainda uma curiosa conexão entre a forma da curva principal e a natureza das excitações das Hamiltonianas de spin-1 associadas aos modelos de vértices.

In this thesis we review some aspects of Yang-Baxter deformations of semi-symmetric space sigma models. We start by giving a short review of the sigma model description of superstrings and then we offer a self contained introduction to the Yang-Baxter deformation technique. We then show how to obtain an integrable deformation of the hybrid sigma model. Also, we show that the gravity dual of beta-deformed ABJM theory can be obtained as a Yang-Baxter deformation. This is done by selecting a convenient combination of Cartan generators in order to construct an Abelian r-matrix satisfying the classical Yang-Baxter equation. / Nesta tese revisamos alguns aspectos das deformações de Yang-Baxter de modelos sigma em espaços semi-simétricos. Damos uma breve revisão do modelo sigma de supercordas e, em seguida, oferecemos uma introdução ao método de deformação de Yang-Baxter. Em seguida, mostramos como obter uma deformação integrável do modelo sigma híbrido. Além disso, mostramos que o dual gravitacional da teoria ABJM beta-deformada pode ser obtida como uma deformação de Yang-Baxter. Isso é feito selecionando-se uma combinação conveniente de geradores de Cartan para construir uma matriz r Abeliana satisfazendo a equação clássica de Yang-Baxter.

L'anneau $\Diff(n)$ des opérateurs différentiels $\h$-déformés apparaît dans la théorie des algèbres de réduction.Dans cette thèse, nous construisons les anneaux des opérateurs différentiels généralisés sur les espaces vectoriels $\h$-déformés de type $\gl$. Contrairement aux espaces vectoriels $q$-déformés pour lequel l'anneau des opérateurs différentiels est unique \`a isomorphisme pr\`es, l'anneau généralisé des opérateurs différentiels $\h$-déformés $\Diffs(n)$ est indexée par une fonction rationnelle $\sigma$ en $n$ variables, solution d'un syst\`eme d\'eg\'en\'er\'e d'\'equations aux diff\'erences finies. Nous obtenons la solution g\'en\'erale de ce syst\`eme. Nous montrons que le centre de $\Diffs(n)$ est un anneau des polynômes en $n$ variables. Nous construisons un isomorphisme entre des localisations de l'anneau $\Diffs(n)$ et de l’algèbre de Weyl $\text{W}_n$ l’étendue par $n$ indéterminés. Nous présentons des conditions irréductibilité des modules de dimension fini de $\Diffs(n)$. Finalement, nous discutons des difficultés a trouver les constructions analogues pour l'anneau $\Diff(n,N)$ correspondant \`a $N$ copies de $\Diff(n)$. / The ring $\Diff(n)$ of $\h$-deformed differential operators appears in the theory of reduction algebras. In this thesis, we construct the rings of generalized differential operators on the $\h$-deformed vector spaces of $\gl$-type. In contrast to the $q$-deformed vector spaces for which the ring of differential operators is unique up to an isomorphism, the general ring of $\h$-deformed differential operators $\Diffs(n)$ is labeled by a rational function $\sigma$ in $n$ variables, satisfying an over-determined system of finite-difference equations. We obtain the general solution of the system. We show that the center of $\Diffs(n)$ is a ring of polynomials in $n$ variables. We construct an isomorphism between certain localizations of $\Diffs(n)$ and the Weyl algebra $\W_n$ extended by $n$ indeterminates. We present some conditions for the irreducibility of the finite dimensional $\Diffs(n)$-modules. Finally, we discuss difficulties for finding analogous constructions for the ring $\Diff(n, N)$ formed by several copies of $\Diff(n)$.