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Treballs Finals de Grau de Matemàtiques, Facultat de Matemàtiques, Universitat de Barcelona, Any: 2018, Director: Toni Gascó, [en] Over the 1850-60’s, twenty years before Sophus Lie and Friedrich Engel started their research on Lie group theory, mathematicians such as A. Cayley, W. R. Hamilton and J. J. Sylvester introduced matrices and matrix groups thinking they had invented something of no possible use for natural scientists. On the contrary, Lie groups have taken an essential role in the theories of modern physics, and many questions have arisen. For example, why do Lie groups have such a fundamental role in physics? Why do they show up so often? How can they be used? The main objective of this research is to study of Lie groups and Lie algebras, based on the works of mathematicians such as Lie, Cartan, Killing and Weyl, in order to identify those features that make them essentially different and that will help us classify them. The second objective is to observe how the systems studied by physics are related to Lie group theory and to see that we can obtain new results thanks to the mathematical approach. This work is based on a bibliographic research to present a description of Lie group theory, with the posterior practical analysis of some specific cases. The first chapter is a historical retrospective to introduce the reader to the issue. We start looking at what the Standard Model of Particle Physics is and how it arose, then we follow with the history of the birth and consolidation of group theory as a mathematical theory, and we finish looking at how it was introduced, during the 20th century, the formulation of group theory in different aspects of science. In the second chapter we introduce the definition of Lie groups and Lie algebras and how they relate through the exponential function. Moreover, we study its basic concepts, such as the reducibility, and others which will be of great use afterwards. In the third chapter we apply these concepts and properties to classify of Lie groups using a simplification of the structure constants, which will allow us to see the underlying structure of the group. This structure will be represented with Root diagrams and Dynkin diagrams, which will reveal some constraints on the configuration of the possible diagrams, limiting the existence of Lie groups essentially different to a reduced number. In the fourth chapter we define the representation of a group as a map from an abstract group to a matrix group. We specifically introduce the adjoint representation of a group as the one that carries the structure constants in its form. In the fifth chapter we apply the concepts studied in Chapter 3 in order to analyze some particular Lie groups. Finally, the sixth chapter shows some examples of the form in which representations of different Lie groups appear in particle physics, evidencing some kind of symmetry that manifests the existence of conserved quantities.

Treballs Finals de Grau de Matemàtiques, Facultat de Matemàtiques, Universitat de Barcelona, Any: 2018, Director: Ricardo García López, [en] There has been an increment on the use of mathematics, beyond practical apliactions, in order to understand physical theories that, originally, were described by a set of equations modeling physical phenomena and nowadays are encompassed in more complex and complete mathematical theories. It is the case, for example, of Maxwell’s electrodinamics, that nowadays is part of the gauge theories that Yang-Mills generalized in 1954 and allowed to create the Standard Model theory, a $U (1) × SU (2) × SU (3)$ gauge theory. Mainly, these theories are based on three mathematical concepts, Lie groups, bundle theory and connections. The objective of this work is to define the necessary mathematical concepts in order to, firstly understand the mathematical theories themselves and then understand the results obtained when applied in a physical context. The idea behind these gauge theories is to study, in first place, the simetries of physical phenomena and relate them to elements of Lie groups. After fixing a diferential manifold where the theory will be build on, we define a principal bundle with the simetry group acting on it. That way we can understand connections over this principal bundle as gauge fields of the model. Matter fileds are then related to the associated vector bundles of the principal bundle. The interaction of all these fields is described by the covariant derivative that we will also define in the following sections. We divide this work in three main chapters. In the first one we will define Lie groups and its representations as well as Lie algebras. In the second one we will focus on bundle theory and on the principal bundle case and its connections. Finally we will study Maxwell’s electrodinamics (a $U (1)$-prinicpal bundle) and classic Yang-Mills theory (a $SU (n)$-principal bundle) and we will obtain their corresponding equations.

Treballs Finals de Grau de Matemàtiques, Facultat de Matemàtiques, Universitat de Barcelona, Any: 2017, Director: Laura Costa Farràs, [en] The present document is a first introduction to the Theory of Lie Groups and Lie Algebras and their representations. Lie Groups verify the characteristics of both a group and a smooth manifold structure. They arise from the need to study continuous symmetries, which is exactly what is needed for some branches of modern Theoretical Physics and in particular for quantum mechanics. The main objectives of this work are the following. First of all, to introduce the notion of a matrix Lie Group and see some examples, which will lead us to the general notion of Lie Group. From there, we will define the exponential map, which is the link to the notion of Lie Algebras. Every matrix Lie Group comes attached somehow to its Lie Algebra. Next we will introduce some notions of Representation Theory. Using the detailed examples of SU(2) and SU(3), we will study how the irreducible representations of certain types of Lie Groups are constructed through their Lie Algebras. Finally, we will state a general classification for the irreducible representations of the complex semisimple Lie Algebras.

Treballs Finals de Grau de Matemàtiques, Facultat de Matemàtiques, Universitat de Barcelona, Any: 2017, Director: Teresa Crespo Vicente, [en] The goal of this project is to classify the real forms of a closed subgroup of the special linear complex group of degree two. Previously we have to study several concepts of algebraic group theory such as galois cohomology. We expose many properties of forms of an algebraic group over a field $k$. We present the Kovacic algorithm for solving second order linear homogeneous differential equation over $\mathbb{C}(x)$.

Treballs Finals de Grau de Matemàtiques, Facultat de Matemàtiques, Universitat de Barcelona, Any: 2016, Director: Ricardo García López, Lie groups and Lie algebras are the basic objects of study of this work. Lie studied them as continuous transformations of partial differential equations, emulating Galois work with polynomial equations. The theory went much further thanks to Killing, Cartan and Weyl and now the wealth of properties of Lie groups makes them a central topic in modern mathematics. This richness comes from the merging of two initially unrelated mathematical structures such as the group structure and the smooth structure of a manifold, which turns out to impose many restrictions. For instance, a closed subgroup of a Lie group is automatically an embedded submanifold of the Lie group. Symmetries are related to groups, in particular continuous symmetries are related to Lie groups and whence, by Noether’s theorem, its importance in modern physics. In this work, we focus on the Lie group - Lie algebra relationship and on the representation theory of Lie groups through the representations of Lie algebras. Especially, we analyze the complex representations of Lie algebras related to compact simply connected Lie groups. With this purpose, we first study the theory of covering spaces and differential forms on Lie groups. Finally, an application to particle physics is presented which shows the role played by the representation theory of SU(3) on flavour symmetry and the theory of quarks.

Treballs Finals de Grau de Matemàtiques, Facultat de Matemàtiques, Universitat de Barcelona, Any: 2015, Director: Pilar Bayer Isant, La teoría de representaciones y la teoría de Lie son dos ramas relativamente modernas de las matemáticas. A pesar de ello, han evolucionado rápidamente, y no hubo de transcurrir mucho tiempo desde los inicios de su desarrollo hasta que se empezaron a encontrar aplicaciones en física teórica. Nosotros presentaremos los conceptos básicos tanto de la teoría de representaciones, desarrollada principalmente por Ferdinand Gerog Frobenius e Issai Schur entre finales del siglo XIX y principios del XX, como de la teoría de Lie, ideada por Sophus Lie a finales del siglo XIX y extendida durante los años posteriores por diversos matemáticos como Wilhelm Killing y Elie Cartan. Una vez provistos con las herramientas necesarias, nos centraremos en el estudio de diversos grupos de Lie específicos, como SO(3) o SL(2, $\mathbb{C}$). Todo ello se hará con un objetivo en mente: encontrar las representaciones irreducibles de dimensión finita del grupo propio de Lorentz, SO$^{0}$ (1, 3), que es el grupo formado por las transformaciones de Lorentz, que describen el movimiento relativo de dos cuerpos de acuerdo con la teoría de la relatividad especial. He aquí la primera relación que hallamos con la física teórica. Pero no es la última, ya que estos resultados acaban dando pie a la obtención de la denominada ecuación de Dirac, cuyas soluciones son funciones de onda de partículas de spin $\frac{1}{2}$, como el electrón. Las páginas de este trabajo incluyen múltiples resultados de gran relevancia en sus campos; desde la unitarización de las representaciones de grupos finitos, cuya generalización para todo tipo de grupos compactos se conoce como unitarian trick, hasta la obtención de las representaciones de dimensión finita del grupo SU(2) mediante el espacio de polinomios homogéneos de dos variables como espacio de representación, debida a Hermann Weyl. Además, también contienen algo de trabajo propio como el cálculo de las matrices que actúan como coeficientes en la ecuación de Dirac.

Treballs Finals de Grau de Matemàtiques, Facultat de Matemàtiques, Universitat de Barcelona, Any: 2015, Director: Vicenç Navarro Aznar, Even though the title of this composition you are reading is Gauge fields, this is just one of the many concepts that appear throughout the work. Moreover, despite the fact that this is rather a physical term (the actual mathematical name would be connections on principal G-bundles), the contents in this exposition are mathematical. As a student of both mathematics and physics, I wanted to choose a topic that could be covered in both aspects, a subject which could be interpreted physically but which had a mathematical background. This composition intends to contain the mathematical background to develop the so-called gauge theories used in modern physics such as quantum physics, particularly quantum field theory (QFT), for example quantum electrodynamics (QED) or quantum chromodynamics (QCD). However, as stated, this contains no physical conclusions, but only mathematical, with maybe some references to their physical applications. Thus, the motivation was a combination of my interest on physics (which may explain the outline and selection of topics) but also my fascination for differential geometry and topology.