As Álgebras Quânticas foram recentemente introduzidas como uma generalização das álgebras de Lie clássicas e estão sendo intensamente investigadas, tanto de um ponto de vista matemático quanto em aplicações envolvendo problemas de Mecânica Estatística e Física Molecular. As representações dessas álgebras podem ser construídas a partir de técnicas tradicionais e apresentam novidades se o parâmetro de deformação q for uma raiz complexa da unidade, e neste caso pode ocorrer perda de irredutibilidade e conseqüentemente alterações nas dimensões dessas representações. Primeiramente, estudamos as representações no caso clássico, a seguir introduzimos as deformações quânticas nas relações de comutação envolvendo os geradores associados as raízes simples. Posteriormente, estudamos especificamente o caso em que q é uma raiz complexa da unidade, à procura de novas reduções dimensionais que não aparecem no caso clássico. Mais precisamente, nos detemos ao estudo das representações das álgebras deformadas Uq(su(2)) e Uq(sl(3)), determinando suas a dimensões, os vetores de base do espaço portador e as suas matrizes irredutíveis. Por fim, calculamos o operador de Casimir quadrático deformado procurando saber como ficam as regras de ramificação da cadeia Uq(sl(3)) &#8835 Uq(sl(2)). / The Quantum Algebras has been recently introduced as a generalization of classical Lie algebras. The representations of these algebras can be built from the traditional techniques and arise novelty, if the parameter of deformation q is a root of unity, in this case, can occur loss of irreducibility and consequently alteration in the dimension of these representations. First of all, we study the representations in the classic case, after that we introduce the quantum deformation in the commuting relations involving the generators associated with the simple roots. Subsequently we studied specifically the case that q is a root of unity, searching for dimensional reduction that do not appear in the classic algebras. More exactly, we studied the deformed representations of Uq(su(2)) e Uq(sl(3)), determining t heir dimensions, t he base vectors of t he carrier space and their irreducible matrices. Finally, we calculated the deformed quadratic Casimir operator in the chain Uq(sl(3)) &#8835 Uq(sl(2)).

We start with the recollection of basic results about differential manifolds and Lie groups. We also recall some preliminary terminologies in Lie algebra. Then we define the Lie algebra corresponding to a Lie group. In the next section, we define a strongly continuous representation of a Lie group on a Banach space. We further define the smooth, analytic and entire vectors for a given representation. Then, we move on to develop some necessary and sufficient criteria to characterize smooth, analytic and entire vectors. We, in particular, take into account of some specific representations of Lie groups like the regular representation of R, the irreducible representations of Heisenberg groups, the irreducible representations of the group of Affine transformations and finally the representations of non-compact simple Lie groups.

京都大学 / 0048 / 新制・課程博士 / 博士(理学) / 甲第21540号 / 理博第4447号 / 新制||理||1639(附属図書館) / 京都大学大学院理学研究科数学・数理解析専攻 / (主査)教授 玉川 安騎男, 教授 向井 茂, 教授 望月 新一 / 学位規則第4条第1項該当 / Doctor of Science / Kyoto University / DGAM

The non-resonant bounded uniserial representations of Vec(R) form a certain class of extensions composed of tensor density modules, all of whose subquotients are indecomposable. The problem of classifying the extensions with a given composition series is reduced via cohomological methods to computing the solution of a certain system of polynomial equations in several variables derived from the cup equations for the extension. Using this method, we classify all non-resonant bounded uniserial extensions of Vec(R) up to length 6. Beyond this length, all such extensions appear to arise as subquotients of extensions of arbitrary length, many of which are explained by the psuedodifferential operator modules. Others are explained by a wedge construction and by the pseudodifferential operator cocycle discovered by Khesin and Kravchenko.

This dissertation studies quantum decoherence in anharmonic oscillator systems to monitor and understand the way the systems evolve. It also explores methods to control the systems' evolution, and the effects of decoherence when applicable. We primarily do this by finding the time evolution of the systems using their Lie algebraic structures. We solve for a generalized Caldirola-Kanai Hamiltonian, and propose a general way to produce a desired evolution of the system. We apply the analysis to the effects of Dirac delta fluctuations in mass and frequency, both separately and simultaneously. We also numerically demonstrate control of the generalized Caldirola-Kanai system for the case of timed Gaussian fluctuations in the mass term. This is done in a way that can be applied to any system that is made up of a Lie algebra. We also explore the evolution of an optomechanical coupled mirror-laser system while maintaining a second order coupling. This system creates anharmonic effects that can produce cat states which can be used for quantum computing. We find that the decoherence in this system causes a rotational smearing effect in the Husimi function which, with the second order term added, causes rotational smearing after a squeezing effect. Finally, we also address the dynamic evolution and decoherence of an anharmonic oscillator with infinite coupling using the Born-Markov master equation. This is done by using the Lie algebraic structure of the Born-Markov master equation's superoperators when applying a strategic mean field approximation to maintain dynamic flexibility. The system is compared to the Born-Markov master equation for the harmonic oscillator, the regular anharmonic oscillator, and the dynamic double anharmonic oscillator. Throughout, Husimi plots are provided to visualize the dynamic decoherence of these systems.

Superconformal field theories (SCFTs) are an important class of quantum field theories. These SCFTs have been a significant component in exploring and comprehending the fundamental framework of quantum field theories. In the vast realm of quantum field theories, integrability plays a crucial role, providing powerful analytic tools that allow us to solve certain physical quantities exactly. In this thesis, we focus on the representation theory of the algebraic structure in six-dimensional (6D) SCFTs and investigate the intricate interplay between 6D SCFTs and integrability. To begin, we delve into the fundamental concepts of representation theory, establishing a solid foundation for our subsequent analysis. The discussion then will move on to all possible generators in the SCFTs, explaining how they are realized in terms of bosonic and fermionic oscillators. Finally, we investigate spin chains and their application in 6D SCFTs. We demonstrate that symmetry arguments derived from representation theory are not sufficient to establish the integrability of the spin chains in 6D SCFTs. This conclusion does not imply the absence of integrable systems within 6D SCFTs; rather, it suggest there are other potential methods available e.g., correlation functions, to explore the appearance of integrable systems in 6D SCFTs. / Superkonforma fältteorier (SCFTs) är en viktig klass av kvantfältteorier. Dessa SCFTs utgör en viktig komponent för att utforska och förstå det fundamentala ramverket för kvantfältteorin. Inom det stora riket av kvantfältteori spelar integrabilitet en avgörande roll, vilket tillhandahåller kraftfulla analytiska verktyg som gör att vi kan lösa vissa fysiska storheter exakt. I denna avhandling fokuserar vi på representationsteorin av den algebraiska strukturen i sexdimensionella (6D) SCFTs och undersöker det intrikat samspelet mellan 6D SCFTs och integrabilitet. Till att börja med kommer vi att fördjupa oss i de grundläggande begreppen inom representationsteori och skapa en gedigen grund för vår efterföljande analys.  Diskussionen kommer sedan att gå vidare till alla möjliga generatorer i SCFTs, och förklarar hur de realiseras i termer av bosoniska och fermioniska oscillatorer. Slutligen kommer spinnkedjor och dess tillämpningar i 6D SCFTs att undersökas. Vi kommer visa att symmetriargument som härleds från representationsteori inte är  tillräckliga för att fastställa integrerbarhet av spinnkedjor i 6D SCFTs. Denna slutsats innebär inte att integrerbara system inte existerar inom 6D SCFTs, utan föreslår att det finns andra potentiella metoder, till exempel korrelationsfunktioner, för att utforska existensen av integrerbara system i 6D SCFTs.

京都大学 / 新制・課程博士 / 博士(理学) / 甲第23685号 / 理博第4775号 / 新制||理||1684(附属図書館) / 京都大学大学院理学研究科数学・数理解析専攻 / (主査)教授 荒川 知幸, 教授 玉川 安騎男, 教授 並河 良典 / 学位規則第4条第1項該当 / Doctor of Science / Kyoto University / DFAM

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京都大学 / 新制・課程博士 / 博士(理学) / 甲第23678号 / 理博第4768号 / 新制||理||1683(附属図書館) / 京都大学大学院理学研究科数学・数理解析専攻 / (主査)教授 加藤 周, 教授 雪江 明彦, 教授 池田 保 / 学位規則第4条第1項該当 / Doctor of Science / Kyoto University / DFAM

We first introduce the octonions as an eight dimensional vector space over a field of characteristic zero with a multiplication defined using a table. We also show that the multiplication rules for octonions can be derived from a special graph with seven vertices call the Fano Plane. Next we explain the Cayley-Dickson construction, which exhibits the octonions as the set of ordered pairs of quaternions. This approach parallels the realization of the complex numbers as ordered pairs of real numbers. The rest of the thesis is devoted to following a paper by N. Jacobson written in 1939 entitled "Cayley Numbers and Normal Simple Lie Algebras of Type G". We prove that the algebra of derivations on the octonions is a Lie algebra of type G_2. The proof proceeds by showing the set of derivations on the octonions is a Lie algebra, has dimension fourteen, and is semisimple. Next, we complexify the algebra of derivations on the octonions and show the complexification is simple. This suffices to show the complexification of the algebra of derivations is isomorphic to g_2 since g_2 is the only semisimple complex Lie algebra of dimension fourteen. Finally, we conclude the algebra of derivations on the octonions is a simple Lie algebra of type G_2. / Master of Science