Made available in DSpace on 2015-03-26T13:45:35Z (GMT). No. of bitstreams: 1 texto completo.pdf: 956809 bytes, checksum: ebf2affe7b281f8af02d3a0fdd8101f6 (MD5) Previous issue date: 2012-02-27 / Coordenação de Aperfeiçoamento de Pessoal de Nível Superior / Special biserial algebras are a class of algebras that appear in many contexts. Butler and Ringel [6] made a description of indecomposable modules and irreducible morphisms of algebras string, a subclass of special biserial algebras. We show that special biserial algebras which are not string, have only one module projective-injective indecomposable for each binomial relation. We are present the Auslander-Reiten sequence in which these modules appear. Then we verify that the remainder of Auslander-Reiten quiver of special biserial algebras is obtained as done by Butler and Ringel [6] for string algebras. We conclude this work by applying the above results for the representations of the algebras of finite cyclic groups and algebras of the Klein group and diedral groups over algebraically closed field of characteristic 2. / Álgebras bisseriais especiais formam uma classe de álgebras que aparecem em diferentes contextos. A aplicabilidade destas álgebras que estamos interessados é no estudo de representações de algumas álgebras de grupo não semissimples sobre corpos algebricamente fechados. Para isso, descrevemos, a menos de isomorfismos, seus módulos indecomponíveis e seus morfismos irredutíveis. Tal descrição é feita através de uma bela apresentação combinatória, dada por Butler e Ringel [6], dos módulos indecomponíveis e dos morfismos irredutíveis de um caso particular de álgebras bisseriais especiais, as álgebras string. No caso geral, de álgebras bisseriais especiais que não são string, mostramos que são acrescentados apenas um módulo projetivo-injetivo indecomponível para cada relação binomial. Apresentamos a sequência de Auslander-Reiten em que estes módulos aparecem e verificamos que, a menos destas sequências, o restante do quiver de Auslander-Reiten é obtido como feito por Butler e Ringel [6] para álgebras string. Para módulos string, apresentamos ainda uma descrição gráfica de uma base dos espaços de morfismos, de acordo com Crawley-Boevey [7]. Finalizamos o trabalho aplicando os resultados acima para obter as representações das álgebras de grupos cíclicos finitos e para as álgebras do grupo de Klein e dos grupos dihedrais sobre corpos algebricamente fechados de característica 2.

京都大学 / 0048 / 新制・課程博士 / 博士(理学) / 甲第21535号 / 理博第4442号 / 新制||理||1638(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)准教授 加藤 周, 教授 重川 一郎, 教授 並河 良典 / 学位規則第4条第1項該当 / Doctor of Science / Kyoto University / DFAM

The Standard model of particle physics provides a successful theory to understand the experimental results of the electroweak and strong interactions. However, it does not have a satisfactory explanation for the hierarchy problem. Many extensions of the Standard Model that solve the hierarchy problem result in new particles. We will study the phenomenology of vector-like fermions resulting in theories where the Higgs boson is typically a pseudo-Nambu-Goldstone boson. In these theories we study the case where a heavy fermion will be heavier than a heavy gluon, and then the channel of a heavy fermion decaying into a color octet is considered. We study this phenomenology at high energy colliders, both the LHC as well as future machines. / O modelo padrão de física de partículas fornece uma teoria bem-sucedida para entender os resultados experimentais das interações eletrofracas e fortes. No entanto, não tem uma explicação satisfatória para o problema de hierarquia. Muitas extensões do Modelo Padrão que resolvem o problema hierarquia resultam em novas partículas. Estudaremos a fenomenologia de férmions vetoriais resultando em teorias onde o bóson de Higgs é tipicamente um bóson pseudo-Nambu-Goldstone. Nessas teorias, estudaremos o caso em que um férmion pesado será mais pesado do que um glúon pesado, e então o canal de um férmion pesado decaindo em um octeto de cor é considerado. Estudamos esta fenomenologia em colisores de alta energia, tanto para o LHC quanto as futuras máquinas.

The Standard model of particle physics provides a successful theory to understand the experimental results of the electroweak and strong interactions. However, it does not have a satisfactory explanation for the hierarchy problem. Many extensions of the Standard Model that solve the hierarchy problem result in new particles. We will study the phenomenology of vector-like fermions resulting in theories where the Higgs boson is typically a pseudo-Nambu-Goldstone boson. In these theories we study the case where a heavy fermion will be heavier than a heavy gluon, and then the channel of a heavy fermion decaying into a color octet is considered. We study this phenomenology at high energy colliders, both the LHC as well as future machines. / O modelo padrão de física de partículas fornece uma teoria bem-sucedida para entender os resultados experimentais das interações eletrofracas e fortes. No entanto, não tem uma explicação satisfatória para o problema de hierarquia. Muitas extensões do Modelo Padrão que resolvem o problema hierarquia resultam em novas partículas. Estudaremos a fenomenologia de férmions vetoriais resultando em teorias onde o bóson de Higgs é tipicamente um bóson pseudo-Nambu-Goldstone. Nessas teorias, estudaremos o caso em que um férmion pesado será mais pesado do que um glúon pesado, e então o canal de um férmion pesado decaindo em um octeto de cor é considerado. Estudamos esta fenomenologia em colisores de alta energia, tanto para o LHC quanto as futuras máquinas.

In this thesis we describe an algorithm and implement a construction of a projective resolution and minimal projective resolution in the representation the- ory of finite-dimensional algebras. In this thesis finite-dimensional algebras are KQ /I where KQ is a path algebra and I is an admissible ideal. To implement the algorithm we use the package QPA [9] for GAP [2]. We use the theory of Gröbners basis of KQ-modules and the theory described in article Minimal Pro- jective Resolutions written by Green, Solberg a Zacharia [5]. First step is find a direct sum such that i∈Tn fn∗ i KQ = i∈Tn−1 fn−1 i KQ ∩ i∈Tn−2 fn−2 i I. Next important step to construct the minimal projective resolution is separate nontri- vial K-linear combinations in i∈Tn−1 fn−1 i I + i∈Tn fn i J from fn∗ i . The Modules of the minimal projective elements are i∈Tn (fn i KQ)/(fn i I). 1

Doctor of Philosophy / Department of Mathematics / Zongzhu Lin / A representation of a quiver [Gamma] over a commutative ring R assigns an R-module to each vertex and an R-linear map to each arrow. In this dissertation, we consider R = k[t]/(t[superscript]n) and all R-free representations of [Gamma] which assign a free R-module to each vertex. The category, denoted by Rep[superscript]f[subscript] R([Gamma]), containing all such representations is not an abelian category, but rather an exact category. In this dissertation, we firstly study the Hall algebra of the category Rep[superscript]f[subscript] R([Gamma]), denote by [Eta](R[Gamma]), for a loop-free quiver [Gamma]. A geometric realization of the composition subalgebra of [Eta](R[Gamma]) is given under the framework of Lusztig's geometric setting. Moreover, the canonical basis and a monomial basis of this subalgebra are constructed by using perverse sheaves. This generalizes Lusztig's result about the geometric realization of quantum enveloping algebra. As a byproduct, the relation between this subalgebra and quantum generalized Kac-Moody algebras is obtained. If [Gamma] is a Jordan quiver, which is a quiver with one vertex and one loop, each representation in Rep[superscript]f[subscript] R([Gamma]), gives a matrix over R when we fix a basis of the free R-module. An interesting case arises when considering invertible matrices. It then turns out that one is dealing with representations of the group GL[subscript]m(k[t]/(t[superscript]n)). Character sheaf theory is a geometric character theory of algebraic groups. In this dissertation, we secondly construct character sheaves on GL[subscript]m(k[t]/(t[superscript]2)). Then we define an induction functor and restriction functor on these perverse sheaves.

This thesis constructs tilting bundles obtained from full strong exceptional collections of line bundles on all smooth toric Fano fourfolds. The tilting bundles lead to a large class of explicit Calabi-Yau-5 algebras, obtained as the corresponding rolled-up helix algebra. We provide two different methods to show that a collection of line bundles is full, whilst the strong exceptional condition is checked using the package QuiversToricVarieties for the computer algebra system Macaulay2, written by the author. A database of the full strong exceptional collections can also be found in this package.

In this thesis, I apply methods from the representation theory of finite dimensional algebras to the study of versal and universal deformation rings. The main idea is that more sophisticated results from representation theory can be used to arrive at a deeper understanding of deformation rings. Such rings arise naturally in a variety of problems in number theory and group representation theory. This thesis has two parts. In the first part, Λ is an arbitrary finite dimensional algebra over a field k. If V is a finitely generated Λ-module, I prove that V has a versal deformation ring R(Λ, V ). Moreover, if Λ is self-injective and the stable endomorphism ring of V is isomorphic to k, then R(Λ, V ) is universal. If additionally A is a Frobenius algebra and Ω(Λ) denotes the syzygy operator over Λ, I show that the universal deformation rings of V and Ω(V) are isomorphic. In the second part, I analyze a particular finite dimensional Frobenius algebra Λ over an algebraically closed field k for which all the finitely generated indecomposable modules can be described combinatorially by using certain words in Λ. I use this description to visualize the indecomposable Λ-modules in the stable Auslander-Reiten quiver of Λ and determine all the components of this stable Auslander-Reiten quiver which contain Λ-modules whose endomorphism ring is isomorphic to k. Finally I determine the universal deformation rings of all the modules in these components whose stable endomorphism ring is isomorphic to k.

We attack the question of BPS occupancy in a wide class of 4d N = 2 quantum field theories. We first review the Seiberg-Witten approach to finding the low energy Wilsonian effective action actions of such theories. In particular, we analyze the case of Gaiotto theories, which provide a large number of non-trivial examples in a unified framework. We then turn to understanding the massive BPS spectrum of such theories, and in particular their relation to BPS quivers. We present a purely 4d characterization of BPS quivers, and explain how a quiver's representation theory encodes the solution to the BPS occupancy problem. Next, we derive a so called mutation method, based on exploiting quiver dualities, to solve the quiver's representation theory. This method makes previously intractable calculations nearly trivial in many examples. As a particular highlight, we apply our methods to understand strongly coupled chambers in ADE SYM gauge theories with matter. Following this, we turn to the general story of quivers for theories of the Gaiotto class. We present a geometric approach to attaining quivers for the rank 2 theories, leading to a very elegant solution which includes a specification of quiver superpotentials. Finally, we solve these theories by an unrelated method based on gauging flavor symmetries in their various dual weakly coupled Lagrangian descriptions. After seeing that this method agrees in the rank 2 case, we will apply our new approach to the case of rank n. / Physics

The object of this work is to exploit various constructions of string theory and M-theory to yield new insights into supersymmetric theories in both four and three dimensions. In 4d, we extend work on Seiberg-Witten theory to study and compute BPS spectra of the class of complete N = 2 theories. The approach we take is based on the program of geometric engineering, in which 4d theories are constructed from compactifications of type IIB strings on Calabi-Yau manifolds. In this setup, the natural candidates for BPS states are D3 branes wrapped on supersymmetric 3-cycles in the Calabi-Yau. Our study makes use of the mathematical structure of quivers, whose representation theory encodes the notion of stability of BPS particles. Except for 11 exceptional cases, all complete theories can be constructed by wrapping stacks of two M5 branes on Riemann surfaces. By exploring the connection between quivers and M5 brane theories, we develop a powerful algorithm for computing BPS spectra, and give an in-depth study of its applications. In particular, we compute BPS spectra for all asymptotically free complete theories, as well as an infinite set of conformal \(SU(2)^k\) theories with certain matter content. From here, we go on to apply the insight gained from our 4d study to 3d gauge theories. We consider the analog of the M5 brane construction in the case of 3d N = 2 theories: pairs of M5 branes wrapped on a 3-manifold. Using the ansantz of R-flow, we study 3-manifolds consisting of Riemann surfaces fibered over R. When the construction is non-singular, the resulting IR physics is described by a free abelian Chern-Simons theory. The mathematical data of a tangle captures the data of the gauge theory, and the Reidemeister equivalances on tangles correspond to dualities of physical descriptions. To obtain interacting matter, we allow singularities in the construction. By extending the tangle description to these singular cases, we find a set of generalized Reidemeister moves that capture non-trivial mirror symmetries of 3d gauge theories. These results give a geometric origin to these well-known 3d dualities. / Physics