The Green ring is a powerful mathematical tool used to codify the interactions between representations of groups and algebras. This ring is spanned by isomorphism classes of finite-dimensional indecomposable representations that are added together via direct sums and multiplied via tensor products. In this thesis, we explore the Green rings of a class of Hopf algebras that form an extension of the Taft algebras. These Hopf algebras are pointed and coserial, meaning their simple comodules are 1-dimensional, and their comodules possess unique composition series respectively. The comodules of these Hopf algebras thus have a particularly well-behaved structure. We present results giving structure to the comodule Green ring of the Hopf algebra Hs and in particular fully classify the Green rings of Hs where s ≤ 6. More generally, we classify the indecomposable comodules of Hs and their composition series and prove how the composition series may be used to classify the tensor product of indecomposable comodules. Additionally, for these Hopf algebras we classify the Grothendieck rings, the subrings of the corresponding Green rings consisting only of isomorphism classes of projective indecomposable comodules. We describe a simpler presentation of these Grothendieck rings and the multiplication in the ring.

Dissertação (mestrado) - Universidade Federal de Santa Catarina, Centro de Ciências Físicas e Matemáticas, Programa de Pós-Graduação em Matemática Pura e Aplicada, Florianópolis, 2017. / Made available in DSpace on 2018-01-23T03:17:00Z (GMT). No. of bitstreams: 1 349629.pdf: 753442 bytes, checksum: 3f93341c06c259c81b20e21cdfa6368d (MD5) Previous issue date: 2017 / Neste trabalho exploraremos alguns conceitos e resultados relativosa representações de grupos e álgebras de Hopf. Como resultado principal,apresentaremos a demonstração do Teorema de Dualidade deTannaka-Krein. Tal resultado trata do isomorfismo entre uma álgebrade Hopf comutativa real H, munida de uma integral não degeneradae a álgebra das funções representativas do grupo compacto G(H), dosmorfismos de álgebra de H, no corpo dos reais. / Abstract : In this work, we will explore some concepts and results related torepresentations of groups and Hopf algebras. As main result, we willpresent the proof of the Tannaka-Krein Duality Theorem. This theoremdeals with the isomorphism between a real commutative Hopf algebraH, provided with a non-degenerate integral and algebra of representativefunctions of the compact group G(H) of the algebra morphismsfrom H, on the field of real numbers.

In this thesis I will study noncommutative differential geometry, after the style of Connes and Woronowicz. In particular two examples of differential calculi on Hopf algebras are considered, and their associated covariant derivatives and Riemannian geometry. These are on the Heisenberg group, and on the finite group A4. I consider bimodule connections after the work of Madore. In the last chapter noncommutative fibrations are considerd, with an application to the Leray spectral sequence. NOTATION. In this thesis equations are numbered as round brackets (), where (a.b) denotes equation b in chapter a, and references are indicated by square brackets []. This thesis has been typeset using Latex, and some figures using the Visio program.

This thesis contains some results on the behavior of a delay differential equation (DDE) with two delays, at a Hopf bifurcation, for the nonzero equilibrium, using the growth rate, r, as bifurcation parameter. This DDE is a model for population growth, incorporating a maturation delay, and a second delay in the harvesting term. Considering a Taylor expansion of the non-dimensionalized model, we find a region of stability for the nonzero equilibrium, after which we find a pair of ODEs which help define the flow on the center manifold. We then find an expression for the first Lypapunov coefficient, which changes sign, so we also find the second Lyapunov coefficient, allowing us to predict multi-stability in the model. Numerical simulations provide examples of the behavior expected. For a similar model with one delay (PMC model), we prove the Hopf bifurcation at the nonzero equilibrium is always supercritical.

In this paper we explore bifurcations, in particular the Hopf bifurcation. We study this especially in connection with the Brusselator, which is a model of certain chemical reaction-diffusion systems. After a thorough exploration of what a bifurcation is and what classifications there are, we give graphic representations of an occurring Hopf bifurcation in the Brusselator. When an additional forcing term is added, behavior changes dramatically. This includes the introduction of a horseshoe in the time map as well as a strange attractor in the system.

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We study the Hopf-Galois module structure of algebraic integers in some finite extensions of $ p $-adic fields and number fields which are at most tamely ramified. We show that if $ L/K $ is a finite unramified extension of $ p $-adic fields which is Hopf-Galois for some Hopf algebra $ H $ then the ring of algebraic integers $ \OL $ is a free module of rank one over the associated order $ \AH $. If $ H $ is a commutative Hopf algebra, we show that this conclusion remains valid in finite ramified extensions of $ p $-adic fields if $ p $ does not divide the degree of the extension. We prove analogous results for finite abelian Galois extensions of number fields, in particular showing that if $ L/K $ is a finite abelian domestic extension which is Hopf-Galois for some commutative Hopf algebra $ H $ then $ \OL $ is locally free over $ \AH $. We study in greater detail tamely ramified Galois extensions of number fields with Galois group isomorphic to $ C_{p} \times C_{p} $, where $ p $ is a prime number. Byott has enumerated and described all the Hopf-Galois structures admitted by such an extension. We apply the results above to show that $ \OL $ is locally free over $ \AH $ in all of the Hopf-Galois structures, and derive necessary and sufficient conditions for $ \OL $ to be globally free over $ \AH $ in each of the Hopf-Galois structures. In the case $ p = 2 $ we consider the implications of taking $ K = \Q $. In the case that $ p $ is an odd prime we compare the structure of $ \OL $ as a module over $ \AH $ in the various Hopf-Galois structures.

After presenting some preliminary information, this paper presents two proofs regarding group schemes. The first relates the category of affine group schemes to the category of commutative Hopf algebras. The second shows that a commutative group scheme of finite order is in fact killed by its order. / text

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