The representation theory of profinite algebras: Interactions with category theory, algebraic topology and compact groups
Abstract
In the present text, we examine current trends in the theory of profinite algebras, and applications, connections and interactions with other fields of mathematics. The thesis consists of one introductory and motivational chapter and two parts afterwards, each consisting of three chapters. Each chapter has its own introduction detailing the results and explaining the context of the work, and each of the chapters 27 is based on the research in the papers [IM, I1, I2, I3, I4, I5], and partially on [I]. The basics of the mathematical theories involved here are most of the time ommited and explained brie y, and the reader is referred to the literature; we concentrate more on the original part of the research, which is more than 90% of this text. In the first chapter, we present the generals of the representation theory of profinite algebras, as dual of coalgebras, and the support for the category of finite dimensional representations of an arbitrary algebra. We also present a summary of the results, as well as various interconnections with other fields of mathematics, such as Hopf Algebras, Category Theory, Locally Compact Groups, Algebraic Topology, Homological Algebra. The first part  chapters 24  concern a type of problem called splitting problem. Given abelian categories [Special characters omitted.] <math> <f> <sc>A</sc></f> </math> ⊆ [Special characters omitted.] <math> <f> <sc>C</sc></f> </math> with suitable properties, define the [Special characters omitted.] <math> <f> <sc>A</sc></f> </math> torsion functor t : [Special characters omitted.] <math> <f> <sc>C</sc></f> </math> [arrow right] [Special characters omitted.] <math> <f> <sc>A</sc></f> </math> as t(X) = the largest subobject of X belonging to [Special characters omitted.] <math> <f> <sc>A</sc></f> </math> ; the splitting problem asks when is t(X) a direct summand of X for all X . We solve this problem for [Special characters omitted.] <math> <f> <sc>C</sc></f> </math> = category of (finitely generated) modules over a profinite algebra and [Special characters omitted.] <math> <f> <sc>A</sc></f> </math> = the subcategory of rational modules, and also for [Special characters omitted.] <math> <f> <sc>A</sc></f> </math> = the category of semiartinian modules (chapter 4), which gives a positive answer to a conjecture of Faith for this case. The second part concerns the development of the theory of infinite dimensional (quasi) Frobenius algebras, which are the dual of (Quasi)coFrobenius coalgebras. We prove various theorems regarding the left and right (quasi)coFrobenius coalgebras, which explain why these are a generalization of the finite dimensional Frobenius algebras, and also reveal how they appear as a leftright symmetric concept. They turn out to have a very interesting "abstract integral theory" which generalize that from Hopf algebras and compact groups. Moreover, these nontrivial generalizations have applications to proving many of the foundational results in Hopf algebras, and have many connections to compact groups. We give categorical results which reveal all the connections between these notions and their finite dimensional counterparts, as well as the previously unknown connections between these notions and various categorical properties.
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