Coalgebras, Hopf Algebras and Combinatorics

In loving memory of my mother " A mathematician is a machine for converting coffee into theorems. " —Alfréd Rényi " A comathematician, by categorical duality, is a machine for converting cotheorems into ffee. " —anonymous Preface Hopf algebras are a relatively new concept in algebra, first encountered by their namesake Heinz Hopf in 1941 in the field of algebraic topology. In the 1960s, study of Hopf algebras increased, perhaps spurred by the findings of Moss Eisenberg Sweedler, and by the late 1980s and early 1990s, Hopf algebras gained further interest from mathematicians and even scientists in other fields, as connections with quantum mechanics became clearer. Hopf algebras are a particularly interesting algebraic concept in that they turn up in almost any field of study – number theory, algebraic geometry and algebraic topology, Lie theory and representation theory, Galois theory and the theory of separable field extensions, operator theory, distribution theory, quantum mechanics and last but not least combinatorics. The aim of my diploma thesis is to give an introduction into the notation and paradigm of coalgebras and Hopf algebras and to present applications in and connections with combinatorics. In Chapter 1, we briefly revisit well-known definitions and results from basic algebra and tensor products to lay the groundwork for the following chapters. Apart from a basic understanding of algebraic thinking and some basic notions from linear algebra, no knowledge on the part of the reader is assumed. The presentation of the content is along the lines of [Abe80], with some elements inspired by [DNR01]. In Chapter 2, we define coalgebras by dualizing the definition of algebras and introduce the sigma notation (or Sweedler notation, after the aforementioned algebraist), which simplifies the written treatment of coalgebras significantly. We then consider dual constructions of algebras and coalgebras and finally discuss a special subset of elements in a coalgebra, the grouplike elements. In Chapter 3, we present definitions and basic properties of bialgebras and Hopf algebras, as well as some examples. We follow the presentation of [DNR01]. Chapter 4 presents an important natural occurrence of Hopf algebras in the theory of com-binatorial classes. The concepts of composition and decomposition of objects are linked to the multiplication and comultiplication maps. The results are due to [Bla10]. In Chapter 5, we consider applications of coalgebras and Hopf algebras in treating enumeration problems, as well as in formulating relations between polynomial sequences on the one …