A BRIEF INTRODUCTION TO COALGEBRA REPRESENTATION THEORY William Chin DePaul University Chicago, Illinois 60614 USA wchin@condor.depaul.edu Abstract In this survey, we review fundamental properties of coalgebras and their representation theory. Following J.A. Green we present the block theory of coalgebras using indecomposable injectives comodules. Using the cohom and cotensor functors we state Takeuchi-Morita equivalence and use it to sketch the proof of existence of “basic” coalgebras, due to the author and S. Montgomery. This leads to a discussion of theory of path coalgebras, quivers and representations. Some quantum and algebraic group examples are given. 1 Introduction This survey article is aimed at algebraists who are not neccesarily specialists in coalgebras and Hopf algebras. As coalgebras are the unions of their, finite dimensional subcoalgebras, their representation theory can be viewed as a gen- eralization of the theory of finite dimensional algebras. We will see that many fundamental results extend to coalgebras. We begin by reviewing some of the most basic definitions and properties of coalgebras and their representations, with the nonspecialist in mind. Some of this material is covered in standard texts [Abe, Mo, Sw], though perhaps in different ways. We mainly follow the treatment in [Gr], with updated termi- nology. We discuss local finiteness, simple comodules, the coradical filtration and coradically graded coalgebras, and pointed coalgebras in section 2. We proceed in section 3 is to see how the structure theory for finite dimensional algebras extends to coalgebras, with injectives comodules playing a role closely analogous to the role of projectives in module theory. We see that block theory extends to coalgebras, and then discuss the Ext-quivers of coalgebras, and path coalgebras of arbitrary quivers. In a final subsection we describe a special case of the Brauer correspondence for modular coalgebras. 1