Special biserial coalgebras and representations of quantum SL ( 2 )

We develop the theory of special biserial and string coalgebras and other concepts from the representation theory of quivers. These theoretical tools are then used to describe the finite dimensional comodules and Auslander-Reiten quiver for the coordinate Hopf algebra of quantum SL(2) at a root of unity. We also compute quantum dimensions and describe the stable Green ring. Let C = k ζ [SL(2)] denote the quantized coordinate Hopf algebra of SL(2) as defined in [APW] and [Lu2] at a root of unity of odd order over a field of characteristic zero. In this article we study the category of finite-dimensional comodules, a category that is equivalent to the category of finite-dimensional modules over (a suitable quotient of) the quantized hyperalgebra U ζ. Our approach uses representations of the Gabriel quiver associated to C, methods from the representation theory of quivers, and most notably, the theory string algebras and special biserial algebras. The methods of this paper demonstrate the utility of coalgebraic methods, which will hopefully be applicable to other categories of interest. We discuss some required coalgebra representation theory in section 1 and then use the more general results to concentrate on the case of the quantized coordinate Hopf algebra C at a root of unity over a field of characteristic zero in section 2. In [CK] we determined the structure of the injective in-decomposable comodules along with the block decomposition of C. Here we completely determine the finite-dimensional indecomposable comodules, the Auslander-Reiten quiver and almost split sequences. We introduce the notions of a special biserial coalgebra and string coalge-bra, the latter modifying the definition in [Sim2], [Sim3]. These notions are coalgebraic versions of machinery known for algebras [Ri], [BR], [Erd]. It turns out that C is special biserial and that its representations are closely related to an associated string subcoalgebra, and this allows a listing of indecompos-ables and a description of the Auslander-Reiten quiver. One consequence is that the coalgebra C is of tame of discrete comodule type [Sim2] and is the 1