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 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 should be applicable to other categories of interest. We discuss some general 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. Notions dual to the standard notions of quasi-Frobenius (selfinjective), Frobenius and symmetric algebras for finite-dimensional algebras have been extended in natural ways to coalgebras with dual terms quasi-coFrobenius (selfprojective), coFrobenius and symmetric coalgebras, see [DNR][DIN]. For a Hopf algebra, the first two notions are equivalent to underlying coalgebra being semiperfect, which is equivalent to it having a nonzero integral in the dual algebra. Being symmetric as coalgebra implies the other conditions. It turns out that C is a symmetric coalgebra, and its basic coalgebra is also 1