The Representations Of Pointed Hopf Algebras

The quantized enveloping algebras and their finite dimensional quotients, i.e., small quantum groups, are a class of pointed Hopf algebras which are very important and typical. The research for them promoted greatly the development of the theory of Hopf algebras. By the lifting method, Andruskiewitsch and Schneider classified (finite dimensional) pointed Hopf algebras with coradicals being group algebras of abelian groups and found many new examples of Hopf algebras which can be regarded as a natural generalization of the quantized enveloping algebras and their small quantum groups([2, 3]). Consequently, it becomes an interesting and urgent task to study the properties and the representation theory of these Hopf algebras. In this thesis we study three kinds of pointed Hopf algebras: the Drinfeld quantum doubles of Taft algebras; the half quantum groups; the pointed Hopf algebras with Cartan matrices of type A₂×A₂, all of which fall among those Hopf algebras classified by Andruskiewitsch and Schneider. Our results also provide a recipe for computing the representations over the pointed Hopf algebras with coradicals being group algebras of abelian groups. In fact, many results here can be generalized to the more general cases.First, given a Taft algebra A_n, Chen constructed its Drinfeld quantum double H_n(p,q) and described all finite dimensional modules over H_n(p,q) in [8, 9, 10, 11]. In this thesis, we examine the infinite dimensional representations of H_n(p, q). We describe explicitly all indecomposable and algebraically compact modules over H_n(p, q) and classify them up to the elementary equivalence.Then, for a semisimple Lie algebra g, we define the half quantum group U^≥0 to be the Borel subalgebra of the quantized enveloping algebra U_q(g). We prove that the Hopf algebra U^≥0 is not quasi-cocommutative, hence the category of left U^≥0-modules is not a braided monoidal category. When q is a root of unity, we also give a sufficient-necessary condition for the finite dimensional quotient u^≥0 of U^≥0to be quasi-cocommutative. In the category of weight modules over U^≥0, we describe all the simple objects, the projective objects and the decomposition of their tensor products. We describe all simple Yetter-Drinfeld U^≥0-weight modules, which are equal to all simple weight modules over the quantum double D(U^≥0). Hence we recover a result of Hu and Zhang about the relation between the representations of D(U^≥0) and U^q(g) ([25]). Our results also provide a way to study the general pointed Hopf algebras constructed by Andruskiewitsch and Schneider which are twist-equivalent to D(U^≥0).Next, according to the classification results of Andruskiewitsch and Schneider ([2, 3]), we define a class of special pointed Hopf algebras U(D,λ) whose Cartan matrices are of type A₂×A₂. There are three cases according to the choice of the linking parameterλ: U(D,λ₁), U(D,λ₂), U(D,λ₃). Note that U(D,λ₁) just is a half quantum group and U(D,λ₁) is the quantized enveloping algebra of sl(3). Hence we concentrate on U(D,λ₂). Our results illustrate that the representations of U(D,λ₂) have a close connection with those of U(D,λ₁) and U(D,λ₃). In particular, we obtain an equation set which can be used to compute idempotent elements. Thus we get a decomposition of the regular module as a direct sum of indecomposable projective modules. This method is valid for the small quantum group of sl(2) too.In the last chapter, we discuss the relations between the pointed Hopf algebra U(D,λ) and some Hopf subalgebras of U(D,λ). We construct several functors between the categories of modules over these Hopf algebras. The representation theory of generalization of many famous Hopf algebras can be recovered from our results combining the representations of the original Hopf algebras. For example, many results in [19] and [27] can be obtained.