2013/09/28 by Wang, Zhen, You, Lan, Chen, Hui-Xiang
#16G30 #16T99 #FOS: Mathematics #Representation Theory (math.RT)
paper · doi:10.48550/arxiv.1309.7442
In this paper, we study the representation theory of Hopf-Ore extensions of group algebras and pointed Hopf algebras of rank one over an arbitrary field k. Let H=kG(χ, a,\d) be a Hopf-Ore extension of kG and H' a rank one quotient Hopf algebra of H, where k is a field, G is a group, a is a central element of G and χ is a k-valued character for G with χ(a)≠ 1. We first show that the simple weight modules over H and H' are finite dimensional. Then we describe the structures of all simple weight modules over H and H', and classify them. We also consider the decomposition of the tensor product of two simple weight modules over H' into the direct sum of indecomposable modules. Furthermore, we describe the structures of finite dimensional indecomposable weight modules over H and H', and classify them. Finally, when χ(a) is a primitive n-th root of unity for some n>2, we determine all finite dimensional indecomposable projective objects in the category of weight modules over H'.