Green Rings Of Hopf-Ore Extensions Of Group Algebras

Hopf algebra is one of important research fields of algebras.The category of the representations of a Hopf algebra is a tensor category(or a monoidal category).The known research results show that tensor categories have wide applications in many fields of mathematics and physics and so on.It has been found that it is an effective approach to classify Hopf algebras by tensor categories.However,the Green ring is an important invariant of a tensor category,the multiplication structure of the Green ring of a tensor category reflects exactly the tensor product structure of the tensor category.Therefore,the representation theory of Hopf algebras,in particular,the tensor product structure and Green rings of the representation categories of Hopf algebras are important subjects worthy to study.In this doctoral thesis,we study the tensor product structure of the finite dimensional representation categories of a class of Hopf-Ore extensions kG(X,?.0)of group algebras KG,where k is an algebraically closed field,G is a group,X is a k-linear character,a is a central element of G satisfyingX(a)? 1.The thesis is divided into five chapters,which is orgazined as follows.In Chapter 1,we introduce some notations and basic concepts,including the struc-ture of the Hopf-Ore extensions of group algebras,the Green ring and Grothendieck ring of a Hopf algebra.We also recall the classification of finite dimensional indecomposable weight modules over kG(X,a,0).In Chapter 2,we study the decomposition rules of the tensor products of indecom-posable weight modules over kG(X,?,0),where char(k)?0.We consider the decompo-sition rules in three cases:|X|=|X(?)|=?,|X|=|X(?)| ?? and |X(?)| ?|X|??.For the tensor products of indecomposable weight modules of different types,we discuss the decomposition rules respectively.The decomposition formulae of the tensor product of any two indecomposable weight modules into the direct sum of indecomposable modules are given.In Chapter 3,based on the decomposition rules given in Chapter 2,we investigate the Green ring r(W)of the category W of the finite dimensional weight modules over kG(X,?,0).Let G be the group of lk-linear characters of G.Then the Green ring r(W)is described as follows.When |X|=|X(?)|??,r(W)is commutative and isomorphic to the polynomial ring ZG[y]in one variable y over the group ring ZG.When |X(?)|<|X|=?,r(W)is also commutative and isomorphic to a factor ring of the polynomial ring ZG[y,z]in two variables y,zover ZG.When |X|<?,r(W)is isomorphic to the factor ring of the skew group ring Z[X]#G of the polynomial ring Z[X]in infinitely many variables X with the linear character group G modulo some ideal.In Chapter 4,we study the finite dimensional representations of KG(X,a,0).At first,for |X|?|q|<? respectively,the structures and classification of finite dimen-sional simple modules are given.Then under the cas sumption that kG is finite dimensional semisimple and[X|=|X(?)|,we describe the structures of all finite dimensional indecom-posable modules and classify them up to isomorphism.Finally,under the assumption that char(K)?0,G is a finite group and |X|=|X|(?)|,it is shown that each finite dimensional indecomposable module is isomorphic to the tensor product of a simple kG-module with an indecomposable weight module.Thus,the decomposition rules for tensor product-s of indecomposable modules follow from those for weight modules given in Chapter 2.Consequently,the Green ring(resp.,Grothendieck ring)of kG(X,a,0)is equal to G0(kG)r(W)(resp.,Go(kG)Go(W),the product of its two subrings:the Grothendieck ring of IkG and the Green ring r(W)(resp.,Grothendieck ring Go(W)of the weight module category w.In Chapter 5,as an application of the previous chapters,we study the representations of the Hopf-Ore extension H:=KDn(x,?m,0)of the group algebra kDn of dihedral group Dn,where k is an algebraically closed field of characteristic zero,n?2m is even and m>1 is odd,the dihedral group Dn is generated by a,b subject to an?b2=(ba)2=1,andX ? Dn is determined byX(?)=-1 and X(b)=1.Firstly,using the results given in Chapter 4,we display the classifications of simple modules and indecomposable modules over H,respectively.It is shown that M(?)N(?)N(?)M for any finite dimensional H-modules M and N.Then it is shown that Go(kDn)is isomorphic to a factor ring of the polynomial ring ZDn[x]in one variable over the group ring ZDn,and the Grothendieck ring G0(H)is isomorphic to a factor ring of the polynomial ring in infinitely many variables over G0(kDn).Finally,it is shown that the Green ring r(H)is isomorphic to a factor ring of the polynomial ring in finitely many variables over G0(kDn)modulo some ideal.