The Representation Rings Of Quantum Algebras And Their Properties

In the category of finite dimensional modules of Hopf algebras,the decompo-sition of the tensor product of arbitrary two indecomposable modules into direct sum of indecomposable modules has attracted wide attention of mathematicians.Many meaningful results have been obtained.Furthermore by studying the rep-resentation rings of Hopf algebras and quantum algebras,we can also understand the properties of these categories.In this dissertation,for finite dimensional quan-tum algebras and weak Hopf algebras,we mainly study the classification of their representations,representation rings and their related properties.The following main results are obtained.?1?Assume that q is a primitive 2p-th root of unity and p?2,???_q?sl₂?is the restricted quantum group of U_q?sl₂?.Applying the classification of all indecom-posable ???_q?sl₂?-modules and decomposition formulas of the tensor product of two arbitrary indecomposable ???_q?sl₂?-modules by Suter,Kondo and Satio et al.,we exactly give the generating relations of representation rings r????_q?sl₂??for p?2.It turns out that the representation ring r????_q?sl₂??is generated by infinitely many generators satisfying to a family of generating relations.Note that if p=2,the representation ring r????_q?sl₂??is a commutative ring,while r????_q?sl₂??is not a commutative ring for p?3.?2?Firstly,two classes of weak Hopf algebras w^s_n,d?s=0,1?corresponding to generalized Taft Hopf algebras H_n,d?q?are introduced.Then their representation rings are described by generators and the generating relations.The results show that the representation rings of weak Hopf algebras w^s_n,dare more complicated than those of Taft algebras and generalized Taft algebras.On the one hand,r(w¹_n,d)is a noncommutative ring.On the other hand,r(w⁰_n,d)is a commutative ring.?3?The representations of a class of finite dimensional non-standard quantum algebra ???_q?A₁?are studied.By the approach of representation theory of algebra,the isomorphism classes of all indecomposable modules are described.Furthermore the decomposition formulas of the tensor product between arbitrary indecompos-able modules and simple?or projective?modules are established.Finally,the projective class rings and Grothendieck rings of ???_q?A₁?are also characterized.?4?Let H₈ be the unique noncommutative and noncocommutative eight di-mensional semi-simple Hopf algebra.The corresponding weak Hopf algebra ???₈based on H₈is constructed.The structure of representation ring of ???₈ is estab-lished explicitly.Finally,it is shown that the automorphism group of r????₈?is just isomorphic to D₆,where D₆is the dihedral group with order 12.?5?A class of neither pointed nor semisimple Hopf algebra H_n,dand its weak forms w^sH_n,d?s=0,1?are studied.A sufficient and necessary condition for H_n,dto be quasitriangular is given.All indecomposable modules of w^sH_n,dare classified and the decomposition formulas of the tensor product of two arbitrary indecomposable w^sH_n,d-modules are established.