The Research On The Restricted Forms Of Non-standard Quantum Groups

In the past 30 years,there have been much achievement about quantum groups theory.In 1991,a class of non-standard quantum groups was constructed when solving the singular solution of Yang-Baxer equations.They can be viewed as a class of non-commutative and non-cocommutative Hopf algebras.There exists great differences in the properties and representations of classical standard quantum groups,such as degen-erate Serre’s relations and zero divisors under some conditions.In 1994,Xq(An)and Xq(Bn)were constructed.It is well known that the research of non-standard quantum group theory has made great progress in recent years,but there are still a lot of problems have not been resolved in this emerging discipline.One of the most basic problems is to determine all the finite dimensional representations of non-standard quantum groups.In this thesis,we study finite dimensional representations of the non-standard quantum groups Xq(A1)with relations K12m = 1,K22m = 1 when q is a root of unity.Firstly,we define the (?)(A1)and give some basic properties.Secondly,the restricted non-standard quantum group (?)(A1)is decomposed into direct summands of indecom-posable two side ideals.The decomposition of the indecomposable two side ideals of (?)(A1)is established along the Suter’s method.The indecomposable projective mod-ules and their isomorphism classes are given by using K1,K2 eigenvectors.In order to be more intuitive and clear,we used the diagram to describe these modules.Then,(?)(A1)is decomposed into direct summands of principal indecomposable modules.By using composition factors of the principal indecomposable modules,(?)(A1)is decom-posed into blocks.Finally,the decomposition of the tensor product of (?)(A1)-module is given.