The Structure And Classification Of New Hopf Algebras Originated From Two-parameter Quantum Groups

The classification of finite-dimensional Hopf algebras has developed rapidly in the past two decades.One of the main methods is the lifting method developed by Andruskiewitsch and Schneider.In 2010,Andruskiewitsch and Schneider established a rough classification theorem of finite-dimensional pointed Hopf algebras,which re-quires that the order of group generated by group-like elements is prime to 210（notice that the prime factors of 210 are 2,3,5,7）.In the first part of this paper,we consider a class of pointed Hopf algebras:restricted two-parametric quantum groups,whose or-der are not prime to 210.We firstly make use of the isomorphism theorem of restricted two-parameter quantum groups established by N.Hu et al.When the parameters are 4th,5th,6th,7th,8th primitive roots of 1,a partial classification of type A,B,C,D,F₄,G₂restricted quantum groups is given.When the parameters are 6th or 8th primitive roots of 1,for some candidates of type A that cannot be applied with the isomorphism theo-rem,we use the isomorphism class of simpe Yetter-Drinfeld’s module and its dimension distribution to make a further distinction.Thus we can get a series of representatives of new isomorphism classes.230 types of isomorphism classes can be completely distin-guished,and another 48 types cannot be completely distinguished for the time being.In the second part,we firstly give the vector representation of two-parameter quan-tum groups of type B.Also,the two-parameter basic R-matrix of type B is calculated.Finally,we determine the structure of the two-parameter quantum coordinate ring O(SO_r,s（2n+1）),according to the basic R-matrix of type B.