Quantum Algebra H (d) And U < Sub > Q, M < / Sub > (sl < Sub > 2 < / Sub >)

Recently, Kashina, Montgomery and Ng introduced the n-th indicator of a finite dimensional Hopf algebra and showed that the n-th indicator enjoys several important properties. On this base, Kenichi Shimizu gave the computational formulas of the n-th indicator of the Taft algebra and the small quantum group associated with simple lie algebra sl2. It turns out that indicators of them can be expressed by special values of the generating function of certain type of partitions at roots of unity. The aim of this paper is to compute the n-th indicator vn(H) of quantum algebras H(d)(Definition3.1) and uq.m(sl2)(Definitional).Explicitly, in the first part, we introduce the reasearch background of the n-th indicator vn(H) and further lead to the object of study in this paper:the n-th indicator of quantum algebras H(d) and uq,m(sl2).In the second part, we list some main results of the n-th indicator:The sequence of the n-th indicator vn(H) n∈N is an invariant of the gauge equivalence class of Hopf algebras of H. If we know a left integral Λ of H and a right integral λ of H*such that λ(Λ)=1, then vn(H)=λ(S(Λ[n])) for all positive integer n(Lemma2.4). If Λ∈H and λ∈H*are both left integrals (or both right integrals)such that λ(Λ)=1. Then vn(H)=λ(Λ[n]) for all positive integer n (Lemma2.5). Let H be a finite dimensional filtered Hopf algebra, then vn(H)=vn(grH) for every n≥1, Where grH is the graded Hopf algebra of H(Lemma2.6).In the third part, we mainly compute the n-th indicator of quantum algebra H(d), the main conclusion is:Theorem3.4The n-th indicator of H(d) is given by where the sum is taken over all integers a, b and i satisfying0≤a,b≤(p-1)(n-1), 0≤i≤N-1,a+b≡ni(mod p2).In the last part,we mainly compute the n-th indicator of quantum algebra uq,m(sl2),the following are the main conclusions:Theorem4.6The n-th indicator of uq,m(sl2)is given by where the sum is taken over all integers a,b and i satisfying0≤a,b≤(N-1)(N-1),0≤i≤N-1,m(a+b)≡ni(mod N).Corollary4.7Let q be a primitive third root of unity,then we have...