Quantum Group U_q (f (k, H)) Center, Said

Quantum group developing in the mid of the eighties in the last century is one of the important branches of algebra. And the theory of quantum groups has been widely studied over the past two decades. The aim of the thesis is to study the center and the simple representations of quantum algebraU_q(f(K,H )) when q is a primitive e-th root of unity with e being odd.Precisely, in the first part, we introduce the background of the quantum algebra Uq ( g), and recall the center and the representations of quantum group Uq (s l(2 )) when the q is a root of unity. Moreover, we lead to the object of the thesis: quantum algebra U = U q( f ( K ,H)). In the second part, we list some well-known results on our quantum group:U = U_q(f(K ,H)) is an algebra over k generated by six generators with the relations admits a Hopf algebra structure (Proposition 2.2.); We have some equalities of our generators by the induction(Lemma 2.3., Lemma 2.4.);U_q(f(K,H ))is a noetherian domain with a basis In the third part, we mainly discuss the center of quantum groupU_q(f(K,H )), the main conclusions are:Lemma 3.1. belongs to the center ofU_q(f(K,H )), where C q denotes the quantum Casimir element .Let U qK be the subalgerba ofU_q(f(K,H )) of all elements commuting with K . Then we have the following results:Lemma 3.2. ,其中[ ]Lemma 3.3. We denoteLemma 3.4. The center Z (U ) of U is a subalgebra ofU_q(f(K,H )) generated by the elements , ,E e F e andFinally, we arrived at this article the first major conclusion:Theorem 3.5. Z (U ) is a subalgebra generated by for s∈.In the fourth, we begin with the concept of weight vector, and then give the classification of simple modules.First of all, we discuss theU_q(f(K,H ))- modules whose dimension is less than e , and prove the following conclusions: Proposition 4.2. Let V be aU_q(f(K,H ))-module with dimension d + 1< e. Then(1) V contains a highest weight vector.(2) The actions induced by E and F on V are nilpotent. Lemma 4.3. Let v be a highest weight vector with weight (a , b). Set v0 =v and v p = Fpv for p≥0. Then Kv p = q ?2 iavp, Hv p = q 2 ibvp, Fv p = v p+1, ( ) ( )Ev p = f ?ia ,b v p?1. Lemma 4.4. Let V be a finite-dimensionalU_q(f(K,H ))-module generated by a highest weight vector v with weight (a , b), dim V = d + 1< e. Then(1) The weight ( a ,b ) satisfies the equation ( ) ( )f ? d+1 a. b = 0.(2) Any other highest weight vector in V is a scalar multiple of v .(3) V is simpleU_q(f(K,H ))-module.Now we can easily come to the first main conclusion in this part:Theorem 4.5. Let V be a finite-dimensionalU_q(f(K,H ))-module with dimension d + 1< e. Then there exists a ,b∈k such that V is isomorphism to Va , b ,d, where Va , b ,d is simpleU_q(f(K,H ))-module with dimension d +1. Next, we consider the simple module with dimension larger than e , we proof the simple module with dimension larger than e does not exist, that is:Theorem 4.6. There is no simpleU_q(f(K,H ))-module with dimension more than e . Finally we have the classification of simple-modules with dimension e :Theorem 4.7. Let V be a vector space with dimV = e, and with basis Then V has simple U -module structures with three types determined by the follows: (Ⅰ) Fixing scalars a , b≠0,λ∈k,Theorem 4.8. Let V be a simple U -module with dimV = e. Then V is isomorphic to one of the simple modules listed above.