摘要
本文假定基础域k为复数域C,N为非负整数集,Z~+为正整数集,0≠q∈k不是单位根.在[5]中量子代数U_q(f(K))的基础上我们构建了代数U_q(f(K,K)),它是由元E,F,K,K生成的结合代数,且满足下面关系:代数U_q(f(K,K))是将U_q(f(K))中条件KK~(-1)=K~(-1)K=1推广为KK=KK=J,本文中我们给出了U_q(f(K,K))具有弱Hopf代数结构的充要条件.
在李方定义的弱奥尔扩张意义下我们证明了U_q(f(K,K))是诺特环k[K,K]的弱奥尔扩张,从而证明了U_q(f(K,K))是诺特环.
本文中我们找到了所有有限维可积的不可约U_q(f(K,K))-模,是W(n)或V(?)的形式,并讨论了U_q(f(K,K))-模的Clebsch-Gordan公式:
(1)V_a(?)V_b(?)⊙_(l=0)~(min(a,b))V_(a+b-2l),
(2)V_a(?)W(b)(?)(a+1)W(b).
(3)W(b)(?)V_a(?)(a+1)W(b).
(4)W(a)(?)W(b)是平凡模.
最后对wsl_q(2)的左伴随作用进行了讨论,证明了U_q(f(K,K))在左伴随作用下是其自身上的拟模代数,并研究了U_q(f(K,K))的局部有限子模F(U_q(f(K,K)))=(x∈U_q(f(K,K))|dim(ad U_q(f(K,K)))(x)<∞}的子模结构.
In this paper, let basic field k is complex field C, N denotes the set of non-negative integers, Z~+denotes the set of positive integers, let q be a parameter with q being not a root of unity. Based on quantum algebras U_q(f(K)) in [5] we construct a algebras U_q(f(K, K)): it is a associative algebras generated by quadruple E, F, K, K satisfying the following relations:
It is generalizes the condition K K~(-1) = K~(-1) K = 1 as KK = K K = J. we give the necessary and sufficient condition of U_q(f(K, K)) have a structure of weak Hopf algebra.Under the definition of weak Ore extension defined by Li Fang, we prove that U_q(f( K, K)) is weak Ore extension of Noetherian ring k[K,K], so U_q(f(K, K)) is a noetherian ring.
In this paper we find all finite dimensional integrable highest weight U_q(f(K, K))-module, it is formal as W(n) or V_(?) and we discuss U_q(f(K, K))-module's Clebsch-Gordan formula:
(1)V_a(?)V_b(?)⊙_(l=0)~(min(a,b))V_(a+b-2l),
(2)V_a(?)(?)(b)(?)(a+1)W(b).
(3)W(b)(?)V_a(?)(a+1)W(b).
(4)W(a)(?)W(b)是平凡模.
At last we constrast to adjoint action for quantum algebraωsl_q(2), we prove that U_q(f(K, K)) is a left quasi-module algebra over itself and study the structure of the submodules of F (U_q(f(K, K))), which is the locally finite submodule of the quantum algebra U_q(f(K, K)).
引文
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