量子代数U_q（f（K，（?）））的表示及其伴随作用
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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)).
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
[1] Kassel C. Quantun group. New York:spinger-verlag New York Inc,1995.
[2] Li F, Duplij s. Weak Hopf algebra and single solutions of quantum Yang-Baxter equation.comm. Math.Phys.,2002,225:191-217.
[3] Yang S L, Wang H. The Clebsch-Gordan decomposition for quantum algebra wsl_q2. The CJR Ring conference paper,Sep.29,2004.
[4]Hou B.Adjoint action for quantum algebra ωsl_q2.数学学报，May,2006,49(3):651-656.
[5] Ji Qingzhong,Wang Dingguo, and Zhou Xiangquan. Finite dimensional representations of Quantumgroups U_q(f(K)). East-West J. Math,(2000),2(2):201-203.
[6] Xin Tang. Constructing Irreducible Representations of Quantum Groups U_q(f(K)). Math. Subject Classification.Oct,2006.
[7] Li F. Weak Hopf algebras and some new solutions of Yang-Baxter equation. J. Algebra 208,72-100(1998).
[8] S.Majid. Foundations of Quantum Group Theory. Cambridge, Cambridge Univ Press.1995.
[9] S.Montgomery. Hopf algebras and their actions on rings. CBMS Reg.Conf.Series, 1993,82.
[10] M.E.Sweedler. Hopf Algebras. Berjamin,New York,1969.
[11] I.Kaplansky. Bialgebras. University of Chicago Lecture Notes, 1975.
[12]李方．Hopf代数的若干弱结构．浙江大学学报(理学版)，2002，29(3)．
[13] G.Bohm,F.Nill,K.Szlachangyi. Weak Hopf Algebras I: Integral Theory and C~* -structure,J. Algebra(1999)P.385.
[14] Hong Wang, Shilin Yang. The Isomorphisms and The Center of Weak Quantum Algebras ωsl_q(2). The CJR RING-Conference Paper,2004.
[2] Li F, Duplij s. Weak Hopf algebra and single solutions of quantum Yang-Baxter equation.comm. Math.Phys.,2002,225:191-217.
[3] Yang S L, Wang H. The Clebsch-Gordan decomposition for quantum algebra wsl_q2. The CJR Ring conference paper,Sep.29,2004.
[4]Hou B.Adjoint action for quantum algebra ωsl_q2.数学学报，May,2006,49(3):651-656.
[5] Ji Qingzhong,Wang Dingguo, and Zhou Xiangquan. Finite dimensional representations of Quantumgroups U_q(f(K)). East-West J. Math,(2000),2(2):201-203.
[6] Xin Tang. Constructing Irreducible Representations of Quantum Groups U_q(f(K)). Math. Subject Classification.Oct,2006.
[7] Li F. Weak Hopf algebras and some new solutions of Yang-Baxter equation. J. Algebra 208,72-100(1998).
[8] S.Majid. Foundations of Quantum Group Theory. Cambridge, Cambridge Univ Press.1995.
[9] S.Montgomery. Hopf algebras and their actions on rings. CBMS Reg.Conf.Series, 1993,82.
[10] M.E.Sweedler. Hopf Algebras. Berjamin,New York,1969.
[11] I.Kaplansky. Bialgebras. University of Chicago Lecture Notes, 1975.
[12]李方．Hopf代数的若干弱结构．浙江大学学报(理学版)，2002，29(3)．
[13] G.Bohm,F.Nill,K.Szlachangyi. Weak Hopf Algebras I: Integral Theory and C~* -structure,J. Algebra(1999)P.385.
[14] Hong Wang, Shilin Yang. The Isomorphisms and The Center of Weak Quantum Algebras ωsl_q(2). The CJR RING-Conference Paper,2004.