Quantum Group U_q (f (k)) Is Isomorphic To The Automorphism

Quantum group theory is a very important research content, it is being developed in the mid-eighties of last century a very important branch of algebra. Since the last two decades, its theories have been widely discussed. The aim of this thesis is to study the isomorphism and automorphism of quantum group ( ( ))U qf K problem when q is not root of unit,where ( ( ))U qf K is generated by the four variables E , F , K ,K ?1 with the relations:In particular, we only discuss the case when ( )1 , ,f K K m Kmm the quantum group ( ( ))U qf K isomorphism and automorphism of the complete classification.Explicitly, in the first part, we introduce the research background of quantum group U q( f ( K )), and explaine that when q is not root of unit, the quantum group U q( sl ( 2)) isomorphism and self-isomorphism, and further lead to the object of study in this paper: quantum group ( ( ))U qf K isomorphism and and automorphism problem.In the second part, we list some of the main results of quantum groups U q( f ( K )): ( ( ))U qf K admits a Hopf algebra structure(lemma 2.2); we get the equalities that the generatorsof ( ( ))U qf K satisfy by induction(lemma 2.4); ( ( ))U qf K is a Notherian domain with a basis { E i F j K si , j∈,s∈}(lemma 2.5); the center ( ( ( )))Z U qf K generated by analog of the Casimir element C qm is a subalgebra of ( ( ))U qf K , ( ( ( )))Z U qf K = ?? Cqm?? ;each finitedimensional ( ( ))U qf K module is semi-simple(theorem 2.3); all finite-dimensional simple-module classification(theorem 2.6); and so on.In the third part, we mainly discuss the isomorphism and automorphism of quantum group ( ( ))U = U qf K, the main conclusions are:Lemma 3.1 An element ( ( ))u∈U qf K is multiplicative invertiable if and only if there existλ∈?,m∈such that u =λKm.Theorem 3.3 Suppose q∈? is not a root of unity in a field , then ( ( ))U qf K and ( ( ))U pf k are isomorphism as -algebras if and only if p =±q±1.Theorem 4.1 Suppose q∈? is not a root of unity in a field , then ( ( ( )))φ∈Aut U qf K if and only if(1 )φ( K ) =αK,φ( E ) =αm ?tλEKr,( )φF =αtλ?1 K ?rF. or( 2 ) ( )φK =αK?1,φ( E ) =αm ?tλK rF,( )φF =αtλ?1 EK?r.whereαis a 2m -th root of unity, r∈,λ∈?.