Research On Some Structural Problems Of Quantum Enveloping Algebras

A quantum group is a generalization of a Lie algebra,which is closely related to many branches of mathematics and physics.It has been one of the most popular research area since its birth.This thesis mainly focuses on the RTT realization of quantum coordinate algebras,and the structure of low-rank quantum enveloping algebras of finite or affine type.We first considered the coordinate superalgebras of quantum superalgebras based on the RTT relation.A quantum superalgebra can be defined by a R-matrix and its corresponding RTT relation,we construct a suitable R-matrix in the tensor product space EndV(?)EndV,and use this R-matrix to give a definition of quantum superalgebra U(R),then verify that matrix generators of its coordinate superalgebra A(R)also satisfy RTT relation.We also verified these results in the case of quantum superalgebras U_q(gl_m|n)and U_q(q_n).Secondly,we study the structure of low-rank finite dimensional quantum enveloping algebras.We reviewed the four families of classical Lie algebras,and explicitly described all possible isomorphisms among so₃(C),sp₂(C),sp₄(C),so₂(C),so₄(C),so₆(C).We fur-ther study on the structure of the low-rank quantum groups U_q(gl₂),U_s(o₃),U_r(sp₂)and U_q(o₄).By analyzing their R-matrices in the natural representation,the relations the quantum enveloping algebras U_r(sp₂),U_q(o₄)and U_q(gl₂)are obtained.Finally,we study the structure of low-rank affine quantum envelope algebras.Sim-ilar to quantum group of finite type and Yangian algebras,we analyze the structure of affine quantum enveloping algebra based on RTT realizations.By exploring the relations between R-matrices of different algebras,the relations between U_s((?)₃),U_r((?)₂),U_q((?)₄)and U_q((?)₂)are obtained.