Research On Structures Of (Restricted) Quantum Superalgebras And Some Related Topics

This dissertation mainly studies the structures of some quantum superalge-bras. A necessary and sufcient condition for the Drinfel d double D(H) of afinite dimensional Hopf superalgebra H to be a ribbon Hopf superalgera is ob-tained. The structure of the quantum superalgebra uq(sl(m|n)), which is the re-stricted form of the quantum superalgebra Uq(sl(m|n)), is studied. The structuresand properties of the quantum general (special) linear supergroup O_q(GL(m|n))resp.O_q(SL(m|n)) are also studied, which are said to be the restricted forms ofO_q(GL(m|n)) and O_q(SL(m|n)). At last, all the algebra automorphisms of thetwo parameter quantum superalgebra Ur,sosp(1,2, c) are described.(1) We discuss when the Drinfel d double (D(H), R) of a finite dimensionalHopf superalgebra H has ribbon elements. Firstly we consider the relationshipbetween grouplike elements and ribbon elements in a finite dimensional quasi-triangular Hopf superalgebra (H, R). Then we give a necessary and sufcientcondition for (H, R) to have a ribbon element when H is unimodular. Finally byshowing that the Drinfel d double (D(H), R) of a finite dimensional Hopf superal-gebra H is always unimodular, we obtain a necessary and sufcient condition for(D(H), R) to have a ribbon element. As an application, we study the necessaryand sufcient condition for the Drinfel d double D(A) of the Taft superalgebra Ato have a ribbon element. Comparing with the non-super case, the quasi-ribbonelements of D(A) may not be ribbon elements, while they are always the same inthe non-super case. (2) Corresponding to the Lie superalgebra sl(m|n), we construct the restrictedquantum superalgebra uq(sl(m|n)), where q is a root of unity. Then we prove thatit is a Hopf superalgebra, describe its PBW basis, left and right integrals, and atlast we give its Hopf automorphisms.(3) Suppose that q is a primitive N-th root of unity. We construct the quan-tum supergroups O_q(GL(m|n)) and O_q(SL(m|n)), which are said to be therestricted forms of the quantum general linear supergroup O_q(GL(m|n)) and thequantum special linear supergroup O_q(SL(m|n)) respectively. We prove that theyare Hopf superalgebras, describe their PBW basis and as an example, we considerthe case when m=n=1to understand these constructions. At last, we describetheir left and right integrals explicitly.(4) We construct a new kind of two parameter quantum superalgebraUr,sosp(1,2, c) corresponding to the Lie superalgebra osp(1,2). We describeits PBW basis, center, and all the algebra automorphisms. As an application, wealso determine its Hopf automorphisms.