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

Quantum groups were introduced independently by Drinfel'd and Jimbo instudying the quantum Yang-Baxter equation and two-dimensional solvable latticemodels. It provides a three-dimensional numerical manifold and Link invariants(the knot theory, Low dimensional topology), and provides a tool to describe thequantum symmetry in quantum mechanical system. It has a considerable richtheoretical content and applications.In this dissertation, we mainly consider the two-parameter weak Hopf su-peralgebras corresponding to the simple Lie superalgebra osp(1|2n). We give adescription of finite-dimensional Uq(osp(1|2n))-module in terms of (generalized)Young tableaux. And introduce the quantum superdeterminant for the quantumsuper group OSPq(1|2n) and discuss the properties of q-superdeterminant. Thecontents of the dissertation are summarized as follows:Firstly, corresponding to the Hopf superalgebras Ur,s(osp(1|2n)), we constructtwo-parameter weak Hopf superalgebras wUr,s(osp(1|2n)). We show that thereexists Lusztig's symmetries from Ur,s(osp(1|2n)) to its associated quantum super-group Ur((?))1,s((?))1(osp(1|2n)), and give the PBW-base of Ur,s(osp(1|2n)). These resultsgeneralize the Lusztig's symmetry theory of the classical quantum groups natu-rally and the generalization is non-trivial. Furthermore, we give the PBW-base ofwUrd,s(osp(1|2n)).Secondly, we give a description of the crystal graph of the tensor productsof finite-dimensional irreducible Uq(osp(1|2n))-modules in terms of generalized Young tableaux. It is shown that a finite-dimensional simple module L(λ) ofUq(osp(1|2n)) with the highest weightλcorresponds to a (generalized) Young dia-gram Y . More precisely, it is shown that a crystal base for the module L(λ) can beindexed by the set B(Y ) consisting of the semistandard tableaux of shape Y withappropriate conditions. Furthermore, the explicit description of the generalizedLittlewood-Richardson rules is obtained for tensor product decompositions.Finally, we study the quantum superdeterminants for the quantum supergroup OSPq(1|2n). The quantum coordinate function superalgebra OSPq(1|2n)of simple Lie superalgebra osp(1|2n) is characterized by quadratic relations inthe generators Tji. Let T be the matrix composed by these generators Tji whichis called the quantum supermatrix, we get an explicit formula of the quantumsuperdeterminant for the quantum supermatrix T. We found that this quantumsuperdeterminant is di(?)erent from the case of SOq(2n + 1), and show that sdetqTis the central group-like element in OSPq(1|2n), and the square of it equals 1.