Abstract. Let U q (g) be the quantum supergroup of gl m|n or the modified quantum supergroup of osp m|2n over the field of rational functions in q, and let V q be the natural module for U q (g). There exists a unique tensor functor, associated with V q , from the category of ribbon graphs to the category of finite dimensional representations of U q (g), which preserves ribbon category structures. We show that this functor is full in the cases g = gl m|n or osp 2ℓ+1|2n . For g = osp 2ℓ|2n , we show that the space Hom Uq(g) (V ⊗r q , V ⊗s q ) is spanned by images of ribbon graphs if r + s < 2ℓ(2n + 1). The proofs involve an equivalence of module categories for two versions of the quantisation of U(g).

ABSTRACT. An algebraic formulation is given for the embedded noncommutative spaces over the Moyal algebra developed in a geometric framework in [8]. We explicitly construct the projective modules corresponding to the tangent bundles of the embedded noncommutative spaces, and recover from this algebraic formulation the metric, Levi-Civita connection and related curvatures, which were introduced geometrically in [8]. Transformation rules for connections and curvatures under general coordinate changes are given. A bar involution on the Moyal algebra is discovered, and its consequences on the noncommutative differential geometry are described.

We construct a class of exact solutions of the noncommutative Einstein field equations in the vacuum, which are noncommutative analogues of the plane-fronted gravitational waves in classical gravity.

Abstract. Gravitational collapse of a class of spherically symmetric stars are investigated. We quantise the geometries describing the gravitational collapse by a deformation quantisation procedure. This gives rise to noncommutative spacetimes with gravitational collapse.PACS. 04.70.Bw Classical black holes -04.70.Dy Quantum aspects of black holes -11.10.Nx Noncommutative field theory.

Abstract. A string basis is constructed for each subalgebra of invariants of the function algebra on the quantum special linear group. By analyzing the string basis for a particular subalgebra of invariants, we obtain a "canonical basis" for every finite dimensional irreducible U q (sl(n))-module. It is also shown that the algebra of functions on any quantum homogeneous space is generated by quantum minors.