Quantum Superalgebras at Roots of Unity and Topological Invariants of Three-manifolds

Abstract

The general method of Reshetikhin and Turaev is followed to develop topological invariants of closed, connected, orientable 3-manifolds from a new class of algebras called pseudomodular Hopf algebras. Pseudo-modular Hopf algebras are a class of Z_2-graded ribbon Hopf algebras that generalise the concept of a modular Hopf algebra. The quantum superalgebra Uq(osp(1|2n)) over C is considered with q a primitive Nth root of unity for all integers N > = 3. For such a q, a certain left ideal I of U_q(osp(1|2n)) is also a two-sided Hopf ideal, and the quotient algebra U^(N)_q(osp(1|2n)) = U_q(osp(1|2n))/I is a Z_2-graded ribbon Hopf algebra. For all n and all N > = 3, a finite collection of finite dimensional representations of U^(N)_q(osp(1|2n)) is defined. Each such representation of U^(N)_q(osp(1|2n)) is labelled by an integral dominant weight belonging to the truncated dominant Weyl chamber. Properties of these representations are considered: the quantum superdimension of each representation is calculated, each representation is shown to be self-dual, and more importantly, the decomposition of the tensor product of an arbitrary number of such representations is obtained for even N. It is proved that the quotient algebra U(N)^q_(osp(1|2n)), together with the set of finite dimensional representations discussed above, form a pseudo-modular Hopf algebra when N > = 6 is twice an odd number. Using this pseudo-modular Hopf algebra, we construct a topological invariant of 3-manifolds. This invariant is shown to be different to the topological invariants of 3-manifolds arising from quantum so(2n+1) at roots of unity.