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Title:  Quantum Superalgebras at Roots of Unity and Topological Invariants of Threemanifolds 
Authors:  Blumen, Sacha Carl 
Keywords:  quantum superalgebras;topological invariants;threemanifolds 
Issue Date:  27Mar2006 
Publisher:  University of Sydney. School of Mathematics and Statistics 
Abstract:  The general method of Reshetikhin and Turaev is followed to develop topological invariants of closed, connected, orientable 3manifolds from a new class of algebras called pseudomodular Hopf algebras. Pseudomodular Hopf algebras are a class of Z_2graded ribbon Hopf algebras that generalise the concept of a modular Hopf algebra. The quantum superalgebra Uq(osp(12n)) 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(12n)) is also a twosided Hopf ideal, and the quotient algebra U^(N)_q(osp(12n)) = U_q(osp(12n))/I is a Z_2graded ribbon Hopf algebra. For all n and all N > = 3, a finite collection of finite dimensional representations of U^(N)_q(osp(12n)) is defined. Each such representation of U^(N)_q(osp(12n)) 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 selfdual, 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(12n)), together with the set of finite dimensional representations discussed above, form a pseudomodular Hopf algebra when N > = 6 is twice an odd number. Using this pseudomodular Hopf algebra, we construct a topological invariant of 3manifolds. This invariant is shown to be different to the topological invariants of 3manifolds arising from quantum so(2n+1) at roots of unity. 
URI:  http://hdl.handle.net/2123/715 
Rights and Permissions:  Copyright Blumen, Sacha Carl;http://www.library.usyd.edu.au/copyright.html 
Appears in Collections:  Sydney Digital Theses (Open Access)

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