Quantum Superalgebras at Roots of Unity and Topological Invariants of Three-manifolds
| dc.creator | Blumen, Sacha C. | |
| dc.date | 2006-01-09 | |
| dc.date.accessioned | 2026-07-07T06:58:39Z | |
| dc.date.available | 2026-07-07T06:58:39Z | |
| dc.description | 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 pseudo-modular 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 U_q(osp(1|2n)) over C is considered with q a primitive N^th 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_q^(N)(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_q^(N)(osp(1|2n)) is defined. Each such representation of U_q^(N)(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_q^(N)(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. | |
| dc.description | 255 pages, 26 figures, PhD thesis, accepted December 2005, School of Mathematics and Statistics, University of Sydney. Can also be downloaded as a pdf file from www.maths.usyd.edu.au | |
| dc.identifier | https://arxiv.org/abs/math/0601170 | |
| dc.identifier | http://arxiv.org/abs/math/0601170 | |
| dc.identifier.uri | http://salesiana.dossiersoluciones.com/handle/123456789/107443 | |
| dc.subject | Quantum Algebra | |
| dc.subject | Geometric Topology | |
| dc.subject | 17B37, 57M27 | |
| dc.title | Quantum Superalgebras at Roots of Unity and Topological Invariants of Three-manifolds | |
| dc.type | text |