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

Research paper by **Sacha C. Blumen**

Indexed on: **09 Jan '06**Published on: **09 Jan '06**Published in: **Mathematics - Quantum Algebra**

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#### 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 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.