309 research outputs found
Hecke algebras, modular categories and 3-manifolds quantum invariants
We construct modular categories from Hecke algebras at roots of unity. For a
special choice of the framing parameter, we recover the Reshetikhin-Turaev
invariants of closed 3-manifolds constructed from the quantum groups U_q sl(N)
by Reshetikhin-Turaev and Turaev-Wenzl, and from skein theory by Yokota. We
then discuss the choice of the framing parameter. This leads, for any rank N
and level K, to a modular category \tilde H^{N,K} and a reduced invariant
\tilde\tau_{N,K}. If N and K are coprime, then this invariant coincides with
the known PSU(N) invariant at level K. If gcd(N,K)=d>1, then we show that the
reduced invariant admits spin or cohomological refinements, with a nice
decomposition formula which extends a theorem of H. Murakami.Comment: 32 pages. See also http://www.math.sciences.univ-nantes.fr/~blanche
Cabling Burau Representation
The Burau representation enables to define many other representations of the
braid group by the topological operation of ``cabling braids''. We show
here that these representations split into copies of the Burau representation
itself and of a representation of . In particular, we show that
there is no gain in terms of faithfulness by cabling the Burau representation.Comment: 11 page
Skein construction of idempotents in Birman-Murakami-Wenzl algebras
We give skein theoretic formulas for minimal idempotents in the
Birman-Murakami-Wenzl algebras. These formulas are then applied to derive
various known results needed in the construction of quantum invariants and
modular categories. In particular, an elementary proof of the Wenzl formula for
quantum dimensions is given. This proof does not use the representation theory
of quantum groups and the character formulas.Comment: 26 pages, LaTeX with figures; Section 8 and details to the proof of
Theorem 3.1 are adde
Modular categories of types B,C and D
We construct four series of modular categories from the two-variable Kauffman
polynomial, without use of the representation theory of quantum groups at roots
of unity. The specializations of this polynomial corresponding to quantum
groups of types B, C and D produce series of pre-modular categories. One of
them turns out to be modular and three others satisfy Brugui\`eres'
modularization criterion. For these four series we compute the Verlinde
formulas, and discuss spin and cohomological refinements.Comment: 32 pages, LaTeX with figures, Comment. Math. Helv. 200
Logarithmic Hennings invariants for restricted quantum sl(2)
We construct a Hennings type logarithmic invariant for restricted quantum
at a -th root of unity. This quantum group
is not braided, but factorizable. The invariant is defined for a pair: a
3-manifold and a colored link inside . The link is split into
two parts colored by central elements and by trace classes, or elements in the
Hochschild homology of , respectively. The two main
ingredients of our construction are the universal invariant of a string link
with values in tensor powers of , and the modified trace introduced by the
third author with his collaborators and computed on tensor powers of the
regular representation. Our invariant is a colored extension of the logarithmic
invariant constructed by Jun Murakami
Unified quantum invariants and their refinements for homology 3-spheres with 2-torsion
For every rational homology 3-sphere with 2-torsion only we construct a
unified invariant (which takes values in a certain cyclotomic completion of a
polynomial ring), such that the evaluation of this invariant at any odd root of
unity provides the SO(3) Witten-Reshetikhin-Turaev invariant at this root and
at any even root of unity the SU(2) quantum invariant. Moreover, this unified
invariant splits into a sum of the refined unified invariants dominating spin
and cohomological refinements of quantum SU(2) invariants. New results on the
Ohtsuki series and the integrality of quantum invariants are the main
applications of our construction.Comment: 23 pages, results of math.QA/0510382 are include
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