## Algebraic & Geometric Topology

### The universal quantum invariant and colored ideal triangulations

Sakie Suzuki

#### Abstract

The Drinfeld double of a finite-dimensional Hopf algebra is a quasitriangular Hopf algebra with the canonical element as the universal $R$–matrix, and one can obtain a ribbon Hopf algebra by adding the ribbon element. The universal quantum invariant of framed links is constructed using a ribbon Hopf algebra. In that construction, a copy of the universal $R$–matrix is attached to each crossing, and invariance under the Reidemeister III move is shown by the quantum Yang–Baxter equation of the universal $R$–matrix. On the other hand, the Heisenberg double of a finite-dimensional Hopf algebra has the canonical element (the $S$–tensor) satisfying the pentagon relation. In this paper we reconstruct the universal quantum invariant using the Heisenberg double, and extend it to an invariant of equivalence classes of colored ideal triangulations of $3$–manifolds up to colored moves. In this construction, a copy of the $S$–tensor is attached to each tetrahedron, and invariance under the colored Pachner $(2,3)$ moves is shown by the pentagon relation of the $S$–tensor.

#### Article information

Source
Algebr. Geom. Topol., Volume 18, Number 6 (2018), 3363-3402.

Dates
Revised: 17 April 2018
Accepted: 30 April 2018
First available in Project Euclid: 27 October 2018

https://projecteuclid.org/euclid.agt/1540605646

Digital Object Identifier
doi:10.2140/agt.2018.18.3363

Mathematical Reviews number (MathSciNet)
MR3868224

Zentralblatt MATH identifier
06990067

#### Citation

Suzuki, Sakie. The universal quantum invariant and colored ideal triangulations. Algebr. Geom. Topol. 18 (2018), no. 6, 3363--3402. doi:10.2140/agt.2018.18.3363. https://projecteuclid.org/euclid.agt/1540605646

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