Algebraic & Geometric Topology

The universal quantum invariant and colored ideal triangulations

Sakie Suzuki

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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

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

Received: 24 August 2017
Revised: 17 April 2018
Accepted: 30 April 2018
First available in Project Euclid: 27 October 2018

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 16T25: Yang-Baxter equations 57M27: Invariants of knots and 3-manifolds 81R50: Quantum groups and related algebraic methods [See also 16T20, 17B37]

knots and links 3-manifolds Heisenberg double Drinfeld double universal quantum invariant colored ideal triangulation


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.

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  • S Baaj, G Skandalis, Unitaires multiplicatifs et dualité pour les produits croisés de $C^*\mskip-2mu$–algèbres, Ann. Sci. École Norm. Sup. 26 (1993) 425–488
  • S Baseilhac, R Benedetti, Quantum hyperbolic invariants of $3\mskip-1.5mu$–manifolds with ${\rm PSL}(2,\mathbb C)$–characters, Topology 43 (2004) 1373–1423
  • S Baseilhac, R Benedetti, Classical and quantum dilogarithmic invariants of flat ${\rm PSL}(2,\mathbb C)$–bundles over $3\mskip-1.5mu$–manifolds, Geom. Topol. 9 (2005) 493–569
  • S Baseilhac, R Benedetti, Quantum hyperbolic geometry, Algebr. Geom. Topol. 7 (2007) 845–917
  • S Baseilhac, R Benedetti, The Kashaev and quantum hyperbolic link invariants, J. Gökova Geom. Topol. 5 (2011) 31–85
  • S Baseilhac, R Benedetti, Analytic families of quantum hyperbolic invariants, Algebr. Geom. Topol. 15 (2015) 1983–2063
  • S Baseilhac, R Benedetti, Non ambiguous structures on $3\mskip-1.5mu$–manifolds and quantum symmetry defects, Quantum Topol. 8 (2017) 749–846
  • R Benedetti, C Petronio, Branched standard spines of $3\mskip-1.5mu$–manifolds, Lecture Notes in Math. 1653, Springer (1997)
  • R Benedetti, C Petronio, Spin structures on $3\mskip-1.5mu$–manifolds via arbitrary triangulations, Algebr. Geom. Topol. 14 (2014) 1005–1054
  • J Cho, H Kim, S Kim, Optimistic limits of Kashaev invariants and complex volumes of hyperbolic links, J. Knot Theory Ramifications 23 (2014) art. id. 1450049
  • V G Drinfel'd, Quantum groups, from “Proceedings of the International Congress of Mathematicians, I” (A M Gleason, editor), Amer. Math. Soc., Providence, RI (1987) 798–820
  • L D Faddeev, R M Kashaev, Quantum dilogarithm, Modern Phys. Lett. A 9 (1994) 427–434
  • K Hikami, R Inoue, Braiding operator via quantum cluster algebra, J. Phys. A 47 (2014) art. id. 474006
  • K Hikami, R Inoue, Braids, complex volume and cluster algebras, Algebr. Geom. Topol. 15 (2015) 2175–2194
  • M Kapranov, Heisenberg doubles and derived categories, J. Algebra 202 (1998) 712–744
  • R M Kashaev, Quantum dilogarithm as a $6j\mskip-1.5mu$–symbol, Modern Phys. Lett. A 9 (1994) 3757–3768
  • R M Kashaev, A link invariant from quantum dilogarithm, Modern Phys. Lett. A 10 (1995) 1409–1418
  • R M Kashaev, The Heisenberg double and the pentagon relation, Algebra i Analiz 8 (1996) 63–74 In Russian; translated in St. Petersburg Math. J. 8 (1997) 585–592
  • R M Kashaev, The hyperbolic volume of knots from the quantum dilogarithm, Lett. Math. Phys. 39 (1997) 269–275
  • C Kassel, Quantum groups, Graduate Texts in Math. 155, Springer (1995)
  • R J Lawrence, A universal link invariant using quantum groups, from “Differential geometric methods in theoretical physics” (A I Solomon, editor), World Scientific, Teaneck, NJ (1989) 55–63
  • R J Lawrence, A universal link invariant, from “The interface of mathematics and particle physics” (D G Quillen, G B Segal, S T Tsou, editors), Inst. Math. Appl. Conf. Ser. New Ser. 24, Oxford Univ. Press (1990) 151–156
  • J-H Lu, On the Drinfel'd double and the Heisenberg double of a Hopf algebra, Duke Math. J. 74 (1994) 763–776
  • S Majid, Quantum double for quasi-Hopf algebras, Lett. Math. Phys. 45 (1998) 1–9
  • S Majid, Double-bosonization of braided groups and the construction of $U_q(\mathfrak g)$, Math. Proc. Cambridge Philos. Soc. 125 (1999) 151–192
  • H Murakami, J Murakami, The colored Jones polynomials and the simplicial volume of a knot, Acta Math. 186 (2001) 85–104
  • \relax minus 2pt A Ocneanu, Chirality for operator algebras, from “Subfactors” (H Araki, Y Kawahigashi, H Kosaki, editors), World Scientific, River Edge, NJ (1994) 39–63
  • T Ohtsuki, Colored ribbon Hopf algebras and universal invariants of framed links, J. Knot Theory Ramifications 2 (1993) 211–232
  • N Y Reshetikhin, V G Turaev, Ribbon graphs and their invariants derived from quantum groups, Comm. Math. Phys. 127 (1990) 1–26
  • M A Semenov-Tian-Shansky, Poisson Lie groups, quantum duality principle, and the quantum double, from “Mathematical aspects of conformal and topological field theories and quantum groups” (P J Sally, Jr, M Flato, J Lepowsky, N Reshetikhin, G J Zuckerman, editors), Contemp. Math. 175, Amer. Math. Soc., Providence, RI (1994) 219–248
  • S Suzuki, On the universal $\mathrm{sl}_2$ invariant of boundary bottom tangles, Algebr. Geom. Topol. 12 (2012) 997–1057
  • V G Turaev, O Y Viro, State sum invariants of $3\mskip-1.5mu$–manifolds and quantum $6j\mskip-1.5mu$–symbols, Topology 31 (1992) 865–902
  • J Weeks, Computation of hyperbolic structures in knot theory, from “Handbook of knot theory” (W Menasco, M Thistlethwaite, editors), Elsevier, Amsterdam (2005) 461–480
  • Y Yokota, On the complex volume of hyperbolic knots, J. Knot Theory Ramifications 20 (2011) 955–976