# zbMATH — the first resource for mathematics

Hopf diagrams and quantum invariants. (English) Zbl 1116.57011
The paper discusses certain aspects of an abstract general construction of quantum manifold invariants. Its main goal is to introduce a monoidal category of Hopf diagrams and to describe a universal encoding of ribbon strings as Hopf diagrams. By definition, a Hopf diagram is a morphism of the category Diag, which is the convolution category $$\text{Conv }_{\mathcal D}(*,{\mathbf 1})$$ of the category $${\mathcal D}$$ generated by one object $$*$$ and the morphisms $$\Delta :*\rightarrow *\otimes *,\epsilon :*\rightarrow {\mathbf 1}, \omega_+:*\otimes *\rightarrow {\mathbf 1}, \omega_{-}:*\otimes *\rightarrow {\mathbf 1}, \theta _+:*\rightarrow {\mathbf 1}, \theta_{-}:*\rightarrow {\mathbf 1},$$ where $${\mathbf 1}$$ denotes the unit object of the monoidal category, subject to the relations $$(\text{id}_*\otimes \Delta )\Delta=(\Delta \otimes \text{id}_*)\Delta$$ and $$(\text{id}_*\otimes \epsilon )\Delta=\text{id}_*=(\epsilon \otimes \text{id}_*)\Delta$$. The universal encoding introduced by the authors is an injective monoidal functor and admits a straightforward monoidal retraction. Any Hopf diagram with $$n$$ legs yields an $$n$$-form on the coend of a ribbon category in a completely explicit way. So the computation of a quantum invariant of a $$3$$-manifold is reduced to a two stage process: first the formal computation of the associated Hopf diagram, and second, the evaluation of this diagram in the given category. The latter relies on the so-called Kirby elements, which are also discussed in the paper.

##### MSC:
 57M27 Invariants of knots and $$3$$-manifolds (MSC2010) 18D10 Monoidal, symmetric monoidal and braided categories (MSC2010) 81R50 Quantum groups and related algebraic methods applied to problems in quantum theory
Full Text:
##### References:
 [1] A Bruguières, Tresses et structure entière sur la catégorie des représentations de $$\mathrm{SL}_N$$ quantique, Comm. Algebra 28 (2000) 1989 · Zbl 0951.18003 [2] A Bruguières, Double braidings, twists and tangle invariants, J. Pure Appl. Algebra 204 (2006) 170 · Zbl 1091.57007 [3] T Kerler, V V Lyubashenko, Non-semisimple topological quantum field theories for 3-manifolds with corners, Lecture Notes in Mathematics 1765, Springer (2001) · Zbl 0982.57013 [4] R Kirby, A calculus for framed links in $$S^3$$, Invent. Math. 45 (1978) 35 · Zbl 0377.55001 [5] W B R Lickorish, An introduction to knot theory, Graduate Texts in Mathematics 175, Springer (1997) · Zbl 0886.57001 [6] V Lyubashenko, Modular transformations for tensor categories, J. Pure Appl. Algebra 98 (1995) 279 · Zbl 0823.18003 [7] V V Lyubashenko, Invariants of 3-manifolds and projective representations of mapping class groups via quantum groups at roots of unity, Comm. Math. Phys. 172 (1995) 467 · Zbl 0844.57016 [8] S Mac Lane, Categories for the working mathematician, Graduate Texts in Mathematics 5, Springer (1998) · Zbl 0906.18001 [9] S Majid, Braided groups, J. Pure Appl. Algebra 86 (1993) 187 · Zbl 0797.17004 [10] A Markoff, Foundations of the algebraic theory of tresses, Trav. Inst. Math. Stekloff 16 (1945) 53 · Zbl 0061.02507 [11] M H A Newman, On theories with a combinatorial definition of “equivalence.”, Ann. of Math. $$(2)$$ 43 (1942) 223 · Zbl 0060.12501 [12] N Reshetikhin, V G Turaev, Invariants of 3-manifolds via link polynomials and quantum groups, Invent. Math. 103 (1991) 547 · Zbl 0725.57007 [13] V G Turaev, Quantum invariants of knots and 3-manifolds, de Gruyter Studies in Mathematics 18, Walter de Gruyter & Co. (1994) · Zbl 0812.57003 [14] A Virelizier, Kirby elements and quantum invariants, Proc. London Math. Soc. $$(3)$$ 93 (2006) 474 · Zbl 1114.57016
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.