nLab Grothendieck-Teichmüller tower

Idea

The Grothendieck-Teichmüller tower construction involves moduli spaces outlined in Grothendieck‘s Esquisse d'un programme and then developed by Vladimir Drinfel'd and others. It involves the Teichmüller groupoids which are the fundamental groupoids of moduli stacks of genus $g$ curves with $n$ points removed. It is a basis for the definition of the Grothendieck-Teichmüller group which is by the definition inertia-preserving automorphism group of the Grothendieck-Teichmüller tower.

Definitions

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Grothendieck-Teichmüller Lie algebra

GT Lie algebra appears as the tangent Lie algebra to the GT group.

Drinfel’d defined it more explicitly as follows.

Let $\mathfrak{lie}_n$ be the degree completion of the free Lie algebra on $n$-generators over a fixed ground field $K$ of characteristic $0$. Let $\mathfrak{der}_n$ be the space of $K$-linear derivations $\mathfrak{lie}_n\to \mathfrak{lie}_n$. A derivation $u\in\mathfrak{der}_n$ is tangential if there exist $a_i\in\mathfrak{lie}_n$ for $i=1,\ldots,n$, such that $u(x_i)=[x_i,a_i]$. In particular, $u$ is determined by elements $a_1,\ldots,a_n$ and is below denoted by $(a_1,\ldots,a_n)$. The tangential derivations form a Lie subalgebra $\mathfrak{tder}_n\subset\mathfrak{der}_n$. The GT Lie algebra is the subspace $\mathfrak{grt}\subset\mathfrak{tder}_2$ consisting of the tangential derivations of the form $(0,\psi)$, where the identities

$$\psi(x,y)=-\psi(y,x),\,\,\,for\,\,\,all\,\,\,x,y,$$

$$\psi(x,y)+\psi(y,z)+\psi(z,x) = 0,\,\,\,whenever\,\,\,\,x+y+z=0,$$

and the pentagon-type identity

$$\psi(t^{1,2},t^{2,34})+\psi(t^{12,3},t^{3,4})= \psi(t^{2,3},t^{3,4})+\psi(t^{1,23},t^{23,4})+\psi(t^{1,2},t^{2,3}),$$

hold, and which is equipped with the Ihara Lie bracket

$$[\psi_1,\psi_2]_{Ihara} = (0,\psi_1)(\psi_2)-(0,\psi_2)(\psi_1)+[\psi_1,\psi_2].$$

Properties

Relation to Drinfeld associators

The GT group acts freely on the set of Drinfeld associators.

Relation to the absolute Galois group of the rational numbers

Theorem

(Drinfeld, Ihara, Deligne)

There is an inclusion of the absolute Galois group of the rational numbers into the Grothendieck-Teichmüller group (recalled e.g. Stix 04, theorem 6).

Relation to the motivic Galois group

The Grothendieck-Teichmüller group is supposed to be a quotient of the motivic Galois group. This is a conjecture due to (Drinfeld 91).

Relation to the graph complex

The Grothendieck-Teichmüller Lie algebra is isomorphic to the 0th cohomology of Kontsevich‘s graph complex (Willwacher 10).

Relation to deformation quantization and the cosmic Galois group

Grothendieck predicted that the GT group is closely related to the absolute Galois group. Maxim Kontsevich later conjectured its action on certain spaces of quantum field theories and outlined its motivic aspects.

This was later proven by Vasily Dolgushev, see at formal deformation quantization – Motivic Galois group action on the space of quantizations for details and pointers.

For more see also at cosmic Galois group for more on this.

Related concepts

• stable symplectic category,

• motivic Galois group, cosmic Galois group

References

General

The Grothendieck-Teichmüller group $GRT$ was originally introduced in

• Vladimir Drinfel'd, On quasitriangular quasi-Hopf algebras and on a group that is closely connected with $Gal(\overline{\mathbb{Q}/\mathbb{Q}})$, abs, Rossiĭskaya Akademiya Nauk. Algebra i Analiz (in Russian) 2 (4): 149–181, ISSN 0234-0852, MR1080203 translation in Leningrad Math. J. 2 (1991), no. 4, 829–860

inspired by

• Alexander Grothendieck, Sketch of a program, London Math. Soc. Lect. Note Ser., 242, Geometric Galois actions, 1, 5–48, Cambridge Univ. Press, Cambridge, 1997.

Review:

• Benoit Fresse, Little discs operads, graph complexes and Grothendieck–Teichmüller groups, in Haynes Miller (ed.) Handbook of Homotopy Theory (arXiv:1811.12536)

See also:

• Leila Schneps, Pierre Lochak, Geometric Galois actions, 1, London Math. Soc. Lecture Note Ser. 242 (doi)– it contains a reprint of Groethendieck’s Esquisse and some surveys by contributors, including Leila Schneps, The Grothendieck-Teichmueller group GT: a survey, pdf

• Maxim Kontsevich, Operads and motives in deformation quantization, Lett. Math. Phys. 48 (1999), 35-72, math.QA/9904055 doi

• Benoit Fresse, Homotopy of Operads and Grothendieck-Teichmüller Groups (Volumes 1 & 2), Mathematical Surveys and Monographs 217 (2017) [ISBN:978-1-4704-3480-9, website, pdf (draft of Part I)]

The second purpose of the book is to explain, from a homotopical viewpoint, a deep relationship between operads and Grothendieck-Teichmüller groups. This connection, which has been foreseen by M. Kontsevich (from researches on the deformation quantization process in mathematical physics), gives a new approach to understanding internal symmetries of structures occurring in various constructions of algebra and topology. In the book, we set up the background required by an in-depth study of this subject, and we make precise the interpretation of the Grothendieck-Teichmüller group in terms of the homotopy of operads. The book is actually organized for this ultimate objective, which readers can take either as a main motivation or as a leading example to learn about general theories.

Videos from a seminar at the Newton Institute:

• Newton Institute Programme Jan-April 2013: Grothendieck-Teichmüller Groups, Deformation and Operads, seminars (some with videos)

Relation to automorphism groups of the profinite completion of braid groups:

• Pierre Lochak, Leila Schneps, The Grothendieck-Teichmüller group and automorphisms of braid groups, in: The Grothendieck Theory of Dessins d’Enfant, Cambridge University Press (1994, 2011) [pdf, doi:10.1017/CBO9780511569302]

The Drinfeld conjecture is stated in

• Vladimir Drinfel'd, On quasi-triangular quasi-Hopf algebras and a group closely related with $Gal(\overline{\mathbb{Q}/\mathbb{Q}})$, Leningrad Math. J. 2 (1991) 829–860.

See also:

• Jakob Stix, The Grothendieck-Teichmüller group and Galois theory of the rational numbers, 2004 (pdf)

• Anton Alekseev, Charles Torossian, The Kashiwara-Vergne conjecture and Drinfeld’s associators, arxiv/0802.4300

Relation to graph complexes

Relation to graph complexes:

$H^0\big( Graphs_0(\mathbb{R}^2) \big)$ is the Lie algebra of the Grothendieck-Teichmüller group:

• Thomas Willwacher, M. Kontsevich’s graph complex and the Grothendieck-Teichmueller Lie algebra, Invent. math. (2015) 200: 671 (arxiv:1009.1654)

• M. Kontsevich, Derived Grothendieck–Teichmüller group and graph complexes [after T. Willwacher], Seminaire Bourbaki 1126 (2016-2017) pdf

• Vasily Dolgushev, Christopher Rogers, Notes on algebraic operads, graph complexes, and Willwacher’s construction, In: Mathematical aspects of quantization 583 (2012): 25–145. (arXiv:1202.2937)

• V. A. Dolgushev, C. L. Rogers, T. H. Willwacher, Kontsevich’s graph complex, GRT, and the deformation complex of the sheaf of polyvector fields, Ann. Math. 182:3 (2015) 855–943 doi

• Vasily Dolgushev, A manifestation of the Grothendieck-Teichmueller group in geometry (slides pdf)

See also:

• Kevin Morand, A note on multi-oriented graph complexes and deformation quantization of Lie bialgebroids (arXiv:2102.07593)