nLab
Grothendieck-Teichmüller tower

Contents

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 gg curves with nn 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

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 𝔩𝔦𝔢 n\mathfrak{lie}_n be the degree completion of the free Lie algebra on nn-generators over a fixed ground field KK of characteristic 00. Let 𝔡𝔢𝔯 n\mathfrak{der}_n be the space of KK-linear derivations 𝔩𝔦𝔢 n𝔩𝔦𝔢 n\mathfrak{lie}_n\to \mathfrak{lie}_n. A derivation u𝔡𝔢𝔯 nu\in\mathfrak{der}_n is tangential if there exist a i𝔩𝔦𝔢 na_i\in\mathfrak{lie}_n for i=1,,ni=1,\ldots,n, such that u(x i)=[x i,a i]u(x_i)=[x_i,a_i]. In particular, uu is determined by elements a 1,,a na_1,\ldots,a_n and is below denoted by (a 1,,a n)(a_1,\ldots,a_n). The tangential derivations form a Lie subalgebra 𝔱𝔡𝔢𝔯 n𝔡𝔢𝔯 n\mathfrak{tder}_n\subset\mathfrak{der}_n. The GT Lie algebra is the subspace 𝔤𝔯𝔱𝔱𝔡𝔢𝔯 2\mathfrak{grt}\subset\mathfrak{tder}_2 consisting of the tangential derivations of the form (0,ψ)(0,\psi), where the identities

ψ(x,y)=ψ(y,x),forallx,y, \psi(x,y)=-\psi(y,x),\,\,\,for\,\,\,all\,\,\,x,y,
ψ(x,y)+ψ(y,z)+ψ(z,x)=0,wheneverx+y+z=0, \psi(x,y)+\psi(y,z)+\psi(z,x) = 0,\,\,\,whenever\,\,\,\,x+y+z=0,

and the pentagon-type identity

ψ(t 1,2,t 2,34)+ψ(t 12,3,t 3,4)=ψ(t 2,3,t 3,4)+ψ(t 1,23,t 23,4)+ψ(t 1,2,t 2,3), \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

[ψ 1,ψ 2] Ihara=(0,ψ 1)(ψ 2)(0,ψ 2)(ψ 1)+[ψ 1,ψ 2]. [\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 space 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.

References

General

The Grothendieck-Teichmüller group GRTGRT was originally introduced in

  • Vladimir Drinfel'd, On quasitriangular quasi-Hopf algebras and on a group that is closely connected with Gal(/¯)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.

See also

  • Leila Schneps, Pierre Lotchak, 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

The following monograph is in progress (and as of early 2016, recently completed):

  • Benoit Fresse, Homotopy of Operads and Grothendieck-Teichmüller groups, website

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

References

The Drinfeld conjeture is stated in

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

Relation to the graph complex

  • 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

Revised on January 31, 2016 05:36:28 by Noam Zeilberger (176.189.43.179)