nLab motivic Galois group

Idea

Simply by changing the symmetry in the symmetric monoidal category of pure numerical motives $Mot_{num}(k,\mathbb{Q})$, we can obtain a Tannakian category $\tilde Mot_{num}(k,\mathbb{Q})$, which is thus equivalent to a category of representations of some algebraic group $GMot_k$:

$$\tilde Mot_{num}(k,\mathbb{Q}) \simeq Rep(GMot_k) \,.$$

In the sense of Galois theory, this algebraic group is called the motivic Galois group for pure motives. There is also a motivic Galois group of mixed motives.

This group is, or is closely related to, the group of algebraic periods, and as such is related to expressions appearing in deformation quantization and in renormalization in quantum field theory, whence it is also sometimes referred to as the cosmic Galois group. See there for more on this.

(Mixed) Motivic Galois groups

Nori’s Motivic Galois group

M. Levine, Mixed motives. (3.3 Motives by Tannakian formalism) In, Handbook of K-theory, Vol. 1, Friedlander and Grayson, eds., p. 429–521, Springer Verlag (2005).(pdf).

Ayoub’s Motivic Galois group

• Joseph Ayoub, L’algèbre de Hopf et le groupe de Galois motiviques d’un corps de caractéristique nulle I,

  (pdf).

• Joseph Ayoub, L’algèbre de Hopf et le groupe de Galois motiviques d’un corps de caractéristique nulle II, (pdf).

• Joseph Ayoub, From motives to comodules over the motivic Hopf algebra, (pdf).

Joseph Ayoub, Lecture 1,Lecture 2,Lecture 3, The lectures were held within the framework of the (Junior) Hausdorff Trimester Program Topology and the Workshop: Interactions between operads and motives.

Mathoverflow, Derived version of equivalence between motives and representations of Motivic galois groups?.

Comparison

Ayoub’s weak Tannakian formalism applied to the Betti realization for Voevodsky motives yields a pro-algebraic group, a candidate for the motivic Galois group. In particular, each Voevodsky motive gives rise to a representation of this group. On the other hand Nori motives are just representations of Nori’s motivic Galois group. These groups are isomorphic.

• Utsav Choudhury, Martin Gallauer Alves de Souza, An isomorphism of motivic galois groups, (arXiv:1410.6104).

the existence of a motivic t-structure (which renders the Betti realization t-exact) would imply the isomorphism of motivic Galois groups.

• J.P.Pridham?, Tannaka duality for enhanced triangulated categories, (arXiv:1309.0637).

Noncommutative Motivic Galois groups

• Matilde Marcolli, Gonçalo Tabuada Unconditional noncommutative motivic galois groups,(arXiv:1112.5422).

• Matilde Marcolli, Gonçalo Tabuada Noncommutative numerical motives, Tannakian structures, and motivic Galois groups. J. Eur. Math. Soc. 18 (2016), 623-655. doi: 10.4171/JEMS/598

References

General

A quick survey is in

• Bruno Kahn, Motivic Galois groups (pdf)

More is in

• Jean-Pierre Serre, Propriétés conjecturales des groups de Galois motiviques et des représentations l-adiques, in U. Jannsen, S. Kleiman, J.-P. Serre (eds) Motives, Proceedings of Symposia in Pure Mathematics, AMS, vol. 55, part 1, 377 - 400 (1994).

• Yves André, Groupes de Galois motiviques et périodes, pdf, Séminaire Bourbaki, 68ème année, 2015-2016 n°1104.

• Yves André, What is…a motivic Galois group, writeup of a seminar, pdf

• AB Goncharov, Polylogarithms and Motivic galois groups, (pdf).

• R Sujatha, Motives from a categorical point of view, (pdf).

• Minhyong Kim, (Videos) Motivic fundamental groups and Diophantine geometry, I, II.

• Annette Huber, Stefan Müller-Stach, On the relation between Nori motives and Kontsevich periods, arxiv/1105.0865

We show that the spectrum of Kontsevich’s algebra of formal periods is a torsor under the motivic Galois group for mixed motives over the rational numbers. This assertion is stated without proof by Kontsevich and originally due to Nori. In a series of appendices, we also provide the necessary details on Nori’s category of motives.

Mathoverflow, What are the possible motivic Galois groups over Q ?.

Mathoverflow, Why would the category of Motives be Tannakian?.

Relation to Grothendieck-Teichmüller group

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

• 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.

In quantum field theory

A conjecture that the motivic Galois group naturally acts on the space of deformation quantizations of free field theories is in section 5 of

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

Regarding that this approach is related to Hodge theory periods, and that QFT can be attacked using the Batalin-Vilkovsky formalism, Barannikov has developed further the Batalin-Vilkovisky geometry and related Hodge theory in a direction which is probably a ground for better understanding of motivic Galois group as well.

An analogous statement, that a motivic Galois group naturally acts on structures in renormalization is in

• Alain Connes, Matilde Marcolli, Renormalization and motivic Galois theory, arXiv:math/0409306

• Alain Connes, Symetries Galoisiennes et Renormalisation, arXiv:math/0211199

For more along these lines see at cosmic Galois group.

Grothendieck’s Yoga Remarks

Related concepts

• Galois theory

• motive

• cosmic Galois group, Grothendieck-Teichmüller group