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Homology of depth-graded motivic Lie algebras and koszulity. (English. French summary) Zbl 1414.17012

Summary: The Broadhurst-Kreimer (BK) conjecture describes the Hilbert series of a bigraded Lie algebra \(\mathfrak{a}\) related to the multizeta values. Brown proposed a conjectural description of the homology of this Lie algebra (homological conjecture (HC)), and showed it implies the BK conjecture. We show that a part of HC is equivalent to a presentation of \(\mathfrak{a}\), and that the remaining part of HC is equivalent to a weaker statement. Finally, we prove that granted the first part of HC, the remaining part of HC is equivalent to either of the following equivalent statements: (a) the vanishing of the third homology group of a Lie algebra with quadratic presentation, constructed out of the period polynomials of modular forms; (b) the koszulity of the enveloping algebra of this Lie algebra.

MSC:

17B55 Homological methods in Lie (super)algebras
11M32 Multiple Dirichlet series and zeta functions and multizeta values
16E40 (Co)homology of rings and associative algebras (e.g., Hochschild, cyclic, dihedral, etc.)
16S10 Associative rings determined by universal properties (free algebras, coproducts, adjunction of inverses, etc.)
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[1] Y. André, Une introduction aux motifs (motifs purs, motifs mixtes, périodes), Panoramas et Synthèses [Panoramas and Syntheses], vol. 17, Société Mathématique de France, Paris, 2004, xii+261 pages. · Zbl 1060.14001
[2] N. Bourbaki, Éléments de mathématique. Algèbre. Chapitres 1 à 3, Hermann, Paris, 1970, xiii+635 pp. pages. · Zbl 0211.02401
[3] D. J. Broadhurst & D. Kreimer, « Association of multiple zeta values with positive knots via Feynman diagrams up to \(9\) loops », Phys. Lett. B393 (1997), no. 3-4, p. 403-412. · Zbl 0946.81028
[4] F. Brown, « Depth-graded motivic multiple zeta values », . · Zbl 1481.11084
[5] —, « Motivic periods and the projective line minus three points », .
[6] —, « Mixed Tate motives over \(\mathbb{Z} \) », Ann. of Math. (2)175 (2012), no. 2, p. 949-976. · Zbl 1278.19008
[7] S. Carr, H. Gangl & L. Schneps, « On the Broadhurst-Kreimer generating series for multiple zeta values », preprint. · Zbl 1382.11065
[8] P. Deligne & A. B. Goncharov, « Groupes fondamentaux motiviques de Tate mixte », Ann. Sci. École Norm. Sup. (4)38 (2005), no. 1, p. 1-56. · Zbl 1084.14024
[9] V. G. Drinfelʼd, « On quasitriangular quasi-Hopf algebras and on a group that is closely connected with \({\rm Gal}(\overline{\bf Q}/{\bf Q})\) », Algebra i Analiz2 (1990), no. 4, p. 149-181. · Zbl 0718.16034
[10] J. Ecalle, « ARI/GARI, la dimorphie et l’arithmétique des multizêtas: un premier bilan », J. Théor. Nombres Bordeaux15 (2003), no. 2, p. 411-478. · Zbl 1094.11032
[11] —, « The flexion structure and dimorphy: flexion units, singulators, generators, and the enumeration of multizeta irreducibles », in Asymptotics in dynamics, geometry and PDEs; generalized Borel summation. Vol. II, CRM Series, vol. 12, Ed. Norm., Pisa, 2011, With computational assistance from S. Carr, p. 27-211. · Zbl 1253.11083
[12] H. Furusho, « Pentagon and hexagon equations », Ann. of Math. (2)171 (2010), no. 1, p. 545-556. · Zbl 1257.17019
[13] —, « Double shuffle relation for associators », Ann. of Math. (2)174 (2011), no. 1, p. 341-360. · Zbl 1321.11088
[14] H. Gangl, M. Kaneko & D. Zagier, « Double zeta values and modular forms », in Automorphic forms and zeta functions, World Sci. Publ., Hackensack, NJ, 2006, p. 71-106. · Zbl 1122.11057
[15] A. B. Goncharov, « Multiple polylogarithms, cyclotomy and modular complexes », Math. Res. Lett.5 (1998), no. 4, p. 497-516. · Zbl 0961.11040
[16] —, « Multiple \(\zeta \)-values, Galois groups, and geometry of modular varieties », in European Congress of Mathematics, Vol. I (Barcelona, 2000), Progr. Math., vol. 201, Birkhäuser, Basel, 2001, p. 361-392. · Zbl 1042.11042
[17] R. Hain, « Genus 3 mapping class groups are not Kähler », J. Topol.8 (2015), no. 1, p. 213-246. · Zbl 1318.14028
[18] K. Ihara, M. Kaneko & D. Zagier, « Derivation and double shuffle relations for multiple zeta values », Compos. Math.142 (2006), no. 2, p. 307-338. · Zbl 1186.11053
[19] T. T. Q. Le & J. Murakami, « Kontsevich’s integral for the Kauffman polynomial », Nagoya Math. J.142 (1996), p. 39-65. · Zbl 0866.57008
[20] J.-L. Loday & B. Vallette, Algebraic operads, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 346, Springer, Heidelberg, 2012, xxiv+634 pages. · Zbl 1260.18001
[21] A. Polishchuk & L. Positselski, Quadratic algebras, University Lecture Series, vol. 37, American Mathematical Society, Providence, RI, 2005, xii+159 pages. · Zbl 1145.16009
[22] G. Racinet, « Doubles mélanges des polylogarithmes multiples aux racines de l’unité », Publ. Math. Inst. Hautes Études Sci. (2002), no. 95, p. 185-231. · Zbl 1050.11066
[23] L. Schneps, « On the Poisson bracket on the free Lie algebra in two generators », J. Lie Theory16 (2006), no. 1, p. 19-37. · Zbl 1120.17004
[24] J.-P. Serre, Cohomologie galoisienne, fifth ed., Lecture Notes in Mathematics, vol. 5, Springer-Verlag, Berlin, 1994, x+181 pages. · Zbl 0812.12002
[25] C. A. Weibel, An introduction to homological algebra, Cambridge Studies in Advanced Mathematics, vol. 38, Cambridge University Press, Cambridge, 1994, xiv+450 pages. · Zbl 0797.18001
[26] D. Zagier, « Values of zeta functions and their applications », in First European Congress of Mathematics, Vol. II (Paris, 1992), Progr. Math., vol. 120, Birkhäuser, Basel, 1994, p. 497-512. · Zbl 0822.11001
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