Double shuffle relation for associators

Furusho, Hidekazu

doi:10.4007/annals.2011.174.1.9

Cited by 47 publications

(43 citation statements)

References 13 publications

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“…A numerical experiment of [11] shows that up to degree 19 the Lie algebra dmr 0 coincides with grt 1 and is freely generated by σ 2k+1 's. Recently, Furusho showed (see [13]) that there is an injective Lie homomorphism grt 1 → dmr 0 . A numerical computation by Albert and the second author [1] shows that up to degree 16 the dimensions of graded components of krv 2 coincide with those of Kt ⊕ lie(σ 3 , σ 5 , .…”

Section: The Grothendieck-teichmüller Lie Algebramentioning

confidence: 99%

The Kashiwara-Vergne conjecture and Drinfeld's associators

2012

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The Kashiwara-Vergne (KV) conjecture is a property of the CampbellHausdorff series put forward in 1978. It has been settled in the positive by E. Meinrenken and the first author in 2006. In this paper, we study the uniqueness issue for the KV problem. To this end, we introduce a family of infinite-dimensional groups KRV 0 n , and a group KRV2 which contains KRV 0 2 as a normal subgroup. We show that KRV2 also contains the Grothendieck-Teichmüller group GRT1 as a subgroup, and that it acts freely and transitively on the set of solutions of the KV problem SolKV. Furthermore, we prove that SolKV is isomorphic to a direct product of affine line A 1 and the set of solutions of the pentagon equation with values in the group KRV 0 3 . The latter contains the set of Drinfeld's associators as a subset. As a by-product of our construction, we obtain a new proof of the Kashiwara-Vergne conjecture based on the Drinfeld's theorem on existence of associators.

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Section: The Grothendieck-teichmüller Lie Algebramentioning

confidence: 99%

The Kashiwara-Vergne conjecture and Drinfeld's associators

2012

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“…It is known by the work of Racinet that the image of i is contained in the Lie algebra dmr 0 of solutions to the double shuffle equations. It is also contained in the space of solutions to Drinfeld's associator equations, which by a result of Furusho [21], are contained in dmr 0 . It is well-known that (1.3) i(σ 2n+1 ) = ad(x 0 ) 2n x 1 + terms of degree ≥ 2 in x 1 , This can be seen as follows.…”

Section: Introductionmentioning

confidence: 99%

Zeta Elements in Depth 3 and the Fundamental Lie Algebra of The infinitesimal Tate Curve

Brown¹

2017

Forum of Mathematics, Sigma

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This paper draws connections between the double shuffle equations and structure of associators; Hain and Matsumoto's universal mixed elliptic motives; and the Rankin-Selberg method for modular forms for SL 2 (Z). We write down explicit formulae for zeta elements σ 2n−1 (generators of the Tannaka Lie algebra of the category of mixed Tate motives over Z) in depths up to four, give applications to the Broadhurst-Kreimer conjecture, and solve the double shuffle equations for multiple zeta values in depths two and three.

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“…Here, the implication (Associator) ⇒ (RDS) was proved by Furusho [2], and the implication (RDS + duality) ⇒ (Kawashima) was proved by Kaneko-Yamamoto [6].…”

Section: Confluencementioning

confidence: 92%