Abstract
We derive from the compatibility of associators with the module harmonic coproduct, obtained in Part I of the series, the inclusion of the torsor of associators into that of double shuffle relations, which completes one of the aims of this series. We define two stabilizer torsors using the module and algebra harmonic coproducts from Part I. We show that the double shuffle torsor can be described using the module stabilizer torsor, and that the latter torsor is contained in the algebra stabilizer torsor.
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Notes
The statements of [4], §2.4 rely on the statement from loc. cit., §2.4.1 that there is an isomorphism of graded algebras \(\mathrm {gr}({\mathcal {V}}^{\mathrm {B}}){\mathop {\rightarrow }\limits ^{\sim }}{\mathcal {V}}^{\mathrm {DR}}\) induced by \([X_i-1]\mapsto e_i\) for \(i=0,1\), which is proved in [4] by quoting reference [Bou] from loc. cit.. This quotation does not prove the claimed statement. A correct proof can be done by applying the functor \(\mathrm {gr}\) to the isomorphism of filtered algebra \(\varphi \) from [8], Example A2.11, S being equal to the set \(\{0,1\}\) and K to \({\mathbb {Q}}\), and then tensoring the resulting isomorphism with \({\mathbf {k}}\).
The definition of the right action of \({\mathbf {k}}^\times \subset {\textsf {GRT} }({\mathbf {k}})\) on \({{\textsf {M} }}({\mathbf {k}})\) given by \((\mu ,\varphi )\cdot c=(\mu /c,\varphi (A/c,B/c))\) (p. 852) does not appear to be compatible with the definition of the group structure on \({\textsf {GRT} }({\mathbf {k}})\) as the semidirect product of \({\textsf {GRT} }_1({\mathbf {k}})\) with the action of \({\mathbf {k}}^\times \) by \((c\cdot g)(A,B):=g(A/c,B/c)\) ( [2], p. 851), since these formulas lead to \((((\mu ,\varphi )\cdot c)\cdot g)\cdot c^{-1}= (\mu ,\varphi )\cdot (c^{-1}\cdot g\cdot c)\). This compatibility is restored, and Proposition 5.5 in [2] is correct, if the right action is defined by \((\mu ,\varphi )\cdot c=(c\mu ,\varphi (cA,cB))\); we work with this convention.
The map \((-)\cdot 1_{\mathrm {DR}}:\hat{{\mathcal {V}}}^{\mathrm {DR}}\rightarrow \hat{{\mathcal {M}}}^{\mathrm {DR}}\) is not compatible with the coproducts \({\hat{\Delta }}^{{\mathcal {V}},{\mathrm {DR}}}\) and \({\hat{\Delta }}^{{\mathcal {M}},{\mathrm {DR}}}\), so that this map does not induce a group morphism \({\mathcal {G}}(\hat{{\mathcal {V}}}^{\mathrm {DR}})\rightarrow {\mathcal {G}}(\hat{{\mathcal {M}}}^{\mathrm {DR}})\).
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Acknowledgements
The collaboration of both authors has been supported by grant JSPS KAKENHI as well as HighAGT ANR-20-CE40-0016.
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Enriquez, B., Furusho, H. The Betti side of the double shuffle theory. II. Double shuffle relations for associators. Sel. Math. New Ser. 29, 3 (2023). https://doi.org/10.1007/s00029-022-00807-w
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DOI: https://doi.org/10.1007/s00029-022-00807-w