On ranks of the stable derivation algebra and Deligne's problem
The author reports some computational results on Deligne's problem on the estimation of the Galois image in the pro-\(l\) fundamental group of the projective line minus three points. The problem is to compare certain two graded Lie algebras, one comes from the above Galois representation and the other is Ihara's stable derivation algebra [\textit{Y. Ihara}, Isr. J. Math. 80, 135-153 (1992; Zbl 0782.20034)] linearizing the Grothendieck-Teichmüller group. Over the \(l\)-adic rationals, these two are conjectured to be equal and freely generated by the Ihara-Soulé elements in degrees \(2k+1\) \((k=1,2,\dots)\). The author reports that this conjecture is valid up to degree 12 and that certain iterated brackets of the Ihara-Soulé elements in degrees up to 17 are linearly independent as expected by the conjecture. The method uses previous studies by \textit{Y. Ihara} and \textit{M. Matsumoto} [Contemp. Math. 186, 201-213 (1995; Zbl 0848.11057)] and the author himself [Publ. Res. Inst. Math. Sci. 31, 113-134 (1995; Zbl 0838.11040)]. The updated web-site providing the computer program used in the study is located at ftp://tnt.math.metro-u.ac.jp/pub/program/lie-1.0.tar.gz.
scientific article; zbMATH DE number 1022289
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