On some derivations of Lie algebras related to Galois representations
In this paper, a systematic way of introducing multi-degrees in the derivation algebras of free Lie algebras is presented by using a Hall basis, and is applied to the theory of exterior Galois representations. Motivated from Matsumoto's work (cf. below), the author applies this device especially to showing that a certain infinite sequence of derivations \(D_m\) of even degrees \((m \geq 4)\) of a free Lie algebra of rank 2 produce nontrivial derivations \([D_{m_1}, [D_{m_2}, [\dots [D_{m_{k - 1}}, D_{m_k}] \dots]]]\) whenever \(m_{k - 1} \neq m_k\). In the case of the graded Lie algebra associated with the pro-\(l\) fundamental group of an elliptic curve minus one point defined over a number field, the reviewer showed that these \(D_i\) correspond to certain exterior Galois images arising from special values of theta functions [On exterior Galois representations associated with open elliptic curves, J. Math. Sci., Univ. Tokyo 2, 197-231 (1995)]. From this viewpoint, the above result gives a quantitative estimate of the nonabelian part of the exterior Galois images originated from theta values in the genus one case. In the genus zero case, analogous estimates have been investigated by \textit{Y. Ihara} [in Galois groups over \(\mathbb{Q}\), Publ., Math. Sci. Res. Inst. 16, 299-313 (1989; Zbl 0706.14018)] and \textit{M. Matsumoto} [On the Galois image in the derivation algebra of \(\pi_1\) of the projective line minus three points, Contemp. Math. 186, 201-213 (1995)].
pro-\(l\) fundamental group of an elliptic curve
scientific article; zbMATH DE number 785369
scientific article