On some derivations of Lie algebras related to Galois representations
type: Item
instance of: scholarly article
title: On some derivations of Lie algebras related to Galois representations
zbMATH Open document ID: 0838.11040
author: Hiroshi Tsunogai
published in: Publications of the Research Institute for Mathematical Sciences, Kyoto University
publication date: 1996-06-03T00:00:00Z
review text: 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)].
Mathematics Subject Classification ID: 11G05
Mathematics Subject Classification ID: 11R32
Mathematics Subject Classification ID: 14H30
Mathematics Subject Classification ID: 17B01
Mathematics Subject Classification ID: 17B40
zbMATH DE Number: 785369
zbMATH Keywords: derivations of Lie algebras
zbMATH Keywords: pro-\(l\) fundamental group of an elliptic curve
zbMATH Keywords: free Lie algebras
zbMATH Keywords: Hall basis
zbMATH Keywords: exterior Galois representations
reviewed by: Hiroaki Nakamura
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.2977/prims/1195164794
OpenAlex ID: W2063637204
cites work: Q3804906
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cites work: Q3487541
cites work: Q3784221
cites work: Q3033957
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DOI: 10.2977/PRIMS/1195164794
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description: scientific article; zbMATH DE number 785369