We introduce a Lie algebra of initial terms of logarithmic vector fields along a hypersurface singularity. We show that the completely reducible part of its linear projection lifts formally to a linear Lie algebra of logarithmic vector fields. For quasihomogeneous singularities, we prove convergence of this linearization. We relate our construction to the work of Hauser and Müller on Levi subgroups of automorphism groups of singularities, which proves convergence even for algebraic singularities. Based on the initial Lie algebra, we introduce a notion of reductive hypersurface singularity and show that any reductive free divisor is linear. As an application, we describe a lower bound for the dimension of hypersurface singularities in terms of the semisimple part of their initial Lie algebra.
1
Dép. de Mathématiques, Université d'Angers, 2 Bd Lavoisier, 49045 Angers, France
2
Oklahoma State University, 401 Mathematical Sciences, Stillwater, OK 74078, U.S.A.
Michel Granger; Mathias Schulze. Initial Logarithmic Lie Algebras of Hypersurface Singularities. Journal of Lie Theory, Tome 19 (2009) no. 2, pp. 209-221. http://geodesic.mathdoc.fr/item/JOLT_2009_19_2_a0/
@article{JOLT_2009_19_2_a0,
author = {Michel Granger and Mathias Schulze},
title = {Initial {Logarithmic} {Lie} {Algebras} of {Hypersurface} {Singularities}},
journal = {Journal of Lie Theory},
pages = {209--221},
year = {2009},
volume = {19},
number = {2},
url = {http://geodesic.mathdoc.fr/item/JOLT_2009_19_2_a0/}
}
TY - JOUR
AU - Michel Granger
AU - Mathias Schulze
TI - Initial Logarithmic Lie Algebras of Hypersurface Singularities
JO - Journal of Lie Theory
PY - 2009
SP - 209
EP - 221
VL - 19
IS - 2
UR - http://geodesic.mathdoc.fr/item/JOLT_2009_19_2_a0/
ID - JOLT_2009_19_2_a0
ER -
%0 Journal Article
%A Michel Granger
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%T Initial Logarithmic Lie Algebras of Hypersurface Singularities
%J Journal of Lie Theory
%D 2009
%P 209-221
%V 19
%N 2
%U http://geodesic.mathdoc.fr/item/JOLT_2009_19_2_a0/
%F JOLT_2009_19_2_a0
