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Article Dans Une Revue Journal of Lie Theory Année : 2009

Initial logarithmic Lie algebras of hypersurface singularities

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Résumé

We introduce a Lie algebra of initial terms of logarithmic vector fields along a hypersurface singularity. Extending the formal structure theorem in [GS06, Thm. 5.4], 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 M"uller [M"ul86, HM89] 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.

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Dates et versions

hal-03040198 , version 1 (04-12-2020)

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  • HAL Id : hal-03040198 , version 1
  • OKINA : ua134

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Jean-Michel Granger, Mathias Schulze. Initial logarithmic Lie algebras of hypersurface singularities. Journal of Lie Theory, 2009, 19 (2), pp.209 - 221. ⟨hal-03040198⟩
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