Journal of Lie Theory

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.