Journal of Lie Theory 16 (2006), No. 4, 621--649
Copyright Heldermann Verlag 2006
Lie Algebras of Simple Hypersurface Singularities
A. G. Elashvili
Razmadze Mathematical Institute, M. Alexidze St. 1, Tbilisi 0193, Georgia
alela@rmi.acnet.ge
Giorgi Khimshiashvili
Razmadze Mathematical Institute, M. Alexidze St. 1, Tbilisi 0193, Georgia
khimsh@rmi.acnet.ge
We investigate structural properties and numerical invariants of the finite-dimensional solvable Lie algebras naturally associated with simple hypersurface singularities. In particular, we establish that the analytic isomorphism class of a simple hypersurface singularity is determined by the Lie algebra of derivations of its moduli algebra if the dimension of the latter algebra is not less than 6. We also describe natural gradings on the Lie algebras of simple singularities and show that all roots of their Poincaré polynomials lie on the unit circle. Moreover, the indices of those Lie algebras are calculated and existence of maximal commutative polarizations is established.
Keywords: Isolated hypersurface singularity, moduli algebra, derivation, vector field, index of Lie algebra, maximal commutative polarization.
MSC: 32S25, 17B30, 17B40
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