Classification of the nilradical of k-th Yau algebras arising from singularities

Every Lie algebra is a semi-direct product of semi-simple Lie algebra and a solvable Lie algebra. Brieskorn gave the connection between simple Lie algebras and simple singularities. Simple Lie algebras have been well understood, but not the solvable (nilpotent) Lie algebras. Classification of nilpotent Lie algebras with dimension up to 7 is known, but not for dimension greater than 7. Therefore it is important to establish connections between theory of singularities and theory of nilpotent Lie alge-bras. Let (V, 0) be an isolated hypersurface singularity defined by the holomorphic function f: (Cⁿ, 0) → (C, 0). The k-th Yau algebras L^k (V), k ≥ 0 were introduced by the authors. It was defined to be the Lie algebra of derivations of the k-th moduli algebra A^k (V). These Lie algebras are solvable in general and play an important role in the study of singularities. In this paper, we investigate the new connection between the nilpotent Lie algebras of dimension less than or equal to 7 and the nilradical of k-th Yau algebras.