Survey on derivation Lie algebras of isolated singularities

Abstract

. Let V be a hypersurface with an isolated singularity at the origin defined by the holomorphic function f : ( C n , 0) → ( C , 0). Let L ( V ) be the Lie algebra of derivations of the moduli algebra A ( V ) := O n / ( f,∂f/∂x 1 , ··· ,∂f/∂x n ), i.e., L ( V ) = Der( A ( V ) ,A ( V )). The Lie algebra L ( V ) is finite dimensional solvable algebra and plays an important role in singularity theory. According to Elashvili and Khimshiashvili ([15], [23]) L ( V ) is called Yau algebra and the dimension of L ( V ) is called Yau number. The studies of finite dimensional Lie algebras L ( V ) that arising from isolated singularities was started by Yau [44] and has been systematically studied by Yau, Zuo and their coauthors. Most studies of Lie algebras L ( V ) were oriented to classify the isolated singularities. This work surveys the researches on Yau algebras L ( V ) of isolated singularities.