GENERALIZED CARTAN MATRICES ARISING FROM NEW DERIVATION LIE ALGEBRAS OF ISOLATED HYPERSURFACE SINGULARITIES

Let V be a hypersurface with an isolated singularity at the origin defined by the holomorphic function f: [Formula Presented]The Yau algebra L(V) is defined to be the Lie algebra of derivations of the moduli algebra A(V) := [Formula Presented]i.e., L(V) = Der(A(V), A(V)). It is known that L(V) is finite dimensional and its dimension λ(V) is called the Yau number. We introduced a new Lie algebra L*(V) which was defined to be the Lie algebra of derivations of [Formula Presented]i.e., L*(V) = Der(A*(V), A*(V)). L*(V) is finite dimensional and [Formula Presented]is the dimension of L*(V). In this paper we compute the generalized Cartan matrix C(V) and other various invariants arising from the new Lie algebra L*(V) for simple elliptic singularities and simple hypersurface singularities. We use the generalized Cartan matrix to characterize the ADE singularities.