New Invariants of Singularities in Terms of k-TH Moduli and Derivation Lie Algebras

The k-th Yau algebra Lk(V),k≥0 defined to be the Lie algebra of derivations of the k-th moduli algebras Ak(V)=On/(f,mkJ(f)),k≥0 and where m is the maximal ideal of On. The k-th Milnor number and k-th Tjurina number are defined as follows: μk:=dimOn/(mkJ(f)),τk:=dimOn/(f,mkJ(f)). The dimension of Lk(V) denoted as λk(V). In this paper, we propose two questions for μk,τk and λk (see Conjecture 1.10 and Question 1.8) and answer these two questions for simple singularities when k is small. We also verified the sharp upper estimate Conjecture 1.3 and the inequality Conjecture 1.1 for binomial singularities.