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

The k-th Yau algebra $$L^k(V), k\ge 0$$ defined to be the Lie algebra of derivations of the k-th moduli algebras $$A^k(V)= \mathcal {O}_n/(f, m^k J(f)), k\ge 0$$ and where m is the maximal ideal of $$\mathcal {O}_n$$ . The k-th Milnor number and k-th Tjurina number are defined as follows: $$\mu ^k:= \text {dim}\mathcal {O}_n/( {m}^kJ(f)), \;\tau ^k:=\text {dim} \mathcal {O}_n/(f, {m}^kJ(f)).$$ The dimension of $$L^k(V)$$ denoted as $$\lambda ^k(V)$$ . In this paper, we propose two questions for $$\mu ^k, \tau ^k$$ and $$\lambda ^k$$ (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.