On the $k$th higher Nash blow-up derivation Lie algebras of isolated hypersurface singularities

Many physical questions such as $4d$ $N=2$ superconformal field theories, the Coulomb branch spectrum, and the Seiberg–Witten solutions are related to singularities. In this paper, we introduce some new invariants $\mathcal L^k_n(V)$, $\rho_n^k$, and $d_n^k(V)$ of isolated hypersurface singularities $(V,0)$. We give a new conjecture for the characterization of simple curve singularities using the $k$th higher Nash blow-up derivation Lie algebra $\mathcal L^k_n(V)$. This conjecture is verified for small $n$ and $k$. A inequality conjecture for $\rho_n^k$ and $d_n^k(V)$ is proposed. These two conjectures are verified for binomial singularities.