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.