ON THE HIGHER NASH BLOW-UP DERIVATION LIE ALGEBRAS OF ISOLATED HYPERSURFACE SINGULARITIES

Introduction: Isolated hypersurface singularities arise naturally in singularity theory and in mathematical physics through deformation theory and related invariants. Since the classical Yau derivation Lie algebra does not always separate simple (ADE) singularities, this paper investigates higher Nash blow-up local algebras and the Lie algebras of their derivations as finer analytic invariants.

Methods: Let $(V,0)\subset (\mathbb{C}^s,0)$ be an isolated hypersurface singularity defined by a polynomial $F$. Using the higher-order Jacobian matrix $\mathrm{Jac}_n(F)$ and the ideal $J_n$ generated by its maximal minors, we define the higher Nash blow-up local algebra
\[
M_n(V):=\mathcal{O}_s/\langle F, J_n\rangle.
\]
For each $k$, we study the derivation Lie algebra $L_k(V):=\mathrm{Der}(M_k(V))$. To obtain computable Lie-theoretic data, we construct a generalized Cartan matrix $C_k(V)$ from the nilradical of $L_k(V)$ via maximal tori and the associated root space decomposition, following a Mostow-type structural framework.

Results: Concentrating on the case $k=2$, we explicitly compute $C_2(V)$ for all ADE hypersurface singularities, including the infinite families $A_k$ and $D_k$ and the exceptional types $E_6$, $E_7$, and $E_8$. We prove that $C_2(V)$ distinguishes analytic isomorphism classes within the ADE category: for ADE singularities $X$ and $Y$, one has $C_2(X)=C_2(Y)$ if and only if $X \cong Y$ analytically.

Conclusions: These results confirm, for $k=2$, a Lie-theoretic characterization of ADE singularities through derivations of higher Nash blow-up algebras. The invariant $C_2(V)$ refines the classical derivation-based approach and suggests a systematic route to separating broader classes of singularities.