Variation of complex structures and variation of derivation Lie algebras III: Yau algebras arising from singularities

Let On=Cx1, …, xn be the ring of convergent power series. An isolated hypersurface singularity (V, 0) is defined by f∈On. Its moduli algebra A (V) ≔On/ (f, ∂f/∂x1, …, ∂f/∂xn) is a finite-dimensional algebra that determines the complex analytic structure of (V, 0). The Yau algebra L (V) ≔DerC (A (V), A (V) ) is a solvable Lie algebra associated with the singularity. As the singularity deforms, the structure of L (V) also varies, leading to a Torelli-type problem: can the Yau algebra distinguish the different analytic structures within a deformation family? This paper investigates a crucial subalgebra, the liftable Yau subalgebra L̃ (V), which consists of derivations that can be lifted along the deformation. We focus on a one-parameter (μ, τ) -invariant deformation family Vt generated by f0 = x4 + y4 + z4 (whose projectivization in CP2 defines a smooth plane quartic curve of genus 3). We compute the 37-dimensional liftable Yau subalgebra family L̃ (Vt) for this deformation. A key technical breakthrough of this paper is the development of new computational tools and algorithms to calculate a series of Lie algebra isomorphism invariants (namely, cross-ratios) for its high-dimensional (36-dim) nilradical Nt=L̃ (Vt), L̃ (Vt). This extends the invariant-based method, previously used for lower-dimensional algebras, to a significantly more complex case. By applying this new computational framework, we prove that L̃ (Vt) ≅L̃ (Vs) implies t2 = s2 or t2 = −324 (s2 − 36) / (1015s2 + 324). Comparing these two results, we demonstrate that although the liftable Yau subalgebra L̃ (Vt) is itself a well-behaved family of solvable Lie algebras constructed geometrically, it fails to be a complete invariant for the singularity isomorphism. Indeed, it fails to be even a basic invariant, as there exist cases where Vt ≅ Vs (isomorphic singularities) but L̃ (Vt) ≇L̃ (Vs) (non-isomorphic Lie algebras). This contrasts sharply with the known cases of Ẽ7 and Ẽ8 singularities. An important part of this work is the development of computational algorithms that make the analysis of these high-dimensional Lie algebras feasible on a computer.