New \(k\)-th Yau algebras of isolated hypersurface singularities and weak Torelli-type theorem
Let \(V\) be a hypersurface with an isolated singularity at the origin and \(L(V)\) be the Yau algebra which is defined to be the Lie algebra of derivations of the moduli algebra. In the paper under review, the authors generalize the original Yau algebra \(L(V)\) to the \(k\)-th Yau algebra \(L^k(V)\). They obtain the weak Torelli-type theorems of simple elliptic singularities using Lie algebras \(L^1(V)\) and \(L^2(V)\). More precisely, they prove that \(L^2(\tilde{E}_6)\), \(L^1(\tilde{E}_7)\), \(L^2(\tilde{E}_7)\), \(L^1(\tilde{E}_8)\) and \(L^2(\tilde{E}_8)\) are non-trivial one-parameter families. Thus the weak Torelli-type theorems hold for simple elliptic singularities \(\tilde{E}_6\), \(\tilde{E}_7\) and \(\tilde{E}_8\). They also show that if \(X\) and \(Y\) are two simple hypersurface singularities, then \(L^1(X)=L^1(Y)\) as Lie algebras, if and only if \(X\) and \(Y\) are analytically isomorphic.
scientific article; zbMATH DE number 7595084