Groupoids derived from the simple elliptic singularities

TL;DR

This paper generalizes Eastwood's method to recover K. Saito's $j$-functions from moduli algebras of simple elliptic singularities, analyzing their automorphism structures and establishing Torelli-type theorems for certain families.

Contribution

It extends the explicit recovery of $j$-functions from moduli algebras to all simple elliptic singularities and clarifies automorphism structures using Yau algebras.

Findings

Recovered $j$-functions from $k$-th moduli algebras.

Established Torelli-type theorem for $ ilde{E}_7$ when $k=1$.

Identified limitations of Torelli theorem in the $ ilde{E}_6$ case.

Abstract

K. Saito's classification of simple elliptic singularities includes three families of weighted homogeneous singularities: $ \tilde{E}_{6}, \tilde{E}_7$, and $ \tilde{E}_8 $. For each family, the isomorphism classes can be distinguished by K. Saito's $j$-functions. By applying the Mather-Yau theorem, which states that the isomorphism class of an isolated hypersurface singularity is completely determined by its $k$-th moduli algebra, M. Eastwood demonstrated explicitly that one can directly recover K. Saito's $j$-functions from the zeroth moduli algebras. This research aims to generalize M. Eastwood's result through meticulous computation of the groupoids associated with simple elliptic singularities. We not only directly retrieve K. Saito's $j$-functions from the $k$-th moduli algebras but also elucidate the automorphism structure within the $k$-th moduli algebras. We derive the automorphisms using the methodology of the $k$-th Yau algebra and establish a Torelli-type theorem for the $\tilde{E}_7 $-family when $k=1$. In contrast, we find that the Torelli-type theorem is inapplicable for the first Yau algebra in the $ \tilde{E}_6 $-family. By considering the first Yau algebra as a module rather than solely as a Lie algebra, we can impose constraints on the coefficients of the transformation matrices, which facilitates a straightforward identification of all isomorphisms. Our new approach also provides a simple verification of the result by Chen, Seeley, and Yau concerning the zeroth moduli algebras.