Noncommutative Tangent Cones and Calabi Yau Algebras

TL;DR

This paper extends the concept of local quivers and tangent cones to broader classes of algebras, providing new tools to analyze their local structure, with applications to Calabi-Yau algebras and their classification.

Contribution

It introduces a generalized framework for local models and tangent cones for finitely generated algebras, connecting these to Calabi-Yau properties and superpotentials.

Findings

Tangent cones of Calabi-Yau 2 algebras are preprojective

Local models for Calabi-Yau 3 algebras often derive from superpotentials

The framework aids in classifying algebras by local behavior

Abstract

We study the generalization of the idea of a local quiver of a representation of a formally smooth algebra, to broader classes of finitely generated algebras. In this new setting we can construct for every semisimple representation $M$ a local model and a non-commutative tangent cone. The representation schemes of these new algebras model the local structure and the tangent cone of the representation scheme of the original algebra at $M$. In this way one can try to classify algebras according to their local behavior. As an application we will show that the tangent cones of Calabi Yau 2 Algebras are always preprojective algebras. For Calabi Yau 3 Algebras the corresponding statement would be that the local model and the tangent cones derive from superpotentials. Although we do not have a proof in all cases, we will show that this will indeed hold in many cases.