Differential Gerstenhaber algebras associated to nilpotent algebras

TL;DR

This paper classifies differential Gerstenhaber algebras for all nilpotent complex structures on six-dimensional nilpotent algebras, revealing self-mirror pseudo-Kähler structures in a weak mirror symmetry sense.

Contribution

It provides a complete description of these algebras and classifies pseudo-Kähler structures with self-mirror properties on six-dimensional nilpotent algebras.

Findings

Complete classification of differential Gerstenhaber algebras for nilpotent complex structures.

Identification of pseudo-Kähler structures with self-mirror properties.

Insights into mirror symmetry in the context of nilpotent algebras.

Abstract

This article provides a complete description of the differential Gerstenhaber algebras of all nilpotent complex structures on any real six-dimensional nilpotent algebra. As an application, we classify all pseudo-K\"ahlerian complex structures on six-dimensional nilpotent algebras such that the differential Gerstenhaber algebra of its complex structure is quasi-isomorphic to that of its symplectic structure. In a weak sense of mirror symmetry, it is a classification of pseudo-K\"ahler structures on six-dimensional nilpotent algebras whose mirror images are themselves.