Weak mirror symmetry of Lie algebras

TL;DR

This paper explores weak mirror symmetry in Lie algebras, showing that certain algebraic structures induce isomorphic deformation theories, with explicit classifications and examples in low dimensions.

Contribution

It demonstrates that flat torsion-free connections on Lie algebras lead to isomorphic deformation complexes, providing new insights into weak mirror symmetry and classifying related structures.

Findings

Isomorphism of differential Gerstenhaber algebras for complex and symplectic structures

Classification of semi-direct products for nilpotent algebras in dimensions 4 and 6

Construction of a one-parameter family of pseudo-Kähler structures

Abstract

The existence of a flat torsion-free connection, or left symmetric algebra structure on a Lie algebra g gives rise to a canonically defined complex structure on g+g and a symplectic structure on g+g^*. We verify that the associated differential Gerstenhaber algebras controlling the deformation theories of the complex and symplectic form are isomorphic. This provides a class of examples of "weak mirror symmetry" as suggested by Merkulov. For nilpotent algebras in dimension 4 and 6 the isomorphism classes of the semi-direct products g+g and g+g^* are listed. A one-parameter family of inequivalent pseudo-K\"ahler structures is given.