Weak Mirror Symmetry of Complex Symplectic Algebras

TL;DR

This paper demonstrates a form of weak mirror symmetry in complex symplectic Lie algebras, showing their self-duality and linking their geometry to torsion-free flat symplectic connections, with a method to construct higher-dimensional examples.

Contribution

It establishes that certain semi-direct product complex symplectic Lie algebras are their own weak mirror images and connects their structure to flat symplectic connections, providing a way to generate higher-dimensional examples.

Findings

Semi-direct product Lie algebras are their own weak mirror images.

Geometry of $( abla, J)$ relates to torsion-free flat symplectic connections.

An inductive process constructs higher-dimensional complex symplectic algebras.

Abstract

A complex symplectic structure on a Lie algebra $\lie h$ is an integrable complex structure $J$ with a closed non-degenerate $(2,0)$-form. It is determined by $J$ and the real part $\Omega$ of the $(2,0)$-form. Suppose that $\lie h$ is a semi-direct product $\lie g\ltimes V$, and both $\lie g$ and $V$ are Lagrangian with respect to $\Omega$ and totally real with respect to $J$. This note shows that $\lie g\ltimes V$ is its own weak mirror image in the sense that the associated differential Gerstenhaber algebras controlling the extended deformations of $\Omega$ and $J$ are isomorphic. The geometry of $(\Omega, J)$ on the semi-direct product $\lie g\ltimes V$ is also shown to be equivalent to that of a torsion-free flat symplectic connection on the Lie algebra $\lie g$. By further exploring a relation between $(J, \Omega)$ with hypersymplectic algebras, we find an inductive process to build families of complex symplectic algebras of dimension $8n$ from the data of the $4n$-dimensional ones.