Weak Mirror Symmetry of Complex Symplectic Algebras

A complex symplectic structure on a Lie algebra $\text{lie}h$ is an integrable complex structure $J$ with a closed non-degenerate $(2,0)$-form. It is determined by $J$ and the real part $\mathrm{\Omega }$ of the $(2,0)$-form. Suppose that $\text{lie}h$ is a semi-direct product $\text{lie}g⋉V$, and both $\text{lie}g$ and $V$ are Lagrangian with respect to $\mathrm{\Omega }$ and totally real with respect to $J$. This note shows that $\text{lie}g⋉V$ is its own weak mirror image in the sense that the associated differential Gerstenhaber algebras controlling the extended deformations of $\mathrm{\Omega }$ and $J$ are isomorphic. The geometry of $(\mathrm{\Omega },J)$ on the semi-direct product $\text{lie}g⋉V$ is also shown to be equivalent to that of a torsion-free flat symplectic connection on the Lie algebra $\text{lie}g$. By further exploring a relation between $(J,\mathrm{\Omega })$ 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.