Weak Mirror Symmetry of 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.Comment: 22 page