Weak mirror symmetry of complex symplectic Lie algebras

A complex symplectic structure on a Lie algebra hh is an integrable complex structure JJ with a closed non-degenerate (2,0)(2,0)-form. It is determined by JJ and the real part ΩΩ of the (2,0)(2,0)-form. Suppose that hh is a semi-direct product g⋉Vg⋉V, and both gg and VV are Lagrangian with respect to ΩΩ and totally real with respect to JJ. This note shows that g⋉Vg⋉V is its own weak mirror image in the sense that the associated differential Gerstenhaber algebras controlling the extended deformations of ΩΩ and JJ are isomorphic.The geometry of (Ω,J)(Ω,J) on the semi-direct product g⋉Vg⋉V is also shown to be equivalent to that of a torsion-free flat symplectic connection on the Lie algebra gg. By further exploring a relation between (J,Ω)(J,Ω) with hypersymplectic Lie algebras, we find an inductive process to build families of complex symplectic algebras of dimension 8n8n from the data of the 4n4n-dimensional ones.