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.

22 pages

References in corpus (4)

• Mirror Symmetry is T-Duality
• Frobenius Manifolds and Formality of Lie Algebras of Polyvector Fields
• Mirror Symmetry and Generalized Complex Manifolds
• Double products and hypersymplectic structures on $R^{4n}$

Cited by in corpus (8)

• Symplectic, product and complex structures on 3-Lie algebras
• Symmetric and skew-symmetric complex structures
• Mirrors, Functoriality, and Derived Geometry
• Complex Symplectic Lie Algebras with Large Abelian Subalgebras
• Differential Gerstenhaber Algebras of Generalized Complex Structures
• Abelian Complex Structures and Generalizations
• Pseudo-Kähler and hypersymplectic structures on semidirect products
• Frobenius Structures and Generalized Deformation of Kodaira Manifolds