A complex symplectic structure on a Lie algebra h is an integrable complex structure J with a closed non-degenerate (2, 0)-form. It is determined by J and the real part of the (2, 0)-form. Suppose that h is a semi-direct product g V, and both g and V are Lagrangian with respect to and totally real with respect to J. This note shows that g V is its own weak mirror image in the sense that the associated differential Gerstenhaber algebras controlling the extended deformations of and J are isomorphic. The geometry of (, J) on the semi-direct product g V is also shown to be equivalent to that of a torsion-free flat symplectic connection on the Lie algebra g. By further exploring a relation between (J, ) with hypersymplectic Lie algebras, we find an inductive process to build families of complex symplectic algebras of dimension 8 n from the data of the 4 n-dimensional ones.
It is shown that an HKT space with closed parallel potential 1-form has D (2, 1; 1) symmetry. Every locally conformally hyper-Khler manifold generates this type of geometry. The HKT spaces with closed parallel potential 1-form arising in this way are characterized by their symmetries and an inhomogeneous cubic condition on their torsion.
This article provides a complete description of the differential Gerstenhaber algebras of all nilpotent complex structures on any real six-dimensional nilpotent algebra. As an application, we classify all pseudo-Khlerian complex structures on six-dimensional nilpotent algebras whose differential Gerstenhaber algebra is quasi-isomorphic to that of the symplectic structure. In a weak sense of mirror symmetry, this gives a classication of pseudo-Khler structures on six-dimensional nilpotent algebras whose mirror images are themselves.
The geometry arising from Michelson & Strominger's study of =4<i>B</i> supersymmetric quantum mechanics with superconformal <i>D</i>(2, 1; )-symmetry is a hyperKhler manifold with torsion (HKT) together with a special homothety. It is shown that different parameters are related via changes in potentials for the HKT target spaces. For 0, 1, we describe how each such HKT manifold <i>M</i> <sup> <i>4m</i> </sup> is derived from a space <i>N</i> <sup> <i>4m4</i> </sup> which is quaternionic Khler with torsion and carries an Abelian instanton.
Weak mirror symmetry relates a manifold with complex structure to another manifold equipped with a symplectic structure through a quasiisomorphism of associated differential Gerstenhaber algebras. The two manifolds are then mirror partners. In this paper we consider the analogous problem on Lie algebras. In particular we show that the semi-direct product of a Lie algebra equipped with a torsion-free flat connection with itself is a mirror partner of a semi-direct product of the same Lie algebra with its dual space. For nilpotent algebras this analysis on Lie algebras can be applied to the compact quotients of the underlying nilpotent group. We classify mirror pairs among 6-dimensional nilpotent Lie algebras that have the semi-direct product structure as well as mirror pairs admitting the more involved special Lagrangian structure, namely compatible complex and symplectic structures on the same space.