Weak mirror symmetry of complex symplectic Lie algebras

Richard Cleyton Yat Sun Poon Gabriela P Ovando

Complex Variables and Complex Analysis mathscidoc:1910.43825

Journal of Geometry and Physics, 61, (8), 1553-1563, 2011.8
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.
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  title={Weak mirror symmetry of complex symplectic Lie algebras},
  author={Richard Cleyton, Yat Sun Poon, and Gabriela P Ovando},
  booktitle={Journal of Geometry and Physics},
Richard Cleyton, Yat Sun Poon, and Gabriela P Ovando. Weak mirror symmetry of complex symplectic Lie algebras. 2011. Vol. 61. In Journal of Geometry and Physics. pp.1553-1563. http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20191020221342674832354.
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