Weak mirror symmetry of Lie algebras

Richard Cleyton Jorge Lauret Yat Sun Poon

Complex Variables and Complex Analysis mathscidoc:1910.43813

Journal of Symplectic Geometry, 8, (1), 37-55, 2010
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.
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@inproceedings{richard2010weak,
  title={Weak mirror symmetry of Lie algebras},
  author={Richard Cleyton, Jorge Lauret, and Yat Sun Poon},
  url={http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20191020220955367304342},
  booktitle={Journal of Symplectic Geometry},
  volume={8},
  number={1},
  pages={37-55},
  year={2010},
}
Richard Cleyton, Jorge Lauret, and Yat Sun Poon. Weak mirror symmetry of Lie algebras. 2010. Vol. 8. In Journal of Symplectic Geometry. pp.37-55. http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20191020220955367304342.
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