Weak mirror symmetry of Lie algebras

RICHARD CLEYTON, JORGE LAURET, YAT-SUN POON

JOURNAL OF SYMPLECTIC GEOMETRY

INT PRESS BOSTON, INC

Año: 2010 vol. 8 p. 37 - 55

1527-5256

Weak mirror symmetry relates a manifold with complex structure toanother manifold equipped with a symplectic structure through a quasiisomorphismof associated differential Gerstenhaber algebras. The twomanifolds are then mirror partners. In this paper we consider the analogousproblem on Lie algebras. In particular we show that the semi-directproduct of a Lie algebra equipped with a torsion-free flat connectionwith itself is a mirror partner of a semi-direct product of the sameLie algebra with its dual space. For nilpotent algebras this analysis onLie algebras can be applied to the compact quotients of the underlyingnilpotent group. We classify mirror pairs among 6-dimensionalnilpotent Lie algebras that have the semi-direct product structure aswell as mirror pairs admitting the more involved special Lagrangianstructure, namely compatible complex and symplectic structures onthe same space.