Weak mirror symmetry of Lie algebras

Details Weak mirror symmetry of Lie algebras Weak mirror symmetry of Lie algebras

2008-04-30

arxiv.org/abs/0804.4787v1

Mathematics

Algebraic Geometry

36 pages, 3 tables

Scientific paper

The existence of a flat torsion-free connection, or left symmetric algebra structure on a Lie algebra g gives rise to a canonically defined complex structure on g+g and a symplectic structure on g+g^*. We verify that the associated differential Gerstenhaber algebras controlling the deformation theories of the complex and symplectic form are isomorphic. This provides a class of examples of "weak mirror symmetry" as suggested by Merkulov. For nilpotent algebras in dimension 4 and 6 the isomorphism classes of the semi-direct products g+g and g+g^* are listed. A one-parameter family of inequivalent pseudo-K\"ahler structures is given.

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