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arXiv · 1206.3464

Pseudo-K\"ahler Lie algebras with abelian complex structures

Ignacio Bajo·Esperanza Sanmartín
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Abstract

We study Lie algebras endowed with an abelian complex structure which admit a symplectic form compatible with the complex structure. We prove that each of those Lie algebras is completely determined by a pair (U,H) where U is a complex commutative associative algebra and H is a sesquilinear hermitian form on U which verifies certain compatibility conditions with respect to the associative product on U. The Riemannian and Ricci curvatures of the associated pseudo-K\"ahler metric are studied and a characterization of those Lie algebras which are Einstein but not Ricci flat is given. It is seen that all pseudo-K\"ahler Lie algebras can be inductively described by a certain method of double extensions applied to the associated complex asssociative commutative algebras.

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Ignacio Bajo, Esperanza Sanmartín. 2012-06-15. Pseudo-K\"ahler Lie algebras with abelian complex structures. https://doi.org/10.1088/1751-8113/45/46/465205

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