Complex Structures on nilpotent Lie algebras Lucia Garcia-Vergnolle ∗ Derpartamento de Geometr´ıa y Topolog´ıa, Universiversidad Complutense de Madrid, 3, Plaza de Ciencias, 28040 Madrid, Spain.
Abstract The study of complex manifolds has interested many authors in different fields in mathematics and physics. A complex structure J over a real even-dimensional Lie algebra g is an endomorphism of g which satisfies the Nijenhuis condition and J 2 = −Id. The classification of complex structures has only been completely obtained in dimensions 2 and 4. In dimension 6, Salamon found all the nilpotent Lie algebras provided with a complex structure. The first general result is the nonexistence of complex structures over nilpotent Lie algebras maximal nilindex, also called filiform. Generalized complex geometry introduced by Hitchin and developed by Gualtieri and Cavalcanti contains complex and symplectic geometry as extremal special cases. We will show that generalized complex structures represent an important tool in the study of complex structures over nilmanifolds.
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