Complex structures on nilpotent Lie algebras

Salamon, Simon

doi:10.1016/s0022-4049(00)00033-5

Cited by 254 publications

(388 citation statements)

References 19 publications

(56 reference statements)

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“…The main result that will be used in the classification is Theorem 4.1, which states that any 6-dimensional nilpotent Lie algebra admitting a complex product structure can be decomposed as a direct sum (as vector spaces) of a 4-dimensional subspace and a 2-dimensional central ideal, both of them invariant by the complex and the product structures. As a first corollary of this theorem we find two 6-dimensional nilpotent Lie algebras admitting complex structures (according to [26]) that do not admit any complex product structure (see Corollary 4.6). It will turn out later that the 6-dimensional nilpotent Lie algebras which admit complex structures but admit no complex product structures are exactly 3.…”

Section: Introductionmentioning

confidence: 89%

“…The class of 6-dimensional nilpotent Lie algebras has been extensively studied, and there is only a finite number of such Lie algebras, up to isomorphism (see for instance [24]). Furthermore, the family of these Lie algebras admitting a complex structure has also been specified, and their list appears in [26].…”

Section: Introductionmentioning

confidence: 99%

“…Our goal in this work is to obtain the full classification of the 6-dimensional nilpotent Lie algebras admitting a complex product structure, and this classification will be done independently of the results given in [26]. Our approach will make use of properties of simply transitive actions of 3-dimensional nilpotent Lie groups on 3-dimensional Euclidean space or, equivalently, complete left symmetric algebra structures on 3-dimensional nilpotent Lie algebras.…”

Section: Introductionmentioning

confidence: 99%

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Complex product structures on 6-dimensional nilpotent Lie algebras

Andrada

2008

Forum Mathematicum

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Abstract. We study complex product structures on nilpotent Lie algebras, establishing some of their main properties, and then we restrict ourselves to 6 dimensions, obtaining the classification of 6-dimensional nilpotent Lie algebras admitting such structures. We prove that any complex structure which forms part of a complex product structure on a 6-dimensional nilpotent Lie algebra must be nilpotent in the sense of Cordero-Fernández-Gray-Ugarte. A study is made of the torsion-free connection associated to the complex product structure and we consider also the associated hypercomplex structures on the 12-dimensional nilpotent Lie algebras obtained by complexification.

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Section: Introductionmentioning

confidence: 89%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Complex product structures on 6-dimensional nilpotent Lie algebras

Andrada

2008

Forum Mathematicum

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“…This can also be seen from the fact that X 6 is just a warped version of one of the complex nilmanifolds classified in Ref. [76]. On the other hand, the complex structure (6.54) is not the one selected by the supersymmetry conditions, and is incompatible with the physical metric (6.36) except at certain points in moduli space.…”

Section: Nonintegrability Of the Almost Complex Structurementioning

confidence: 96%

Superstring orientifolds with torsion: O5 orientifolds of torus fibrations and their massless spectra

Schulz

2004

Fortschritte der Physik

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Using a "Superstrings with Torsion" type description, we study a class of IIB orientifolds in which spacefilling O5 planes and D5 branes wrap the T 2 fiber in a warped modification of the product of 4D Minkowski space and a T 2 fibration. For the case that the base is T 4 , we provide examples that preserve 4D N = 1, 2, and 3 supersymmetry, both with internal RR flux, and with a combination of internal RR and NS flux. In these examples, the internal geometries admit integrable complex structure; however, the almost complex structure selected by the supersymmetry conditions is nonintegrable in the case that there is NS flux. We indicate explicitly the massless spectrum of gauge fields and moduli in each example. In a previous investigation, this class of orientifolds was studied using T-duality. Here, we extend the previous analysis, first by providing an intrinsic description that does not rely on duality, and then by elaborating on details of the T-duality map, which we use to check our results.

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“…We use the classification of nilpotent Lie algebras given by Salamon [14]. It is based on the Morozov classification of 6 -dimensional nilpotent Lie algebras [12].…”

Section: Theorem 41 There Exist At Least Five 6 -Dimensional Flexibmentioning

confidence: 99%

On symplectically harmonic forms on six-dimensional nilmanifolds

Ibáñez

Rudyak

Tralle

et al. 2001

Commentarii Mathematici Helvetici

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Abstract. In the present paper we study the variation of the dimensions h k of spaces of symplectically harmonic cohomology classes (in the sense of Brylinski) on closed symplectic manifolds. We give a description of such variation for all 6-dimensional nilmanifolds equipped with symplectic forms. In particular, it turns out that certain 6-dimensional nilmanifolds possess families of homogeneous symplectic forms ω t for which numbers h k (M, ωt) vary with respect to t . This gives an affirmative answer to a question raised by Boris Khesin and Dusa McDuff. Our result is in contrast with the case of 4-dimensional nilmanifolds which do not admit such variations by a remark of Dong Yan.

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Complex structures on nilpotent Lie algebras

Abstract: We classify real 6-dimensional nilpotent Lie algebras for which the corresponding Lie group has a left-invariant complex structure, and estimate the dimensions of moduli spaces of such structures.

Cited by 254 publications

References 19 publications

Complex product structures on 6-dimensional nilpotent Lie algebras

Complex product structures on 6-dimensional nilpotent Lie algebras

Superstring orientifolds with torsion: O5 orientifolds of torus fibrations and their massless spectra

On symplectically harmonic forms on six-dimensional nilmanifolds

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