HyperKähler torsion structures invariant by nilpotent Lie groups

Dotti, Isabel G.; Fino, Anna

doi:10.1088/0264-9381/19/3/309

Cited by 39 publications

(63 citation statements)

References 27 publications

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“…Theorem 3.1 in [9] says that the hypercomplex structure of a HKT structure on any 2-step nilpotent Lie algebra is abelian. Thus the hypercomplex structure {J 1 , J 2 } in the above example is not a hypercomplex structure of a HKT structure.…”

Section: Now By Proposition 22 We Havementioning

confidence: 99%

“…Complex structures on solvable and nilpotent Lie algebras have been extensively studied recently (see , [7], [8], [9], Barberis-Dotti [2] and Salamon [13] and references therein). In particular, abelian structures where considered in [8], where it is proved that abelian complex structures occur only on solvable Lie algebras, and in [2] where a characterization of solvable Lie algebras admiting abelian complex structures is given.…”

Section: Introductionmentioning

confidence: 99%

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Lie algebras with complex structures having nilpotent eigenspaces

Santos¹,

Martín

2011

Proyecciones (Antofagasta)

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Let (g, [·, ·]) be a Lie algebra with an integrable complex structure J. The ±i eigenspaces of J are complex subalgebras of g C isomorphic to the algebraWe consider here the case where these subalgebras are nilpotent and prove that the original (g, [·, ·]) Lie algebra must be solvable. We consider also the 6-dimensional case and determine explicitly the possible nilpotent Lie algebras (g, [ * ] J ). Finally we produce several examples illustrating different situations, in particular we show that for each given s there exists g with complex structure J such that (g, [ * ] J ) is s-step nilpotent. Similar examples of hypercomplex structures are also built.

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Section: Now By Proposition 22 We Havementioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Lie algebras with complex structures having nilpotent eigenspaces

Santos¹,

Martín

2011

Proyecciones (Antofagasta)

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“…For a Lie group G with a left-invariant hyper-Hermitian structure ({J α }, g), it was shown in [15] that ({J α }, g) is HKT if and only if…”

Section: Preliminariesmentioning

confidence: 99%

“…where a i , i = 1, 2, 3, b are real numbers and the endomorphisms J For the Lie algebra (3) we can consider the weak HKT structure ({J α }, g) defined by (15) and (16). In this case, we have the following homomorphism D:…”

Section: Weak Hkt Structures On 8-dimensional Tangent Lie Algebrasmentioning

confidence: 99%

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New HKT manifolds arising from quaternionic representations

Barberis

Fino

2009

Math. Z.

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Abstract. We give a procedure for constructing an 8n-dimensional HKT Lie algebra starting from a 4n-dimensional one by using a quaternionic representation of the latter. The strong (respectively, weak, hyper-Kähler, balanced) condition is preserved by our construction. As an application of our results we obtain a new compact HKT manifold with holonomy in SL(n, H) which is not a nilmanifold. We find in addition new compact strong HKT manifolds.We also show that every Kähler Lie algebra equipped with a flat, torsion-free complex connection gives rise to an HKT Lie algebra. We apply this method to two distinguished 4-dimensional Kähler Lie algebras, thereby obtaining two conformally balanced HKT metrics in dimension 8.Both techniques prove to be an effective tool for giving the explicit expression of the corresponding HKT metrics.

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Odd‐dimensional counterparts of abelian complex and hypercomplex structures

Andrada¹,

Dileo²

2022

Mathematische Nachrichten

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We introduce the notion of abelian almost contact structures on an odd‐dimensional real Lie algebra g$\mathfrak {g}$. We investigate correspondences with even‐dimensional Lie algebras endowed with an abelian complex structure, and with Kähler Lie algebras when g$\mathfrak {g}$ carries a compatible inner product. The classification of 5‐dimensional Sasakian Lie algebras with abelian structure is obtained. Later, we introduce abelian almost 3‐contact structures on real Lie algebras of dimension 4n+3$4n+3$, obtaining the classification of these Lie algebras in dimension 7. Finally, we deal with the geometry of a Lie group G endowed with a left invariant abelian almost 3‐contact metric structure. We determine conditions for G to admit a canonical metric connection with skew torsion, which plays the role of the Bismut connection for hyperKähler with torsion (HKT) structures arising from abelian hypercomplex structures. We provide examples and discuss the parallelism of the torsion of the canonical connection.

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HyperKähler torsion structures invariant by nilpotent Lie groups

Cited by 39 publications

References 27 publications

Lie algebras with complex structures having nilpotent eigenspaces

Lie algebras with complex structures having nilpotent eigenspaces

New HKT manifolds arising from quaternionic representations

Odd‐dimensional counterparts of abelian complex and hypercomplex structures

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