Four-dimensional Lie algebras with a para-hypercomplex structure

Blažić, Novica; Vukmirovic, Srdjan

doi:10.1216/rmj-2010-40-5-1391

Cited by 8 publications

(12 citation statements)

References 11 publications

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“…• They generalize previous results concerning complex product structures [2,7], complex and symplectic structures related to tangent algebras [1,6,17], complex and paracomplex structures on homogeneous manifolds [11]. • The existence of LSA structures imposes a clear obstruction.…”

Section: Introductionsupporting

confidence: 79%

From almost (para)-complex structures to affine structures on Lie groups

Calvaruso

Ovando

2017

manuscripta math.

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Let G = H ⋉ K denote a semidirect product Lie group with Lie algebra g = h⊕k, where k is an ideal and h is a subalgebra of the same dimension as k. There exist some natural split isomorphisms S with S 2 = ± Id on g: given any linear isomorphism j : h → k, we get the almost complex structure J(x, v) = (−j −1 v, jx) and the almost paracomplex structure E(x, v) = (j −1 v, jx). In this work we show that the integrability of the structures J and E above is equivalent to the existence of a left-invariant torsionfree connection ∇ on G such that ∇J = 0 = ∇E and also to the existence of an affine structure on H. Applications include complex, paracomplex and symplectic geometries.2010 Mathematics Subject Classification. 53C15, 53C55, 53D05, 22E25, 17B56.

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Section: Introductionsupporting

confidence: 79%

From almost (para)-complex structures to affine structures on Lie groups

Calvaruso

Ovando

2017

manuscripta math.

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“…In [6], N. Blažić and S. Vukmirović classified 4-dimensional Lie algebras admitting left invariant para-hypercomplex structures. H. R. Salimi Moghaddam obtained some curvature properties of left invariant Riemannian metrics on such Lie groups [23].…”

Section: Preliminariesmentioning

confidence: 99%

“…Now, we list all five classes of 4-dimensional Lie algebras admitting an invariant para-hypercomplex structure and non-zero parallel vector fields. These classes of Lie algebras were first introduced in [6].…”

Section: Preliminariesmentioning

confidence: 99%

“…Case 4. [6] In the Lie algebra structure of Case 4, there are two real parameters λ and η. This Lie algebra has the following structure:…”

Section: Casementioning

confidence: 99%

“…M. L. Barberis classified the invariant hypercomplex structures on a simply-connected 4-dimensional real Lie group [3,5]. In [6], N. Blažić and S. Vukmirović classified 4-dimensional Lie algebras admitting a para-hypercomplex structure.…”

Section: Introductionmentioning

confidence: 99%

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LEFT INVARIANT $(\alpha,\beta)$-METRICS ON 4-DIMENSIONAL LIE GROUPS

2020

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Let $G$ be a 4-dimensional Lie group with an invariant para-hypercomplex structure and let $F= \beta+ a\alpha+\beta^2/{\alpha}$ be a left invariant $(\alpha,\beta)$-metric, where $\alpha$ is a Riemannian metric and $\beta$ is a 1-form on $G$, and $a$ is a real number. We prove that the flag curvature of $F$ with parallel 1-form $\beta$ is non-positive, except in Case 2, in which $F$ admits both negative and positive flag curvature. Then, we determine all geodesic vectors of $(G,F)$.

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Generalized geometric structures on complex and symplectic manifolds

Salvai¹

2014

Annali di Matematica

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On a smooth manifold M , generalized complex (generalized paracomplex) structures provide a notion of interpolation between complex (paracomplex) and symplectic structures on M .Given a complex manifold (M, j), we define six families of distinguished generalized complex or paracomplex structures on M . Each one of them interpolates between two geometric structures on M compatible with j, for instance, between totally real foliations and Kähler structures, or between hypercomplex and C-symplectic structures. These structures on M are sections of fiber bundles over M with typical fiber G/H for some Lie groups G and H. We determine G and H in each case.We proceed similarly for symplectic manifolds. We define six families of generalized structures on (M, ω), each of them interpolating between two structures compatible with ω, for instance, between a C-symplectic and a para-Kähler structure (aka biLagrangian foliation).2000 MSC: 22F30, 22F50, 53B30, 53B35, 53C15, 53C56, 53D05

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Four-dimensional Lie algebras with a para-hypercomplex structure

Cited by 8 publications

References 11 publications

From almost (para)-complex structures to affine structures on Lie groups

From almost (para)-complex structures to affine structures on Lie groups

LEFT INVARIANT $(\alpha,\beta)$-METRICS ON 4-DIMENSIONAL LIE GROUPS

Generalized geometric structures on complex and symplectic manifolds

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