Hypercomplex structures on a class of solvable lie groups

Barberis, M. L.; Miatello, Isabel Dotti

doi:10.1093/qjmath/47.188.389

Cited by 19 publications

(21 citation statements)

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“…However, the realm of hypercomplex manifolds is much broader than the one of hyperkähler manifolds. To cite but a few hypercomplex non-hyperkähler manifolds, note that some nilmanifolds, that is quotients of a nilpotent Lie group by a cocompact lattice, admit hypercomplex structures [5]. Furthermore, Dominic Joyce constructed many left-invariant hypercomplex structures on Lie groups [16] and similar ones have been analysed by physicists interested in string theory [30] in the context of N = 4 supersymmetry.…”

Section: 1mentioning

confidence: 99%

Cohomologies on Hypercomplex Manifolds

Lejmi

Weber

2017

Complex and Symplectic Geometry

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Section: 1mentioning

confidence: 99%

Cohomologies on Hypercomplex Manifolds

Lejmi

Weber

2017

Complex and Symplectic Geometry

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“…If dim n = 1, according to [3,Theorem 4.1], n is isomorphic to n 1 or n 2 . If dim n = 2, since z is J-stable, then n = z and we have that dim z = 2.…”

Section: Dimension 6: the Nilpotent Casementioning

confidence: 99%

Classification of abelian complex structures on 6-dimensional Lie algebras

Andrada

Barberis

Dotti

2011

Journal of the London Mathematical Society

128

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We classify the 6-dimensional Lie algebras that can be endowed with an abelian complex structure and parametrize, on each of these algebras, the space of such structures up to holomorphic isomorphism. IntroductionLet g be a Lie algebra, J be an endomorphism of g such that J 2 = −I, and let g 1,0 be the i-eigenspace of J in g C := g ⊗ R C. When g 1,0 is a complex subalgebra, we say that J is integrable; when g 1,0 is abelian, we say that J is abelian; and when g 1,0 is a complex ideal, we say that J is bi-invariant. We note that a complex structure on a Lie algebra cannot be both abelian and bi-invariant, unless the Lie bracket is trivial. If G is a connected Lie group with Lie algebra g, by left translating J one obtains a complex manifold (G, J) such that left multiplication is holomorphic and, in the bi-invariant case, also right multiplication is holomorphic, which implies that (G, J) is a complex Lie group.Our concern here will be the case when J is abelian. In this case the Lie algebra has abelian commutator, thus it is 2-step solvable (see [17]). However, its nilradical need not be abelian (see Remark 8). Abelian complex structures have interesting applications in hyper-Kähler with torsion geometry (see [6]). It has been shown in [9] that the Dolbeault cohomology of a nilmanifold with an abelian complex structure can be computed algebraically. Also, deformations of abelian complex structures on nilmanifolds have been studied in [10].Of importance, when studying complex structures on a Lie algebra g, is the ideal g J := g + Jg constructed from algebraic and complex data. We say that the complex structure J is proper when g J is properly contained in g. Any complex structure on a nilpotent Lie algebra is proper [19]. The 6-dimensional nilpotent Lie algebras carrying complex structures were classified in [19], and those carrying abelian complex structures were classified in [12].There is only one 2-dimensional non-abelian Lie algebra, the Lie algebra of the affine motion group of R, denoted by aff(R). It carries a unique complex structure, up to equivalence, which turns out to be abelian. The 4-dimensional Lie algebras admitting abelian complex structures were classified in [20]. Each of these Lie algebras, with the exception of aff(C), the realification of the Lie algebra of the affine motion group of C, has a unique abelian complex structure up to equivalence. On aff(C) there is a 2-sphere of abelian complex structures, but only two equivalence classes distinguished by J being proper or not. Furthermore, aff(C) is equipped with a natural bi-invariant complex structure. In dimension 6 it turns out that, as a consequence of our results, some of the Lie algebras equipped with abelian complex structures are of the form aff(A), where A is a 3-dimensional commutative associative algebra. Mathematics Subject Classification 17B30 (primary), 53C15 (secondary).The authors were partially supported by CONICET, ANPCyT and SECyT-UNC (Argentina). n 2 . Furthermore, for 0 r, s n 2 , if J r is equivalent to J s , by comparin...

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“…The 4-dimensional compact manifold M = (Γ\N ) × S 1 = (Γ × Z)\(N × R) is known as the Kodaira-Thurston manifold. The Lie group N × R admits a left invariant abelian complex structure (see for instance [5,3]), and therefore M inherits a complex structure J admitting an abelian connection. On the other hand, M is not complex parallelizable.…”

Section: We Begin By Recalling Known Facts On Invariant Complex Strucmentioning

confidence: 99%

“…The connections ∇ 1 and ∇ 2 appearing in the proof of Proposition 2.2 are known, respectively, as the first and second canonical connection associated to the Hermitian manifold (M, J, g). The connection ∇ 2 is also known as the Chern connection, and it is the unique connection on (M, J, g) satisfying (5). In the almost Hermitian case, the Chern connection is the unique complex metric connection whose torsion is of type (2, 0) + (0, 2), equivalently, the (1, 1)-component of the torsion vanishes.…”

Section: Introductionmentioning

confidence: 99%