scholarly journals Deformations of Dolbeault cohomology classes for Lie algebra with complex structures

2021 ◽  
Author(s):  
Wei Xia
Keyword(s):  
Lie Algebra ◽  
Complex Structures ◽  
Dolbeault Cohomology

scholarly journals About Zero Torsion Linear Maps on Lie Algebras

ISRN Algebra ◽  
2011 ◽  
Vol 2011 ◽  
pp. 1-16
Author(s):  
Louis Magnin
Keyword(s):  
Lie Algebra ◽  
Lie Algebras ◽  
Equivalence Classes ◽  
Explicit Description ◽  
Complex Structures ◽  
Linear Map ◽  
3 Dimensional ◽  
Linear Maps

We prove that any zero torsion linear map on a nonsolvable real Lie algebra is an extension of some CR-structure. We then study the cases of (2, ) and the 3-dimensional Heisenberg Lie algebra . In both cases, we compute up to equivalence all zero torsion linear maps on , and deduce an explicit description of the equivalence classes of integrable complex structures on .

scholarly journals Complex structures on the complexification of a real Lie algebra

2018 ◽  
Vol 5 (1) ◽  
pp. 150-157
Author(s):  
Takumi Yamada
Keyword(s):  
Lie Algebra ◽  
Complex Structure ◽  
Complex Structures ◽  
Direct Sum ◽  
Direct Sum Decomposition

AbstractLet g = a+b be a Lie algebra with a direct sum decomposition such that a and b are Lie subalgebras. Then, we can construct an integrable complex structure J̃ on h = ℝ(gℂ) from the decomposition, where ℝ(gℂ) is a real Lie algebra obtained from gℂby the scalar restriction. Conversely, let J̃ be an integrable complex structure on h = ℝ(gℂ). Then, we have a direct sum decomposition g = a + b such that a and b are Lie subalgebras. We also investigate relations between the decomposition g = a + b and dim Hs.t∂̄J̃ (hℂ).

scholarly journals VB-Courant Algebroids, E-Courant Algebroids and Generalized Geometry

2018 ◽  
Vol 61 (3) ◽  
pp. 588-607 ◽  
Author(s):  
Honglei Lang ◽  
Yunhe Sheng ◽  
Aïssa Wade
Keyword(s):  
Lie Algebra ◽  
Complex Structure ◽  
Contact Structures ◽  
Complex Structures ◽  
Courant Algebroids ◽  
Courant Algebroid ◽  
Generalized Complex Structures ◽  
Generalized Complex Structure ◽  
Generalized Geometry ◽  
Generalized Complex

AbstractIn this paper, we first discuss the relation between VB-Courant algebroids and E-Courant algebroids, and we construct some examples of E-Courant algebroids. Then we introduce the notion of a generalized complex structure on an E-Courant algebroid, unifying the usual generalized complex structures on even-dimensional manifolds and generalized contact structures on odd-dimensional manifolds. Moreover, we study generalized complex structures on an omni-Lie algebroid in detail. In particular, we show that generalized complex structures on an omni-Lie algebra gl(V) ⊕ V correspond to complex Lie algebra structures on V.

scholarly journals Extending invariant complex structures

2015 ◽  
Vol 26 (11) ◽  
pp. 1550096 ◽  
Author(s):  
Rutwig Campoamor Stursberg ◽  
Isolda E. Cardoso ◽  
Gabriela P. Ovando
Keyword(s):  
Lie Algebra ◽  
Algebraic Structure ◽  
Symplectic Structure ◽  
Complex Structure ◽  
Complex Structures ◽  
Additional Structure ◽  
Totally Real

We study the problem of extending a complex structure to a given Lie algebra 𝔤, which is firstly defined on an ideal 𝔥 ⊂ 𝔤. We consider the next situations: 𝔥 is either complex or it is totally real. The next question is to equip 𝔤 with an additional structure, such as a (non)-definite metric or a symplectic structure and to ask either 𝔥 is non-degenerate, isotropic, etc. with respect to this structure, by imposing a compatibility assumption. We show that this implies certain constraints on the algebraic structure of 𝔤. Constructive examples illustrating this situation are shown, in particular computations in dimension six are given.

scholarly journals Holomorphic Poisson Cohomology

2015 ◽  
Vol 2 (1) ◽  
Author(s):  
Zhuo Chen ◽  
Daniele Grandini ◽  
Yat-Sun Poon
Keyword(s):  
Spectral Sequence ◽  
Complex Structure ◽  
Complex Structures ◽  
Dolbeault Cohomology ◽  
Double Complex ◽  
Poisson Cohomology ◽  
Compact Complex Surfaces ◽  
Generalized Geometry ◽  
Holomorphic Poisson Cohomology ◽  
Compact Complex Manifolds

AbstractHolomorphic Poisson structures arise naturally in the realm of generalized geometry. A holomorphic Poisson structure induces a deformation of the complex structure in a generalized sense, whose cohomology is obtained by twisting the Dolbeault @-operator by the holomorphic Poisson bivector field. Therefore, the cohomology space naturally appears as the limit of a spectral sequence of a double complex. The first sheet of this spectral sequence is simply the Dolbeault cohomology with coefficients in the exterior algebra of the holomorphic tangent bundle. We identify various necessary conditions on compact complex manifolds on which this spectral sequence degenerates on the level of the second sheet. The manifolds to our concern include all compact complex surfaces, Kähler manifolds, and nilmanifolds with abelian complex structures or parallelizable complex structures.

scholarly journals Some relations between Hodge numbers and invariant complex structures on compact nilmanifolds

2017 ◽  
Vol 4 (1) ◽  
pp. 73-83
Author(s):  
Takumi Yamada
Keyword(s):  
Lie Algebra ◽  
Lie Group ◽  
Complex Structure ◽  
Complex Structures ◽  
Nilpotent Lie Group ◽  
Hodge Numbers ◽  
Compact Nilmanifolds ◽  
Simply Connected ◽  
Invariant Complex Structure

AbstractLet N be a simply connected real nilpotent Lie group, n its Lie algebra, and € a lattice in N. If a left-invariant complex structure on N is Γ-rational, then HƏ̄s,t(Γ/N) ≃ HƏ̄s,t(nC) for each s; t. We can construct different left-invariant complex structures on one nilpotent Lie group by using the complexification and the scalar restriction. We investigate relationships to Hodge numbers of associated compact complex nilmanifolds.

scholarly journals COMPLEX STRUCTURES ON INDECOMPOSABLE 6-DIMENSIONAL NILPOTENT REAL LIE ALGEBRAS

2007 ◽  
Vol 17 (01) ◽  
pp. 77-113 ◽  
Author(s):  
L. MAGNIN
Keyword(s):  
Automorphism Group ◽  
Lie Algebra ◽  
Lie Algebras ◽  
Equivalence Classes ◽  
Complex Structures ◽  
Smooth Submanifold ◽  
Computer Calculations

We show how resorting to dependable computer calculations makes it possible to compute all integrable complex structures on indecomposable 6-dimensional nilpotent real Lie algebras and their equivalence classes under the automorphism group of the Lie algebra. We also prove that the set comprised of all integrable complex structures on such a Lie algebra is a smooth submanifold of ℝ36.

scholarly journals Compact nilmanifolds with nilpotent complex structures: Dolbeault cohomology

2000 ◽  
Vol 352 (12) ◽  
pp. 5405-5433 ◽  
Author(s):  
Luis A. Cordero ◽  
Marisa Fernández ◽  
Alfred Gray ◽  
Luis Ugarte
Keyword(s):  
Complex Structures ◽  
Dolbeault Cohomology ◽  
Compact Nilmanifolds
Sign in / Sign up

Export Citation Format

Share Document