Cocalibrated structures on Lie algebras with a codimension one Abelian ideal

Freibert, Marco

doi:10.1007/s10455-012-9326-0

Cited by 25 publications

(39 citation statements)

References 13 publications

(39 reference statements)

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“…These Lie algebras are often called almost abelian in the literature. It was proved in that there exists an orthonormal basis

{e_{1}, \dots, e_{7}}

frakturg

with respect to

{false⟨ \cdot, \cdot false⟩}_{φ}

such that

h = span {e_{1}, \dots, e_{6}}

and

φ = ω \land e^{7} + ρ^{+} = e^{127} + e^{347} + e^{567} + e^{135} - e^{146} - e^{236} - e^{245},

where

ω : = e^{12} + e^{34} + e^{56}, ρ^{+} : = e^{135} - e^{146} - e^{236} - e^{245} .

We note that each of these Lie algebras is completely determined by the real

6 \times 6

matrix

A : = ad e_{7} |_{frakturh},

and so its Lie bracket will be denoted by

μ_{A}

and the corresponding simply connected Lie group by

G_{A}

. Note Identification of linear maps with matrices will always be done via the ordered basis

{e_{1}, e_{3}, e_{5}, e_{2}, e_{4}, e_{6}, e_{7}}

, unless otherwise stated.…”

Section: Almost Abelian Solvmanifoldsmentioning

confidence: 99%

“…The non‐degenerate 2‐form

ω \in {normalΛ}^{2} {frakturh}^{*}

can be written as

ω = {⟨ J \cdot, \cdot ⟩}_{φ}

, where

Given

A \in {frakturgl}_{6} false(double-struckR false)

, let

(G_{A}, φ)

denote the Lie group

G_{A}

endowed with the left‐invariant

G_{2}

‐structure defined by the fixed positive 3‐form

φ

given in . Proposition

(G_{A}, φ)

is closed if and only if the matrix

A

belongs to

(G_{A}, φ)

is coclosed if and only if

A

is in

…”

Section: Almost Abelian Solvmanifoldsmentioning

confidence: 99%

“…Remark Part (ii) is proved in for the 3‐form

ω \land e^{7} - ρ^{-}

, but it is easily seen to be valid for

φ

as well (see Lemma (iv) for the definition of

ρ^{-}

).…”

Section: Almost Abelian Solvmanifoldsmentioning

confidence: 99%

“…A G 2 -structure on a 7-manifold M is closed if and only if the torsion forms τ 0 , τ 1 and τ 3 all vanish (see (10)). Thus, only the torsion form τ 2 survives for a closed G 2 -structure ϕ.…”

Section: Closed G 2 -Structuresmentioning

confidence: 99%

“…In Section 5, we work in a more explicit way on the class of almost‐abelian (that is, with a codimension‐one abelian normal subgroup) solvable Lie groups. Left‐invariant coclosed and closed

G_{2}

‐structures on these Lie groups have been studied by Freibert in . One attaches to each matrix

A \in sl false(3, double-struckC false) \subset {frakturgl}_{6} false(double-struckR false)

a left‐invariant closed

G_{2}

‐structure on a simply connected solvable Lie group denoted by

G_{A}

.…”

Section: Introductionmentioning

confidence: 99%

See 4 more Smart Citations

Laplacian flow of homogeneous G2-structures and its solitons

Lauret

2017

Proc. London Math. Soc.

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We use the bracket flow/algebraic soliton approach to study the Laplacian flow of $G_2$-structures and its solitons in the homogeneous case. We prove that any homogeneous Laplacian soliton is equivalent to a semi-algebraic soliton (i.e.\ a $G$-invariant $G_2$-structure on a homogeneous space $G/K$ that flows by pull-back of automorphisms of $G$ up to scaling). Algebraic solitons are geometrically characterized among Laplacian solitons as those with a `diagonal' evolution. Unlike the Ricci flow case, where any homogeneous Ricci soliton is isometric to an algebraic soliton, we have found, as an application of the above characterization, an example of a left-invariant closed semi-algebraic soliton on a nilpotent Lie group which is not equivalent to any algebraic soliton. The (normalized) bracket flow evolution of such a soliton is periodic. In the context of solvable Lie groups with a codimension-one abelian normal subgroup, we obtain long time existence for any closed Laplacian flow solution; furthermore, the norm of the torsion is strictly decreasing and converges to zero. We also classify algebraic solitons in this class and exhibit several explicit examples of closed expanding Laplacian solitons.Comment: 33 pages, final version accepted in Proc. London Math. So

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“…These Lie algebras are often called almost abelian in the literature. It was proved in that there exists an orthonormal basis

{e_{1}, \dots, e_{7}}

frakturg

with respect to

{false⟨ \cdot, \cdot false⟩}_{φ}

such that

h = span {e_{1}, \dots, e_{6}}

and

φ = ω \land e^{7} + ρ^{+} = e^{127} + e^{347} + e^{567} + e^{135} - e^{146} - e^{236} - e^{245},

where

ω : = e^{12} + e^{34} + e^{56}, ρ^{+} : = e^{135} - e^{146} - e^{236} - e^{245} .

We note that each of these Lie algebras is completely determined by the real

6 \times 6

matrix

A : = ad e_{7} |_{frakturh},

and so its Lie bracket will be denoted by

μ_{A}

and the corresponding simply connected Lie group by

G_{A}

. Note Identification of linear maps with matrices will always be done via the ordered basis

{e_{1}, e_{3}, e_{5}, e_{2}, e_{4}, e_{6}, e_{7}}

, unless otherwise stated.…”

Section: Almost Abelian Solvmanifoldsmentioning

confidence: 99%

“…The non‐degenerate 2‐form

ω \in {normalΛ}^{2} {frakturh}^{*}

can be written as

ω = {⟨ J \cdot, \cdot ⟩}_{φ}

, where

Given

A \in {frakturgl}_{6} false(double-struckR false)

, let

(G_{A}, φ)

denote the Lie group

G_{A}

endowed with the left‐invariant

G_{2}

‐structure defined by the fixed positive 3‐form

φ

given in . Proposition

(G_{A}, φ)

is closed if and only if the matrix

A

belongs to

(G_{A}, φ)

is coclosed if and only if

A

is in

…”

Section: Almost Abelian Solvmanifoldsmentioning

confidence: 99%

“…Remark Part (ii) is proved in for the 3‐form

ω \land e^{7} - ρ^{-}

, but it is easily seen to be valid for

φ

as well (see Lemma (iv) for the definition of

ρ^{-}

).…”

Section: Almost Abelian Solvmanifoldsmentioning

confidence: 99%

Section: Closed G 2 -Structuresmentioning

confidence: 99%

G_{2}

‐structures on these Lie groups have been studied by Freibert in . One attaches to each matrix

A \in sl false(3, double-struckC false) \subset {frakturgl}_{6} false(double-struckR false)

a left‐invariant closed

G_{2}

‐structure on a simply connected solvable Lie group denoted by

G_{A}

.…”

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

Laplacian flow of homogeneous G2-structures and its solitons

Lauret

2017

Proc. London Math. Soc.

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Purely coclosed G₂‐structures on nilmanifolds

Bazzoni

Garvı́n

Muñoz

2023

Mathematische Nachrichten

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We classify seven‐dimensional nilpotent Lie groups, decomposable or of nilpotency step at most 4, endowed with left‐invariant purely coclosed G2‐structures. This is done by going through the list of all seven‐dimensional nilpotent Lie algebras given by Gong, providing an example of a left‐invariant 3‐form φ which is a pure coclosed G2‐structure (i.e., it satisfies d∗φ=0$d*\varphi =0$, φ∧dφ=0$\varphi \wedge d\varphi =0$) for those nilpotent Lie algebras that admit them; and by showing the impossibility of having a purely coclosed G2‐structure for the rest of them.

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Half-flat structures on indecomposable Lie groups

Freibert

Schulte-Hengesbach

2012

Transformation Groups

Self Cite

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This article can be viewed as a continuation of the articles [12] and [5] where the decomposable Lie algebras admitting half-flat SU(3)-structures are classified. The new main result is the classification of the indecomposable six-dimensional Lie algebras with five-dimensional nilradical which admit a half-flat SU(3)-structure. As an important step of the proof, a considerable refinement of the classification of six-dimensional Lie algebras with five-dimensional non-Abelian nilradical is established. Additionally, it is proved that all non-solvable six-dimensional Lie algebras admit half-flat SU(3)-structures.

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Cocalibrated structures on Lie algebras with a codimension one Abelian ideal

Cited by 25 publications

References 13 publications

Laplacian flow of homogeneous G2-structures and its solitons

Laplacian flow of homogeneous G2-structures and its solitons

Purely coclosed G₂‐structures on nilmanifolds

Half-flat structures on indecomposable Lie groups

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Cocalibrated structures on Lie algebras with a codimension one Abelian ideal

Cited by 25 publications

References 13 publications

Laplacian flow of homogeneous G2-structures and its solitons

Laplacian flow of homogeneous G2-structures and its solitons

Purely coclosed G2‐structures on nilmanifolds

Half-flat structures on indecomposable Lie groups

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Product

Resources

About

Purely coclosed G₂‐structures on nilmanifolds