Determination of 7-Dimensional Indecomposable Nilpotent Complex Lie Algebras by Adjoining a Derivation to 6-Dimensional Lie Algebras

Magnin, Louis

doi:10.1007/s10468-009-9172-3

Cited by 24 publications

(17 citation statements)

References 15 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…It is easy to see that Γ and its finite index subgroups have asymptotically equivalent systolic growth. Date: August 4, 2016. 2010 Mathematics Subject Classification.…”

Section: Introductionmentioning

confidence: 99%

Gradings on Lie algebras, systolic growth, and cohopfian properties of nilpotent groups

Cornulier¹

2016

Bul. Soc. Math. France

View full text Add to dashboard Cite

We study the existence of various types of gradings on Lie algebras, such as Carnot gradings or gradings in positive integers, and prove that the existence of such gradings is invariant under extensions of scalars.As an application, we prove that if Γ is a finitely generated nilpotent group, its systolic growth is asymptotically equivalent to its word growth if and only if the Malcev completion of Γ is Carnot.We also characterize when Γ is non-cohopfian, in terms of the existence of a non-trivial grading in non-negative integers, and deduce that this property only depends on its real (or even complex) Malcev completion. n≥0 φ n (Γ) = {1}; such φ is called a dis-cohopf endomorphism. It appears, for a nontrivial group, as a strong negation of being cohopfian. Contents

show abstract

“…It is easy to see that Γ and its finite index subgroups have asymptotically equivalent systolic growth. Date: August 4, 2016. 2010 Mathematics Subject Classification.…”

Section: Introductionmentioning

confidence: 99%

Gradings on Lie algebras, systolic growth, and cohopfian properties of nilpotent groups

Cornulier¹

2016

Bul. Soc. Math. France

View full text Add to dashboard Cite

show abstract

“…It is easy to verify that F 2,3 has property F , since it is enough to check it just for one specific pair of generators. In dimension 6, the classification shows that the only stem nilpotent Lie algebra satisfying dim g/[g, g] = 2 and dim Z(g) ≥ 2 is given by g 6,14 , in Magnin's notation [15]. The Lie brackets are given by [x 1 , x i ] = x i+1 , 2 ≤ i ≤ 4, [x 2 , x 3 ] = x 6 .…”

Section: Property Fmentioning

confidence: 99%

Commutative post-Lie algebra structures and linear equations for nilpotent Lie algebras

2019

View full text Add to dashboard Cite

We show that for a given nilpotent Lie algebra g with Z(g) ⊆ [g, g] all commutative post-Lie algebra structures, or CPA-structures, on g are complete. This means that all left and all right multiplication operators in the algebra are nilpotent. Then we study CPA-structures on free-nilpotent Lie algebras F g,c and discover a strong relationship to solving systems of linear equations of type [x, u] + [y, v] = 0 for generator pairs x, y ∈ F g,c . We use results of Remeslennikov and Stöhr concerning these equations to prove that, for certain g and c, the free-nilpotent Lie algebra F g,c has only central CPA-structures.

show abstract

“…It is easy to verify the following result for low-dimensional nilpotent Lie algebras. The notation is taken from Magnin [16]. Proposition 4.9.…”

Section: Lie Algebra Identitiesmentioning

confidence: 99%

Derivation double Lie algebras

Burde

2016

J. Algebra Appl.

View full text Add to dashboard Cite

Abstract. We study classical R-matrices D for Lie algebras g such that D is also a derivation of g. This yields derivation double Lie algebras (g, D). The motivation comes from recent work on post-Lie algebra structures on pairs of Lie algebras arising in the study of nil-affine actions of Lie groups. We prove that there are no nontrivial simple derivation double Lie algebras, and study related Lie algebra identities for arbitrary Lie algebras.

show abstract

Determination of 7-Dimensional Indecomposable Nilpotent Complex Lie Algebras by Adjoining a Derivation to 6-Dimensional Lie Algebras

Cited by 24 publications

References 15 publications

Gradings on Lie algebras, systolic growth, and cohopfian properties of nilpotent groups

Gradings on Lie algebras, systolic growth, and cohopfian properties of nilpotent groups

Commutative post-Lie algebra structures and linear equations for nilpotent Lie algebras

Derivation double Lie algebras

Contact Info

Product

Resources

About