Crystallographic actions on Lie groups and post-Lie algebra structures

Burde, Dietrich

doi:10.2478/cm-2021-0003

Cited by 3 publications

(4 citation statements)

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“…For the existence question of post-Lie algebra structures on pairs of Lie algebras (g, n) we have introduced a "table" in [9], where g and n belong to one of the following seven classes, defined by the following properties: abelian, nilpotent, solvable, simple, semisimple, reductive, complete. Here we want to avoid an unnecessary overlap, so we assume that nilpotent means non-abelian, solvable means non-nilpotent, semisimple means non-simple, and reductive respectively complete means non-semisimple.…”

Section: The Existence Tablementioning

confidence: 99%

“…Similarly, a reductive, non-semisimple Lie algebra may be abelian, but has a non-trivial center and hence cannot be complete. The existence table in [9] still has six open cases:…”

Section: The Existence Tablementioning

confidence: 99%

“…Some important examples of such areas are geometric structures on manifolds, affine actions on Lie groups, Rota-Baxter operators, étale and prehomogeneous modules for Lie algebras, decompositions of Lie algebras, crystallographic groups, operad theory, deformation theory, or quantum field theory. There is a large literature on post-Lie algebra structures, see for example the papers [15,4,5,6,7,8,9] and the references given therein.…”

Section: Introductionmentioning

confidence: 99%

“…In section 4 we complete our "existence table" for post-Lie algebra structures on pairs (g, n) from [9]. We only leave one case open, namely where g is reductive and n is semisimple -see Conjecture 3.5.…”

Section: Introductionmentioning

confidence: 99%

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Rigidity results for Lie algebras admitting a post-Lie algebra structure

Burde

Dekimpe

Monadjem

2022

Int. J. Algebra Comput.

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We study rigidity questions for pairs of Lie algebras [Formula: see text] admitting a post-Lie algebra structure. We show that if [Formula: see text] is semisimple and [Formula: see text] is arbitrary, then we have rigidity in the sense that [Formula: see text] and [Formula: see text] must be isomorphic. The proof uses a result on the decomposition of a Lie algebra [Formula: see text] as the direct vector space sum of two semisimple subalgebras. We show that [Formula: see text] must be semisimple and hence isomorphic to the direct Lie algebra sum [Formula: see text]. This solves some open existence questions for post-Lie algebra structures on pairs of Lie algebras [Formula: see text]. We prove additional existence results for pairs [Formula: see text], where [Formula: see text] is complete, and for pairs, where [Formula: see text] is reductive with [Formula: see text]-dimensional center and [Formula: see text] is solvable or nilpotent.

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Section: The Existence Tablementioning

confidence: 99%

“…Similarly, a reductive, non-semisimple Lie algebra may be abelian, but has a non-trivial center and hence cannot be complete. The existence table in [9] still has six open cases:…”

Section: The Existence Tablementioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Rigidity results for Lie algebras admitting a post-Lie algebra structure

Burde

Dekimpe

Monadjem

2022

Int. J. Algebra Comput.

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Post-Lie algebra structures for perfect Lie algebras

Burde,

Dekimpe,

Monadjem

2024

Communications in Algebra

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We study the existence of post-Lie algebra structures on pairs of Lie algebras , where one of the algebras is perfect non-semisimple, and the other one is abelian, nilpotent non-abelian, solvable non-nilpotent, simple, semisimple non-simple, reductive non-semisimple or complete non-perfect. We prove several nonexistence results, but also provide examples in some cases for the existence of a post-Lie algebra structure. Among other results we show that there is no post-Lie algebra structure on , where is perfect non-semisimple, and is . We also show that there is no post-Lie algebra structure on , where is perfect and is reductive with a 1-dimensional center.

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Rigidity results for Lie algebras admitting a post-Lie algebra structure

Burde¹,

Dekimpe²,

Monadjem³

2022

Preprint

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We study rigidity questions for pairs of Lie algebras (g, n) admitting a post-Lie algebra structure. We show that if g is semisimple and n is arbitrary, then we have rigidity in the sense that g and n must be isomorphic. The proof uses a result on the decomposition of a Lie algebra g = s 1 ∔ s 2 as the direct vector space sum of two semisimple subalgebras. We show that g must be semisimple and hence isomorphic to the direct Lie algebra sum g ∼ = s 1 ⊕ s 2 . This solves some open existence questions for post-Lie algebra structures on pairs of Lie algebras (g, n). We prove additional existence results for pairs (g, n), where g is complete, and for pairs, where g is reductive with 1-dimensional center and n is solvable or nilpotent.

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Crystallographic actions on Lie groups and post-Lie algebra structures

Abstract: This survey on crystallographic groups, geometric structures on Lie groups and associated algebraic structures is based on a lecture given in the Ostrava research seminar in 2017.

Cited by 3 publications

References 50 publications

Rigidity results for Lie algebras admitting a post-Lie algebra structure

Rigidity results for Lie algebras admitting a post-Lie algebra structure

Post-Lie algebra structures for perfect Lie algebras

Rigidity results for Lie algebras admitting a post-Lie algebra structure

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