Applying matrix theory to classify real solvable Lie algebras having 2-dimensional derived ideals

Le, Vu A.; Nguyen, Tuan Anh; Nguyen, Tu T. C.; Vo, Thieu N.

doi:10.1016/j.laa.2019.11.031

Cited by 4 publications

(4 citation statements)

References 8 publications

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“…Proposition 2.11 (see [19]). Assume that G belongs to Lie (n, 2) (3 n ∈ N) and is not 2-step nilpotent.…”

Section: Nguyen and Lementioning

confidence: 99%

“…In 2020, V.A. Le et al [19] presented a new approach to the problem of classifying Lie (n, 2) and gave a list of non 2-step nilpotent Lie algebras in Lie (n, 2).…”

Section: Introductionmentioning

confidence: 99%

“…In this paper, we consider the Lie algebras listed in [19]. First, we give in Theorem 3.2 and Corollary 3.4 the upper bound of µ(G) for each Lie algebra G under consideration.…”

Section: Introductionmentioning

confidence: 99%

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Representation of Real Solvable Lie Algebras Having 2-dimensional Derived Ideal and Geometry of Coadjoint Orbits of Corresponding Lie Groups

Nguyen¹,

Le²

2021

Preprint

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Given a Lie algebra G , let µ(G) be the minimal degree of a faithful representation of G . This is an integer valued invariant of G , which has been introduced by D. Burde in 1998. It is not known, in general, how to determine this invariant for a given solvable Lie algebra. Lie (n, k) denotes the class of all n-dimensional real solvable Lie algebras having k -dimensional derived ideals. In 2020 we gave a classification of all non 2-step nilpotent Lie algebras of Lie (n, 2). We propose in this paper to study representations of these Lie algebras. First, we give an upper bound of µ(G) for each G classified. For indecomposable case, we further compute the character afforded by the adjoint representation and describe the picture of coadjoint orbits of Lie groups corresponding to them.

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“…Proposition 2.11 (see [19]). Assume that G belongs to Lie (n, 2) (3 n ∈ N) and is not 2-step nilpotent.…”

Section: Nguyen and Lementioning

confidence: 99%

“…In 2020, V.A. Le et al [19] presented a new approach to the problem of classifying Lie (n, 2) and gave a list of non 2-step nilpotent Lie algebras in Lie (n, 2).…”

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

Representation of Real Solvable Lie Algebras Having 2-dimensional Derived Ideal and Geometry of Coadjoint Orbits of Corresponding Lie Groups

Nguyen¹,

Le²

2021

Preprint

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“…Schur [19] and Jacobson [14] investigated formulas for determining the maximal dimension of a commutative subalgebra of a matrix Lie algebra. Based on the results of [14,19], a full classification for real solvable Lie algebras with 2-dimensional derived algebras was achieved in [23]. To the best of our knowledge, the problem of classifying real solvable Lie algebras with the derived algebras of dimension = 1, 2 still remains open.…”

Section: Introductionmentioning

confidence: 99%

On the problem of classifying solvable Lie algebras having small codimensional derived algebras

Duong¹,

Le²,

Nguyen³

et al. 2020

Preprint

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This paper concerns the problem of classifying finite-dimensional real solvable Lie algebras whose derived algebras are of codimension 1 or 2. On the one hand, we present an effective method to classify all (n + 1)-dimensional real solvable Lie algebras having 1-codimensional derived algebras provided that a full classification of n-dimensional nilpotent Lie algebras is given. On the other hand, the problem of classifying all (n + 2)-dimensional real solvable Lie algebras having 2-codimensional derived algebras is proved to be wild. In this case, we provide a method to classify a subclass of the considered Lie algebras which are extended from their derived algebras by a pair of derivations containing at least one inner derivation.

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On the problem of classifying solvable Lie algebras having small codimensional derived algebras

Duong

Nguyen

et al. 2022

Communications in Algebra

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Applying matrix theory to classify real solvable Lie algebras having 2-dimensional derived ideals

Cited by 4 publications

References 8 publications

Representation of Real Solvable Lie Algebras Having 2-dimensional Derived Ideal and Geometry of Coadjoint Orbits of Corresponding Lie Groups

Representation of Real Solvable Lie Algebras Having 2-dimensional Derived Ideal and Geometry of Coadjoint Orbits of Corresponding Lie Groups

On the problem of classifying solvable Lie algebras having small codimensional derived algebras

On the problem of classifying solvable Lie algebras having small codimensional derived algebras

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