On an algorithm for finding derivations of Lie algebras

Ayala, Víctor; Kizil, Eyüp; Tribuzy, Ivan de Azevedo

doi:10.4067/s0716-09172012000100008

Cited by 32 publications

(23 citation statements)

References 8 publications

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“…On the other hand, algebras are also useful via their derivations. For example one can mention their use in control theory ( [1,2]). One can also mention the higher derived bracket construction of Voronov in [28] that produces out of some special derivation of a graded Lie algebra a homotopy Lie algebra.…”

Section: Introductionmentioning

confidence: 99%

Cohomologies of a Lie algebra with a derivation and applications

2019

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The main object of study of this paper is the notion of a LieDer pair, i.e. a Lie algebra with a derivation. We introduce the concept of a representation of a LieDer pair and study the corresponding cohomologies. We show that a LieDer pair is rigid if the second cohomology group is trivial, and a deformation of order n is extensible if its obstruction class, which is defined to be an element is the third cohomology group, is trivial. We classify central extensions of LieDer pairs using the second cohomology group with the coefficient in the trivial representation. For a pair of derivations, we define its obstruction class and show that it is extensible if and only if the obstruction class is trivial. Finally, we classify Lie2Der pairs using the third cohomology group of a LieDer pair.

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Section: Introductionmentioning

confidence: 99%

Cohomologies of a Lie algebra with a derivation and applications

2019

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“…Let { } { } be basis for . We apply the formula in (Ayala, Kizil, & Tribuzy, 2012) is not equal to zero.…”

Section: Proofmentioning

confidence: 99%

On Properties of the (2n+1)-Dimensional Heisenberg Lie Algebra

Kurniadi¹

2020

JTAM

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In the present paper, we study some properties of the Heisenberg Lie algebra of dimension . The main purpose of this research is to construct a real Frobenius Lie algebra from the Heisenberg Lie algebra of dimension . To achieve this, we exhibit how to compute the derivation of the Heisenberg Lie algebra by following Oom’s result. In this research, we use a literature review method to some related papers corresponding to a derivation of a Lie algebra, Frobenius Lie algebras, and Plancherel measure. Determining a conjecture of a real Frobenius Lie algebra is obtained. As the main result, we prove that conjecture. Namely, for the given the Heisenberg Lie algebra, there exists a commutative subalgebra of dimension one such that its semi direct sum is a real Frobenius Lie algebra of dimension . Futhermore, in the notion of the Lie group of the Heisenberg Lie algebra which is called the Heisenberg Lie group, we compute the generalized character of its group and we determine the Plancherel measure of the unitary dual of the Heisenberg Lie group. As our contributions, we complete some examples of Frobenius Lie algebras obtained from a nilpotent Lie algebra and we also give alternative computations to find the Plancherel measure of the Heisenberg Lie group.

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“…However, such computer programs have reasonably limited applications since some of them do not compute with Lie algebras having pure real or complex non-rational structure constants while others have not been conceived to consider parameters. See the paper by Ayala-Kizil-Tribuzy in [6] where the structure tensor is considered to obtain conditions that a linear transformation g → g must satisfy in order to be a derivation of g.…”

Section: Locally Controllable At E ∈ G If the Reachable Setmentioning

confidence: 99%