Extensions of hom-Lie algebras in terms of cohomology

Abdoreza R. Armakan; Mohammed Reza Farhangdoost

Displaying similar documents to “Extensions of hom-Lie algebras in terms of cohomology”

Universal central extension of direct limits of Hom-Lie algebras

Valiollah Khalili (2019)

Czechoslovak Mathematical Journal

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We prove that the universal central extension of a direct limit of perfect Hom-Lie algebras $(ℒ_{i}, α_{ℒ_{i}})$ is (isomorphic to) the direct limit of universal central extensions of $(ℒ_{i}, α_{ℒ_{i}})$ . As an application we provide the universal central extensions of some multiplicative Hom-Lie algebras. More precisely, we consider a family of multiplicative Hom-Lie algebras ${({sl}_{k} (å), α_{k})}_{k \in I}$ and describe the universal central extension of its direct limit.

The Wells map for abelian extensions of 3-Lie algebras

Youjun Tan, Senrong Xu (2019)

Czechoslovak Mathematical Journal

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The Wells map relates automorphisms with cohomology in the setting of extensions of groups and Lie algebras. We construct the Wells map for some abelian extensions $0 \to A ↪ L \overset{π}{\to} B \to 0$ of 3-Lie algebras to obtain obstruction classes in $H^{1} (B, A)$ for a pair of automorphisms in $Aut (A) \times Aut (B)$ to be inducible from an automorphism of $L$ . Application to free nilpotent 3-Lie algebras is discussed.

The variety of dual mock-Lie algebras

Luisa M. Camacho, Ivan Kaygorodov, Viktor Lopatkin, Mohamed A. Salim (2020)

Communications in Mathematics

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We classify all complex $7$ - and $8$ -dimensional dual mock-Lie algebras by the algebraic and geometric way. Also, we find all non-trivial complex $9$ -dimensional dual mock-Lie algebras.

Algorithmic computations of Lie algebras cohomologies

Šilhan, Josef

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From the text: The aim of this work is to advertise an algorithmic treatment of the computation of the cohomologies of semisimple Lie algebras. The base is Kostant’s result which describes the representation of the proper reductive subalgebra on the cohomologies space. We show how to (algorithmically) compute the highest weights of irreducible components of this representation using the Dynkin diagrams. The software package $L i E$ offers the data structures and corresponding procedures for...

The groups of automorphisms of the Witt $W_{n}$ and Virasoro Lie algebras

Vladimir V. Bavula (2016)

Czechoslovak Mathematical Journal

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Let $L_{n} = K [x_{1}^{\pm 1}, ..., x_{n}^{\pm 1}]$ be a Laurent polynomial algebra over a field $K$ of characteristic zero, $W_{n} : = {Der}_{K} (L_{n})$ the Lie algebra of $K$ -derivations of the algebra $L_{n}$ , the so-called Witt Lie algebra, and let $Vir$ be the Virasoro Lie algebra which is a $1$ -dimensional central extension of the Witt Lie algebra. The Lie algebras $W_{n}$ and $Vir$ are infinite dimensional Lie algebras. We prove that the following isomorphisms of the groups of Lie algebra automorphisms hold: ${Aut}_{Lie} (Vir) ≃ {Aut}_{Lie} (W_{1}) ≃ {\pm 1} ⋉ K^{*}$ , and give a short proof that ${Aut}_{Lie} (W_{n}) ≃ {Aut}_{K - alg} (L_{n}) ≃ {GL}_{n} (ℤ) ⋉ K^{* n}$ .

Local superderivations on Lie superalgebra $𝔮 (n)$

Haixian Chen, Ying Wang (2018)

Czechoslovak Mathematical Journal

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Let $𝔮 (n)$ be a simple strange Lie superalgebra over the complex field $ℂ$ . In a paper by A. Ayupov, K. Kudaybergenov (2016), the authors studied the local derivations on semi-simple Lie algebras over $ℂ$ and showed the difference between the properties of local derivations on semi-simple and nilpotent Lie algebras. We know that Lie superalgebras are a generalization of Lie algebras and the properties of some Lie superalgebras are similar to those of semi-simple Lie algebras, but $𝔭 (n)$ is an exception....

When unit groups of continuous inverse algebras are regular Lie groups

Helge Glöckner, Karl-Hermann Neeb (2012)

Studia Mathematica

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It is a basic fact in infinite-dimensional Lie theory that the unit group $A^{\times}$ of a continuous inverse algebra A is a Lie group. We describe criteria ensuring that the Lie group $A^{\times}$ is regular in Milnor’s sense. Notably, $A^{\times}$ is regular if A is Mackey-complete and locally m-convex.

On some properties of the upper central series in Leibniz algebras

Leonid A. Kurdachenko, Javier Otal, Igor Ya. Subbotin (2019)

Commentationes Mathematicae Universitatis Carolinae

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This article discusses the Leibniz algebras whose upper hypercenter has finite codimension. It is proved that such an algebra $L$ includes a finite dimensional ideal $K$ such that the factor-algebra $L / K$ is hypercentral. This result is an extension to the Leibniz algebra of the corresponding result obtained earlier for Lie algebras. It is also analogous to the corresponding results obtained for groups and modules.

Branching problems and $𝔰𝔩 (2, ℂ)$ -actions

Pavle Pandžić, Petr Somberg (2015)

Archivum Mathematicum

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We study certain $𝔰𝔩 (2, ℂ)$ -actions associated to specific examples of branching of scalar generalized Verma modules for compatible pairs $(𝔤, 𝔭)$ , $(𝔤^{'}, 𝔭^{'})$ of Lie algebras and their parabolic subalgebras.