Extensions and automorphisms of Lie algebras

Bardakov, Valeriy G.; Singh, Mahender

doi:10.1142/s0219498817501626

Cited by 19 publications

(13 citation statements)

References 13 publications

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“…In [7], the authors study extensions of a pair of automorphisms of Lie algebras. Since derivations are infinitesimals of automorphisms, we are interested in extensions of a pair of derivations.…”

Section: Extensions Of a Pair Of Derivationsmentioning

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: Extensions Of a Pair Of Derivationsmentioning

confidence: 99%

Cohomologies of a Lie algebra with a derivation and applications

2019

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“…(i) Let A R and B S be two Rota-Baxter algebras. A non-abelian extension of A R by B S consists of a short exact sequence of Rota-Baxter algebras of the form (5) We denote by Ext(A R , B S ) the set of all equivalence classes of non-abelian extensions of A R by B S . In the following, we will parametrize the set Ext(A R , B S ) by certain non-abelian cohomology.…”

Section: Non-abelian Extensions Of Rota-baxter Algebrasmentioning

confidence: 99%

A unified extension theory of Rota-Baxter algebras, dendriform algebras, and a fundamental sequence of Wells

Das¹,

Rathee²

2022

Preprint

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A Rota-Baxter algebra A R is an algebra A equipped with a distinguished Rota-Baxter operator R on it. Rota-Baxter algebras are closely related to dendriform algebras introduced by Loday. In this paper, we first consider the non-abelian extension theory of Rota-Baxter algebras and classify them by introducing the non-abelian cohomology. Next, given a non-abelian extension 0 → B S → E U → A R → 0 of Rota-Baxter algebras, we construct the Wells type exact sequences and find their role in extending a Rota-Baxter automorphism β ∈ Aut(B S ) and lifting a Rota-Baxter automorphism α ∈ Aut(A R ) to an automorphism in Aut(E U ). We end this paper by considering a similar study for dendriform algebras.

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“…It is well-known that derivations are infinitesimals of automorphisms, and a study [8] has been done on extensions of a pair of automorphisms of Lie-algebras. In this section, we study extensions of a pair of derivations and see how it is related to the cohomology of the 3-LieDer pair.…”

Section: Extensions Of a Pair Of Derivationsmentioning

confidence: 99%

On 3-Lie algebras with a derivation

Guo,

Saha

2021

Preprint

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In this paper, we study 3-Lie algebras with derivations. We call the pair consisting of a 3-Lie algebra and a distinguished derivation by the 3-LieDer pair. We define a cohomology theory for 3-LieDer pair with coefficients in a representation. We study central extensions of a 3-LieDer pair and show that central extensions are classified by the second cohomology of the 3-LieDer pair with coefficients in the trivial representation. We generalize Gerstenhaber's formal deformation theory to 3-LieDer pairs in which we deform both the 3-Lie bracket and the distinguished derivation.

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Extensions and automorphisms of Lie algebras

Cited by 19 publications

References 13 publications

Cohomologies of a Lie algebra with a derivation and applications

Cohomologies of a Lie algebra with a derivation and applications

A unified extension theory of Rota-Baxter algebras, dendriform algebras, and a fundamental sequence of Wells

On 3-Lie algebras with a derivation

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