Cohomology, derivations and abelian extensions of 3-Lie algebras

Xu, Senrong

doi:10.1142/s0219498819501305

Cited by 12 publications

(9 citation statements)

References 17 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Based on deformation theory of 3-Lie algebras [38,39,40], we only need to check that R χ is a Rota-Baxter operator of the 3-Lie algebra A ⊕ V if and only if ∂ R χ + δψ = 0.…”

Section: Extension Of Rota-baxter 3-lie Algebrasmentioning

confidence: 99%

Representations and cohomolgies of Rota-Baxter 3-Lie algebras

Sun¹,

Chen²

2022

Preprint

View full text Add to dashboard Cite

show abstract

“…Based on deformation theory of 3-Lie algebras [38,39,40], we only need to check that R χ is a Rota-Baxter operator of the 3-Lie algebra A ⊕ V if and only if ∂ R χ + δψ = 0.…”

Section: Extension Of Rota-baxter 3-lie Algebrasmentioning

confidence: 99%

Representations and cohomolgies of Rota-Baxter 3-Lie algebras

Sun¹,

Chen²

2022

Preprint

View full text Add to dashboard Cite

show abstract

“…Recently Tang et al studied a Lie algebra with a derivation from the cohomological point of view and construct a cohomology theory that controls, among other things, simultaneous deformations of a Lie algebra with a derivation [11]. These results have been extended to associative algebras [3], Leibniz algebras [4], 3-Lie colour algebras [15], 3-Lie algebras [14], Lie triple systems [13], and n-Lie algebras [10]. Generalized representations of 3-Lie algebras and 3-Lie superalgebras was introduced in [17,18].…”

Section: Introductionmentioning

confidence: 99%

Cohomology, superderivations, and abelian extensions of $3$-Lie superalgebras

Nandi¹,

Padhan²

2022

Preprint

View full text Add to dashboard Cite

The main object of study of this paper is the notion of 3-Lie superalgebras with superderivations. We consider a representation (Φ, P) of a 3-Lie superalgebra Q on P and construct first-order cohomologies by using superderivations of P, Q which induces a Lie superalgebra TΦ and its representation Ψ. Then we consider abelian extensions of 3-Lie superalgebras of the form 0 → P ֒→ L → Q → 0 with [P, P, L] = 0 and construct an obstruction class to extensibility of a compatible pair of superderivation. Moreover we prove that a pair of superderivation is extensible if and only if its obstruction class is trivial under some suitable conditions.

show abstract

“…In recent years, researchers began to investigate noncommutative differential algebras in order to broaden the scope of the theory to include path algebras, for instance, and to have a more meaningful differential Lie algebra theory [23,24] and also from an operadic point of view [19,9]. There are also some recent work dealing with other algebraic structures endowed with derivations [29,8,31].…”

Section: Introductionmentioning

confidence: 99%

Gröbner-Shirshov bases and linear bases for free differential type algebras over algebras

Zhou¹,

Qin²,

Zhou³

2021

Preprint

View full text Add to dashboard Cite

We study a question which can be roughly stated as follows: Given a (unital or nonunital) algebra A together with a Gröbner-Shirshov basis G, consider the free operated algebra B over A, such that the operator satisfies some polynomial identities Φ which are Gröbner-Shirshov in the sense of Guo et al., when does the union Φ ∪ G will be an operated Gröbner-Shirshov basis for B? We answer this question in the affirmative under a mild condition in our previous work with Wang. When this condition is satisfied, Φ ∪ G is an operated Gröbner-Shirshov basis for B and as a consequence, we also get a linear basis of B. However, the condition could not be applied directly to differential type algebras introduced by Guo, Sit and Zhang, including usual differential algebras.This paper solves completely this problem for differential type algebras. Some new monomial orders are introduced which, together with some known ones, permit the application of the previous result to most of differential type algebras, thus providing new operated GS bases and linear bases for these differential type algebras. Versions are presented both for unital and nonunital algebras. However, a class of examples are also presented, for which the natural expectation in the question is wrong and these examples are dealt with by direct inspection.

show abstract

Cohomology, derivations and abelian extensions of 3-Lie algebras

Cited by 12 publications

References 17 publications

Representations and cohomolgies of Rota-Baxter 3-Lie algebras

Representations and cohomolgies of Rota-Baxter 3-Lie algebras

Cohomology, superderivations, and abelian extensions of $3$-Lie superalgebras

Gröbner-Shirshov bases and linear bases for free differential type algebras over algebras

Contact Info

Product

Resources

About