Identical Relations in Lie Algebras

Bahturin, Yuri

doi:10.1515/9783110566659

Cited by 27 publications

(78 citation statements)

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“…Any 0-connected free simplicial module A • is homotopy equivelent to a simplicial module of the form B • [1], where B • is also a free simplicial module (it follows from the same fact for non-negatively graded chain complexes). Moreover, if A • is k-connected, we can chose B • so that Proof.…”

Section: Connectivity Of Functorsmentioning

confidence: 96%

A Simple proof of Curtis' connectivity theorem for Lie powers

Ivanov¹,

Romanovskii²,

Семенов³

2019

Preprint

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Section: Connectivity Of Functorsmentioning

confidence: 96%

A Simple proof of Curtis' connectivity theorem for Lie powers

Ivanov¹,

Romanovskii²,

Семенов³

2019

Preprint

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“…This is the free metabelian Lie algebra over F with free generators the [z, c I ]'s. Now it is well known (see for example [2]), that M (z) has a basis consisting of the left-normed Lie commutators…”

Section: The Lie Algebra Lmentioning

confidence: 99%

Locally finite $p$-groups with a left 3-Engel element whose normal closure is not nilpotent

Hadjievangelou¹,

Noce²,

Traustason³

2020

Preprint

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“…Therefore, either V is an abelian variety, or λ " ˘1, which proves the result. Theorem 1.4 ([29,1]). If V is a variety of algebras over an infinite field, then all of its identities are of the form φpx 1 , .…”

Section: Introductionmentioning

confidence: 99%

“…Theorem 1.5 (Theorem 3 in Section 4.2 of [1]). In a subvariety of Lie K , any nontrivial identity has a nontrivial multilinear consequence.…”

Section: Introductionmentioning

confidence: 99%

Algebras with representable representations

García-Martínez,

Tsishyn,

Van der Linden

et al. 2020

Preprint

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Just like group actions are represented by group automorphisms, Lie algebra actions are represented by derivations: up to isomorphism, a split extension of a Lie algebra B by a Lie algebra X corresponds to a Lie algebra morphism B Ñ Der pXq from B to the Lie algebra Der pXq of derivations on X.In this article, we study the question whether the concept of a derivation can be extended to other types of non-associative algebras over a field K, in such a way that these generalized derivations characterize the K-algebra actions. We prove that the answer is no, as soon as the field K is infinite. In fact, we prove a stronger result: already the representability of all abelian actions-which are usually called representations or Beck modules-suffices for this to be true. Thus we characterize the variety of Lie algebras over an infinite field of characteristic different from 2 as the only variety of non-associative algebras which is a non-abelian category with representable representations. This emphasizes the unique role played by the Lie algebra of linear endomorphisms glpV q as a representing object for the representations on a vector space V .

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Identical Relations in Lie Algebras

Cited by 27 publications

References 0 publications

A Simple proof of Curtis' connectivity theorem for Lie powers

A Simple proof of Curtis' connectivity theorem for Lie powers

Locally finite $p$-groups with a left 3-Engel element whose normal closure is not nilpotent

Algebras with representable representations

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