On the normality of Higgins commutators

Cigoli, Alan S.; Gray, James Richard Andrew; Linden, Tim Van der

doi:10.1016/j.jpaa.2014.05.025

Cited by 28 publications

(68 citation statements)

References 21 publications

(45 reference statements)

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“…A first step towards the main result of this paper is proving that in the context of nonassociative algebras, action representability/representability of representations implies a condition called algebraic coherence [10]. The reason we want this comes from the fact that an algebraically coherent variety satisfies some identities of degree three, useful in the next sections.…”

Section: Algebraic Coherencementioning

confidence: 97%

Algebras with representable representations

García-Martínez

Tsishyn²,

Linden³

et al. 2021

Proceedings of the Edinburgh Mathematical Society

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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 $B\to {\mathit {Der}}(X)$ from $B$ to the Lie algebra ${\mathit {Der}}(X)$ 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 ${\mathbb {K}}$ , in such a way that these generalized derivations characterize the ${\mathbb {K}}$ -algebra actions. We prove that the answer is no, as soon as the field ${\mathbb {K}}$ 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 $\mathfrak {gl}(V)$ as a representing object for the representations on a vector space $V$ .

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Section: Algebraic Coherencementioning

confidence: 97%

Algebras with representable representations

García-Martínez

Tsishyn²,

Linden³

et al. 2021

Proceedings of the Edinburgh Mathematical Society

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“…Note that for the category of groups, Condition (vii) of Proposition 2.1 corresponds to the fact that the commutator rK, Ks K is a characteristic subgroup of K: being invariant under automorphisms means precisely that every action on K restricts to an action on rK, Ks K . This observation can be extended to arbitrary semi-abelian categories when the definition of characteristic subobject from [17] is used; see [16] for the proof under the condition (NH).…”

Section: Preliminariesmentioning

confidence: 98%

“…An example in [15] shows that in the category of non-associative rings the two commutators generally need not coincide. Thus the coincidence rK, Ls " rK, Ls X for all K, L X becomes a basic condition which a semi-abelian category may or may not satisfy; this condition, which we will denote by (NH), was introduced by Cigoli in his Ph.D. thesis [15] and was studied further, by Cigoli together with the present authors, in [16].…”

Section: Preliminariesmentioning

confidence: 99%

“…Theorem 5.3.6 in Cigoli's thesis [15] shows that any category of interest satisfies (WNH), while those categories satisfy (SH) by action accessibility ( [12] combined with [32]). It is known that the condition (SH) fails for the categories of loops and digroups [22,18,2,6], and that the condition (WNH) fails for nonassociative rings [15,Example 5.3.7] [16,Example 5.4]. Note the that the latter category is strongly protomodular.…”

Section: Proposition 22 Every Arithmetical Category Satisfies (Wnh)mentioning

confidence: 99%

“…[16, Lemma 2.6] In a semi-abelian category, consider a point with chosen kernel as in bottom row of the diagram…”

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Peri-abelian categories and the universal central extension condition

Gray

Linden

2015

Journal of Pure and Applied Algebra

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Abstract. We study the relation between Bourn's notion of peri-abelian category and conditions involving the coincidence of the Smith, Huq and Higgins commutators. In particular we show that a semi-abelian category is periabelian if and only if for each normal subobject K X, the Higgins commutator of K with itself coincides with the normalisation of the Smith commutator of the denormalisation of K with itself. We show that if a category is periabelian, then the condition (UCE), which was introduced and studied by Casas and the second author, holds for that category. In addition we show, using amongst other things a result by Cigoli, that all categories of interest in the sense of Orzech are peri-abelian and therefore satisfy the condition (UCE).

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Categorical Commutator Theory

Mantovani

Montoli

2021

New Perspectives in Algebra, Topology and Categories

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On the normality of Higgins commutators

Cited by 28 publications

References 21 publications

Algebras with representable representations

Algebras with representable representations

Peri-abelian categories and the universal central extension condition

Categorical Commutator Theory

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