Metabelian Thin Lie Algebras

Gavioli, Norberto; Monti, Valerio; Young, David S

doi:10.1006/jabr.2001.8752

Cited by 10 publications

(10 citation statements)

References 5 publications

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“…The same argument in the proof of Proposition 6 also shows that a thin Lie algebra over a quadratically closed field (a field without extensions of degree two) cannot have two diamonds occurring as consecutive homogeneous components. In fact, the thin Lie algebras studied in [6,15], where all homogeneous components except L 2 are diamonds, require their base field to have a quadratic field extension.…”

Section: Propositionmentioning

confidence: 99%

A sandwich in thin lie algebras

Mattarei

2022

Proceedings of the Edinburgh Mathematical Society

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A thin Lie algebra is a Lie algebra $L$ , graded over the positive integers, with its first homogeneous component $L_1$ of dimension two and generating $L$ , and such that each non-zero ideal of $L$ lies between consecutive terms of its lower central series. All homogeneous components of a thin Lie algebra have dimension one or two, and the two-dimensional components are called diamonds. Suppose the second diamond of $L$ (that is, the next diamond past $L_1$ ) occurs in degree $k$ . We prove that if $k>5$ , then $[Lyy]=0$ for some non-zero element $y$ of $L_1$ . In characteristic different from two this means $y$ is a sandwich element of $L$ . We discuss the relevance of this fact in connection with an important theorem of Premet on sandwich elements in modular Lie algebras.

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Section: Propositionmentioning

confidence: 99%

A sandwich in thin lie algebras

Mattarei

2022

Proceedings of the Edinburgh Mathematical Society

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“…According to [CJ99] for p = 2 , and to [AJM10] for p = 2, which amends [Jur99], we may assume that L/L k is metabelian, which means (L 2 ) 2 ⊆ L k . Obviously L 2 = [L 1 , L 1 ] cannot be a diamond, but examples in [CMNS96] and [GMY01] show that L 3 can be, hence k = 3 is a possibility. Thus, from now on we assume k > 3, and so L 3 is not a diamond.…”

Section: The Degree Of the Second Diamond In A Thin Lie Algebramentioning

confidence: 99%

“…In particular, the next diamond after L 3 is L 5 , whence h = 2 in those cases. However, for the metabelian thin Lie algebras constructed in [GMY01] every homogeneous component except for L 2 is a diamond, and hence h = 1 for those.…”

Section: The Degree Of the Second Diamond In A Thin Lie Algebramentioning

confidence: 99%

Constituents of graded Lie algebras of maximal class and chain lengths of thin Lie algebras

Mattarei¹

2020

Preprint

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Thin Lie algebras are infinite-dimensional graded Lie algebras L = ∞ i=1 , with dim(L 1 ) = 2 and satisfying a covering property: for each i, each nonzero z ∈ L i satisfies [zL 1 ] = L i+1 . It follows that each homogeneous components L i is either one-or two-dimensional, and in the latter case is called a diamond. Hence L 1 is a diamond, and if there are no other diamonds then L is a graded Lie algebra of maximal class.We present simpler proofs of some fundamental facts on graded Lie algebras of maximal class, and on thin Lie algebras, based on a uniform method. Among else, we determine the possible values for the most fundamental parameter of such algebras, which is the dimension of their largest metabelian quotient.

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“…In this paper we study the class of thin algebras all of whose homogeneous components, except the second, have dimension 2. The only known examples of such algebras are those considered in [GMY01]. There it is proved that all metabelian thin Lie algebras belong to this class, and they are in one-to-one correspondence with the quadratic extensions of the field F. In particular, in one of the constructions in [GMY01], a thin metabelian Lie algebra is realised as a subalgebra over F of the tensor product of the unique infinite-dimensional metabelian Lie algebra of maximal class by a quadratic extension of F.…”

Section: Introductionmentioning

confidence: 99%

Thin subalgebras of Lie algebras of maximal class

Avitabile¹,

Caranti²,

Gavioli³

et al. 2021

Preprint

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For every field F which has a quadratic extension E we show there are non-metabelian infinite-dimensional thin graded Lie algebras all of whose homogeneous components, except the second one, have dimension 2. We construct such Lie algebras as F-subalgebras of Lie algebras M of maximal class over E. We characterise the thin Lie F-subalgebras of M generated in degree 1. Moreover we show that every thin Lie algebra L whose ring of graded endomorphisms of degree zero of L 3 is a quadratic extension of F can be obtained in this way. We also characterise the 2generator F-subalgebras of a Lie algebra of maximal class over E which are ideally r-constrained for a positive integer r.The first four authors are members of the Italian INdAM-GNSAGA, and thank the Department of Mathematics and its Applications of the University of Milano -Bicocca for the hospitality.A. Caranti is grateful to the Department of Mathematics of the University of Trento for its support. E. A. O'Brien has been supported by the Marsden Fund of New Zealand grant UOA 1626.

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Metabelian Thin Lie Algebras

Cited by 10 publications

References 5 publications

A sandwich in thin lie algebras

A sandwich in thin lie algebras

Constituents of graded Lie algebras of maximal class and chain lengths of thin Lie algebras

Thin subalgebras of Lie algebras of maximal class

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