Characterizing<i>n</i>-Isoclinism Classes of Lie Algebras

Salemkar, Ali Reza; Mirzaei, Fateme

doi:10.1080/00927870903117535

Cited by 19 publications

(12 citation statements)

References 9 publications

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“…Hence, we may ask under which conditions the converse to Shcur's and Baer's theorems are still true in case of Lie algebras. In [12], it is proved that if L is a finitely generated Lie algebra, then the converse to Baer's theorem is also true. The authors in [1] generalize this result in the case c = 1 and obtain an upper bound for the central factor of L. Indeed, is it proved that…”

Section: Introduction and Preliminarymentioning

confidence: 88%

A remark on the converse of Baer’s theorem for Lie algebras

Saeedi

2015

Rend. Circ. Mat. Palermo

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The classic theorem of Schur states that for a group G the finiteness of G/Z (G) implies the finiteness of G . Baer extends this theorem and shows that if G/Z n (G) is finite for a group G, the so is γ n+1 (G), where Z n (G) and γ n+1 (G) are the nth term of the upper central series and the (n+1)th term of lower central series of G, respectively. These results are extended to Lie algebras by Salemkar et. al. by showing that the converse of Baer's theorem remains hold when the corresponding Lie algebra is finitely generated. Generalizing this result, we also obtain a bound for the n-central factor of a Lie algebra.

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Section: Introduction and Preliminarymentioning

confidence: 88%

A remark on the converse of Baer’s theorem for Lie algebras

Saeedi

2015

Rend. Circ. Mat. Palermo

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“…Among other results, it is shown that if C is a relative n-isoclinism family of Lie algebras and (M, L) is a pair in C such that L is finitely generated and [M, n L] is of finite dimension, then there exists an n-stem pair (R, S) ∈ C such that and if M i is equal to L i , the map is shown by γ(n, L i ). Clearly, in this case we have the notion of n-isoclinism (see [10]) and when n = 1 one obtains the concept of isoclinism of the pairs of Lie algebras.…”

mentioning

confidence: 98%

Characterization of relativen-isoclinism of a pair of Lie algebras

Moghaddam

Saeedi

Tajnia

et al. 2015

Quaestiones Mathematicae

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In 1994, Moneyhun introduced and studied the concept of isoclinism in Lie algebras. Moghaddam and parvaneh in 2009 introduced and gave some properties of isoclinism of a pair of Lie algebras. In this paper we introduce the concept of relative n-isoclinism between two pairs of Lie algebras. They form equivalence classes and we show that each equivalent class contains an n-stem pair of Lie algebras. We also show that for a relative n-isoclinism family of Lie algebras C say, consists of (M, L) such that L is finitely generated and [M,n L] is finite dimension, then there exists an n-stem pair of Lie algebras (R, S) ∈ C such that dim([R,n−1 S] = min{dim([M,n−1 L]); f or all (M, L) ∈ C}. Mathematics Subject Classification (2010): Primary 17B30, 17B60, 17B99; Secondary 20E99, 20D15.Key words: Lie algebra, n-isoclinism, relative n-isoclinism, n-stem Lie algebra. Introduction. We consider all Lie algebras over a fixed fieldF and assume M to be an ideal of a Lie algebra L, with the Lie product [ , ]. Then (M, L) is said to be a pair of Lie algebras. The lower and the upper central series of L are defined inductively by L 1 = L , L n+1 = [L n , L], for all n ≥ 1, and Z 0 (L) = {0}, Z 1 (L) = Z(L), and Z n+1 (L)/Z n (L) = Z(L/Z n (L)), for all n ≥ 0. Also one may consider Quaestiones Mathematicae is co-published by NISC (Pty) Ltd and Taylor & Francis 28 M.R.R. Moghaddam, F. Saeedi, S. Tajnia and B. Veisi [M, L] = ⟨[m, l]; m ∈ M, l ∈ L⟩, Z(M, L) = {m ∈ M ; [m, l] = 0, ∀l ∈ L} = Z(L) ∩ M , to be the commutator and the centre of the pair of Lie algebras (M, L), respectively. We also denote [M, L, . . . , L n−times

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“…in the top and bottom horizontal maps, respectively (see [8,13] for more details). If there exists such an n-isoclinism, we say that L is n-isoclinic to H and write L ∼ n H.…”

Section: An Upper Bound For the Dimension Of L/z N (L)mentioning

confidence: 99%

“…Note that the first author and Saeedi proved the above result for n = 1 in [2]. Here, we use the idea of n-isoclinism discussed in [13], which gives us a different method from the technique applied in [2].…”

Section: Introductionmentioning

confidence: 99%

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On the Structure of Factor Lie Algebras

Arabyani¹,

Panbehkar²,

Safa³

2017

Bulletin of the Korean Mathematical Society

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Abstract. The Lie algebra analogue of Schur's result which is proved by Moneyhun in 1994, states that if L is a Lie algebra such that dimL/Z(L) = n, then dimL (2) = 1 2 n(n − 1) − s for some non-negative integer s. In the present paper, we determine the structure of central factor (for s = 0) and the factor Lie algebra L/Z 2 (L) (for all s ≥ 0) of a finite dimensional nilpotent Lie algebra L, with n-dimensional central factor. Furthermore, by using the concept of n-isoclinism, we discuss an upper bound for the dimension of L/Zn(L) in terms of dimL (n+1) , when the factor Lie algebra L/Zn(L) is finitely generated and n ≥ 1.

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Characterizingn-Isoclinism Classes of Lie Algebras

Cited by 19 publications

References 9 publications

A remark on the converse of Baer’s theorem for Lie algebras

A remark on the converse of Baer’s theorem for Lie algebras

Characterization of relativen-isoclinism of a pair of Lie algebras

On the Structure of Factor Lie Algebras

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