Proof of a Conjecture of Wiegold for nilpotent Lie algebras (original) (raw)
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Capability of Nilpotent Lie Algebras of Small Dimension
Bulletin of the Iranian Mathematical Society
Given a nilpotent Lie algebra L of dimension ≤ 6 on an arbitrary field of characteristic = 2, we show a direct method which allows us to detect the capability of L via computations on the size of its nonabelian exterior square L ∧ L. For dimensions higher than 6, we show a result of general nature, based on the evidences of the low dimensional case, focusing on generalized Heisenberg algebras. Indeed we detect the capability of L ∧ L via the size of the Schur multiplier M (L/Z ∧ (L)) of L/Z ∧ (L), where Z ∧ (L) denotes the exterior center of L. 1. Statement of the results and terminology Throughout this paper all Lie algebras are considered over a prescribed field F and [ , ] denotes the Lie bracket. Let L and K be two Lie algebras. Following [6, 7], an action of L on K is an F-bilinear map L × K → K given by (l, k) → l k satisfying [l,l ′ ] k = l (l ′ k) − l ′ (l k) and l [k, k ′ ] = [ l k, k ′ ] + [k, l k ′ ], for all l, l ′ ∈ L and k, k ′ ∈ K. These actions are compatible, if k l k ′ = k ′ l k and l k l ′ = l ′ k l for all l, l ′ ∈ L and k, k ′ ∈ K. Now L ⊗ K is the Lie algebra generated by the symbols l ⊗ k subject to the relations c(l ⊗ k) = cl ⊗ k = l ⊗ ck, (l + l ′) ⊗ k = l ⊗ k + l ′ ⊗ k,
New bounds on the betti numbers of nilpotent lie algebras
Communications in Algebra, 1997
We provide new upper and lower bounds for the Betti numbers of nilpotent Lie algebras. As an application, we prove the toral rank conjecture (TRC) for nilmanifolds of dimension at most 14 and for a small class of coformal spaces. We also give a new, direct proof of the result of Deninger and Singhof that the TRC is true for 2-step nilpotent Lie algebras.
On nilpotency in the Ado-Harish-Chandra theorem on Lie algebra representations
Proceedings of the American Mathematical Society, 1986
Let L L be a finite-dimensional Lie algebra, over an arbitrary field, and regard L L as embedded in its enveloping algebra UL. Theorem. If K K is an ideal of L L and K K is nilpotent of class q q , then for any r r there exists a finite-dimensional representation ρ \rho of L L which vanishes on all products (in UL) of ≥ q r + 1 \geq qr + 1 elements of K K and is faithful on the subspace of UL spanned by all products of ≤ r \leq r elements of L L . This result sharpens (with respect to nilpotency) the Ado-Iwasawa theorem on the existence of faithful representations and the Harish-Chandra theorem on the existence of representations separating points of UL.
Characterizing nilpotent Lie algebras that satisfy on converse of the Schur's theorem
arXiv (Cornell University), 2022
Let L be a finite dimensional nilpotent Lie algebra and d be the minimal number generators for L/Z(L). It is known that dim L/Z(L) = d dim L 2 − t(L) for an integer t(L) ≥ 0. In this paper, we classify all finite dimensional nilpotent Lie algebras L when t(L) ∈ {0, 1, 2}. We find also a construction, which shows that there exist Lie algebras of arbitrary t(L).
2016
Let L be an n-dimensional non-abelian nilpotent Lie algebra. Niroomand and Russo (2011) proved that dimM(L) = 1 2 (n − 1)(n − 2) + 1 − s(L), where M(L) is the Schur multiplier of L and s(L) is a non-negative integer. They also characterized the structure of L, when s(L) = 0. Assume that (N, L) is a pair of finite dimensional nilpotent Lie algebras, in which L is non-abelian and N is an ideal in L and also M(N, L) is the Schur multiplier of the pair (N, L). If N admits a complement K say, in L such that dimK = m, then dimM(N, L) = 1 2 (n 2 + 2nm − 3n − 2m + 2) + 1 − (s(L) − t(K)), where t(K) = 1 2 m(m − 1)−dimM(K). In the present paper, we characterize the pairs (N, L), for which 0 t(K) s(L) 3. In particular, we classify the pairs (N, L) such that L is a non-abelian filiform Lie algebra and 0 t(K) s(L) 17. Abstract. Let L be an n-dimensional non-abelian nilpotent Lie algebra. Niroomand and Russo (2011) proved that dimM(L) = 1 2 (n − 1)(n − 2) + 1 − s(L), where M(L) is the...
On Dimension of the Schur Multiplier of Nilpotent Lie Algebras
Central European Journal of Mathematics
Let L be an n-dimensional non-abelian nilpotent Lie algebra and $ s(L) = \frac{1} {2}(n - 1)(n - 2) + 1 - \dim M(L) $ where M(L) is the Schur multiplier of L. In [Niroomand P., Russo F., A note on the Schur multiplier of a nilpotent Lie algebra, Comm. Algebra (in press)] it has been shown that s(L) ≥ 0 and the structure of all nilpotent Lie algebras has been determined when s(L) = 0. In the present paper, we will characterize all finite dimensional nilpotent Lie algebras with s(L) = 1; 2.
On the Tensor Square of Non-Abelian Nilpotent Finite Dimensional Lie Algebras
Arxiv preprint arXiv:1002.2563, 2010
For every finite p-group G of order p n with derived subgroup of order p m , Rocco [N.R. Rocco, On a construction related to the nonabelian tensor square of a group, Bol. Soc. Brasil. Mat. 1 (1991), pp. 63–79] proved that the order of tensor square of G is at most p n(n−m). This upper bound has been improved recently by the author [P. Niroomand, On the order of tensor square of non abelian prime power groups (submitted)]. The aim of this article is to obtain a similar result for a non-abelian nilpotent Lie algebra of finite dimension. More precisely, for any given n-dimensional non-abelian nilpotent Lie algebra L with derived subalgebra of dimension m we have dim(L L) ≤ (n − m)(n − 1) + 2. Furthermore for m = 1, the explicit structure of L is given when the equality holds.