Finite-dimensional Nilpotent Lie Algebra Research Articles

The Nash–Moser theorem of Hamilton and rigidity of finite dimensional nilpotent Lie algebras

We apply the Nash–Moser theorem for exact sequences of R. Hamilton to the context of deformations of Lie algebras and we discuss some aspects of the scope of this theorem in connection with the polynomial ideal associated to the variety of nilpotent Lie algebras. This allows us to introduce the space Hk-nil2(g,g), and certain subspaces of it, that provide fine information about the deformations of g in the variety of k-step nilpotent Lie algebras.Then we focus on degenerations and rigidity in the variety of k-step nilpotent Lie algebras of dimension n with n≤7 and, in particular, we obtain rigid Lie algebras and rigid curves in the variety of 3-step nilpotent Lie algebras of dimension 7. We also recover some known results and point out a possible error in a published article related to this subject.

2-Local automorphisms on finite-dimensional Lie algebras

We prove that every 2-local automorphism on a finite-dimensional semi-simple Lie algebra L over an algebraically closed field of characteristic zero is an automorphism. We also show that each finite-dimensional nilpotent Lie algebra L with dim⁡L≥2 admits a 2-local automorphism which is not an automorphism.

Nilradicals of parabolic subalgebras admitting symplectic structures

In this paper we describe all the nilradicals of parabolic subalgebras of split real simple Lie algebras admitting symplectic structures.The main tools used to obtain this list are Kostant's description of the highest weight vectors (hwv) of the cohomology of these nilradicals and some necessary conditions obtained for the g-hwv's of H2(n) for a finite dimensional real symplectic nilpotent Lie algebra n with a reductive Lie subalgebra of derivations g acting on it.

Some upper bounds on the dimension of the Schur multiplierof a pair of nilpotent Lie algebras

Let $(L,N)$ be a pair of Lie algebras where $N$ is an ideal of the finite dimensional nilpotent Lie algebra $L$. Some upper bounds on the dimension of the Schur multiplier of $(L,N)$ are obtained without considering the existence of a complement for $N$. These results are applied to derive a new bound on the dimension of the Schur multiplier of a nilpotent Lie algebra.

Local derivations on finite-dimensional Lie algebras

We prove that every local derivation on a finite-dimensional semisimple Lie algebra over an algebraically closed field of characteristic zero is a derivation. We also give examples of finite-dimensional nilpotent Lie algebras L with dim⁡L≥3 which admit local derivations which are not derivations.

Grušin operators, Riesz transforms and nilpotent Lie groups

We establish that the Riesz transforms of all orders corresponding to the Grusin operator \(H_N=-\nabla _{x}^2-|x|^{2N}\,\nabla _{y}^2\), and the first-order operators \((\nabla _{x},x^\nu \,\nabla _{y})\) where \(x\in \mathbf{R}^n\), \(y\in \mathbf{R}^m\), \(N\in \mathbf{N}_+\), and \(\nu \in \{1,\ldots ,n\}^N\), are bounded on \(L_p(\mathbf{R}^{n+m})\) for all \(p\in \langle 1,\infty \rangle \) and are also weak-type (1, 1). Moreover, the transforms of order less than or equal to \(N+1\) corresponding to \(H_N\) and the operators \((\nabla _{x}, |x|^N\nabla _{y})\) are bounded on \(L_p(\mathbf{R}^{n+m})\) for all \(p\in \langle 1,\infty \rangle \). But if N is odd all transforms of order \(N+2\) are bounded if and only if \(p\in \langle 1,n\rangle \). The proofs are based on the observation that the \((\nabla _{x},x^\nu \,\nabla _{y})\) generate a finite-dimensional nilpotent Lie algebra, the corresponding connected, simply connected, nilpotent Lie group is isometrically represented on the spaces \(L_p(\mathbf{R}^{n+m})\) and \(H_N\) is the corresponding sublaplacian.

On nilpotent Lie algebras of small breadth

A Lie algebra L is said to be of breadth k if the maximal dimension of the images of left multiplication by elements of the algebra is k. In this paper we give characterization of finite dimensional nilpotent Lie algebras of breadth less than or equal to two. Furthermore, using these characterizations we determined the isomorphism classes of these algebras.

On Borel complexity of the isomorphism problems for graph related classes of Lie algebras and finite p-groups

We reduce the isomorphism problem for undirected graphs without loops to the isomorphism problems for some class of finite-dimensional 2-step nilpotent Lie algebras over a field and for some class of finite p-groups. We show that the isomorphism problem for graphs is harder than the two latter isomorphism problems in the sense of Borel reducibility. A computable analogue of Borel reducibility was introduced by S. Coskey, J. D. Hamkins, and R. Miller (2012). A relation of the isomorphism problem for undirected graphs to the well-known problem of classifying pairs of matrices over a field (up to similarity) is also studied.

2-Local derivations on finite-dimensional Lie algebras

We prove that every 2-local derivation on a finite-dimensional semi-simple Lie algebra L over an algebraically closed field of characteristic zero is a derivation. We also show that a finite-dimensional nilpotent Lie algebra L with dim⁡L≥2 admits a 2-local derivation which is not a derivation.

Verification of Some Properties of the C-nilpotent Multiplier in Lie Algebras

The purpose of this paper is to obtain some inequalities and certain bounds for the dimension of the c-nilpotent multiplier of finite dimensional nilpotent Lie algebras and their factor Lie algebras. Also, we give an inequality for the dimension of the c-nilpotent multiplier of L connected with dimension of the Lie algebras γd (L) and L / Zd−1 (L) . Finally, we compare our results with the previously known result.

A lower bound for faithful representations of nilpotent Lie algebras

Given a finite-dimensional nilpotent Lie algebra , let (resp. ) be the minimal dimension of such that admits a faithful representation (resp. nilrepresentation) on . In this paper, we present a lower bound for for a -step nilpotent Lie algebra over a field of characteristic zero. Our bound is given as the minimum of a quadratically constrained linear optimization problem, it works for arbitrary and takes into account a given filtration of . We present some estimates of this minimum which leads to a very explicit lower bound for that involves the dimensions of and its centre. This bound allows us to obtain for some families of nilpotent Lie algebras.

Some remarks on graded nilpotent Lie algebras and the Toral Rank Conjecture

If 𝔫 is a [Formula: see text]-graded nilpotent finite-dimensional Lie algebra over a field of characteristic zero, a well-known result of Deninger and Singhof states that dim H*(𝔫) ≥ L(p) where p is the polynomial associated to the grading and L(p) is the sum of the absolute values of the coefficients of p. From this result they derived the Toral Rank Conjecture (TRC) for 2-step nilpotent Lie algebras. An algebraic version of the TRC states that dim H*(𝔫) ≥ 2 dim (ℨ) for any finite-dimensional nilpotent Lie algebra 𝔫 with center ℨ. The TRC is more than 25 years old and remains open even for [Formula: see text]-graded 3-step nilpotent Lie algebras. Investigating to what extent the bound given by Deninger and Singhof could help to prove the TRC in this case, we considered the following two questions regarding a nilpotent Lie algebra 𝔫 with center ℨ: (A) If 𝔫 admits a [Formula: see text]-grading [Formula: see text], such that its associated polynomial p satisfies L(p) > 2 dim ℨ, does 𝔫 admit a ℤ+-grading [Formula: see text] such that its associated polynomial p′ satisfies L(p′) > 2 dim ℨ? (B) If 𝔫 is r-step nilpotent admitting a grading 𝔫 = 𝔫1 ⊕ 𝔫2 ⊕ ⋯ ⊕ 𝔫k such that its associated polynomial p satisfies L(p) > 2 dim ℨ, does 𝔫 admit a grading [Formula: see text] such that its associated polynomial p′ satisfies L(p′) > 2 dim ℨ? In this paper we show that the answer to (A) is yes, but the answer to (B) is no.

Characterization of Nilpotent Lie Algebras Pair by Their Schur Multipliers

Let (N, L) be a pair of finite dimensional nilpotent Lie algebras, in which N is an ideal in L and ℳ(N, L) is the Schur multiplier of the pair (N, L). In 2011, the second author and colleagues proved that if N admits a complement K say, in L with dim N = n and dim K = m, then , where t(N, L) is a non-negative integer. In the present article, we characterize the pairs (N, L) for which t(N, L) = 0, 1, 2, 3, and 4. Also, in a special case when L is a filiform Lie algebra, we determine all pairs (N, L), when t(N, L) = 0, 1,…, 10.

Nilpotent Lie and Leibniz Algebras

We extend results on finite dimensional nilpotent Lie algebras to Leibniz algebras and counterexamples to others are found. One generator algebras are used in these examples and are investigated further.

The dimension of the c-nilpotent multiplier

The purpose of this paper is to obtain some inequalities for the dimension of the c-nilpotent multiplier of finite dimensional nilpotent Lie algebras and their factor Lie algebras. Finally, we compare our results to upper bound given in Salemkar et al. (2009) [10].

BOUNDS FOR THE DIMENSION OF THE SCHUR MULTIPLIER OF A PAIR OF NILPOTENT LIE ALGEBRAS

Let (L, N) be a pair of finite-dimensional nilpotent Lie algebras, in which N is an ideal in L. In this paper we derive some inequalities for the dimension of the Schur multiplier of the pair (L, N) in terms of the dimension of the commutator subalgebra [L, N].

Some inequalities for the dimension of the Schur multiplier of a pair of (nilpotent) Lie algebras

The purpose of this paper is to obtain some inequalities for the dimension of the Schur multiplier of a pair of finite dimensional Lie algebras and their factor Lie algebras. Moreover, we present some inequalities for the Schur multiplier of a pair of finite dimensional nilpotent Lie algebras.

Relatively Free Nilpotent Torsion-Free Groups and Their Lie Algebras

Let K be a field of characteristic zero. For a torsion-free finitely generated nilpotent group G, we naturally associate four finite dimensional nilpotent Lie algebras over K, ℒ K (G), grad(ℓ)(ℒ K (G)), grad(g)(exp ℒ K (G)), and L K (G). Let 𝔗 c be a torsion-free variety of nilpotent groups of class at most c. For a positive integer n, with n ≥ 2, let F n (𝔗 c ) be the relatively free group of rank n in 𝔗 c . We prove that ℒ K (F n (𝔗 c )) is relatively free in some variety of nilpotent Lie algebras, and ℒ K (F n (𝔗 c )) ≅ L K (F n (𝔗 c )) ≅ grad(ℓ)(ℒ K (F n (𝔗 c ))) ≅ grad(g)(exp ℒ K (F n (𝔗 c ))) as Lie algebras in a natural way. Furthermore, F n (𝔗 c ) is a Magnus nilpotent group. Let G 1 and G 2 be torsion-free finitely generated nilpotent groups which are quasi-isometric. We prove that if G 1 and G 2 are relatively free of finite rank, then they are isomorphic. Let L be a relatively free nilpotent Lie algebra over ℚ of finite rank freely generated by a set X. Give on L the structure of a group R, say, by means of the Baker–Campbell–Hausdorff formula, and let H be the subgroup of R generated by the set X. We show that H is relatively free in some variety of nilpotent groups; freely generated by the set X, H is Magnus and L ≅ ℒℚ(H) ≅ L ℚ(H) as Lie algebras. For relatively free residually torsion-free nilpotent groups, we prove that ℒ K and L K are isomorphic as Lie algebras. We also give an example of a finitely generated Magnus nilpotent group G, not relatively free, such that ℒℚ(G) is not isomorphic to L ℚ(G) as Lie algebras.

A restriction on the Schur multiplier of nilpotent Lie algebras

An improvement of a bound of Yankosky (2003) is presented in this paper, thanks to a restriction which has been recently obtained by the authors on the Schur multiplier M(L) of a finite dimensional nilpotent Lie algebra L. It is also described the structure of all nilpotent Lie algebras such that the bound is attained. An important role is played by the presence of a derived subalgebra of maximal dimension. This allows precision on the size of M(L). Among other results, applications to the non-abelian tensor square L ⊗ L are illustrated.

On dimension of the Schur multiplier of nilpotent Lie algebras

Abstract 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.