Minimal algebras and 2-step nilpotent Lie algebras in dimension 7

We use the methods of Bazzoni and Muñoz (Trans Am Math Soc 364:1007–1028, 2012) to give a classification of 7-dimensional minimal algebras, generated in degree 1, over any field ${\mathbf{k}}$ of characteristic ${{\rm char}(\mathbf{k})\neq 2}$ , whose characteristic filtration has length 2. Equivalently, we classify 2-step nilpotent Lie algebras in dimension 7. This classification also recovers the real homotopy type of 7-dimensional 2-step nilmanifolds.     

Free 2-step nilpotent Lie algebras and indecomposable representations

Given an algebraically closed field F of characteristic 0 and an F-vector space V, let L(V)?=?V⊕Λ²(V) denote the free 2-step nilpotent Lie algebra associated to V. In this paper, we classify all uniserial representations of the solvable Lie algebra 𝔤?=??x??L(V), where x acts on V via an arbitrary invertible Jordan block.     

A refinement of the toral rank conjecture for 2-step nilpotent Lie algebras

It is known that the total (co)-homoloy of a 2-step nilpotent Lie algebra is at least , where is the center of . We improve this result by showing that a better lower bound is , where and is a complement of in . Furthermore, we provide evidence that this is the best possible bound of the form .

     

Classes of 2-step nilpotent lie algebras

We give a characterization of the Lie algebras of H-type independent of the inner product used in the definition. We classify the real 2-step nilpotent Lie algebras with 2-dimensional center. Using these results we give examples of regular Lie algebras that are not H-type.     

2-step nilpotent Lie groups of higher rank

We construct a family of simply connected 2-step nilpotent Lie groups of higher rank such that every geodesic lies in a flat. These are as Riemannian manifolds irreducible and arise from real representations of compact Lie algebras. Moreover we show that groups of Heisenberg type do not even infinitesimally have higher rank. Received: 2 July 2001 / Revised version: 19 October 2001     

Complete commutative sets of polynomials for solvable Lie algebras of dimension six and nilpotent Lie algebras of dimension seven

A complete commutative set of polynomials is constructed using Sadetov’s method on the coalgebra of each real 6-dimensional solvable non-nilpotent Lie algebra and of each real 7-dimensional nilpotent Lie algebra.     

On dimension of the Schur multiplier of nilpotent Lie algebras

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) $ 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.     

Characteristically nilpotent Lie algebras

Translated from Algebra i Logika, Vol. 28, No. 6, pp. 722–737, November–December, 1989.     

Representations of nilpotent Lie algebras

Let (L,[p]) a finite dimensional nilpotent restricted Lie algebra of characteristic p 3 3, c ? L^*p \geq 3, \chi \in L^* a linear form. In this paper we study the representation theory of the reduced universal enveloping algebra u(L,c)u(L,\chi ). It is shown that u(L,c)u(L,\chi ) does not admit blocks of tame representation type. As an application, we prove that the nonregular AR-components of u(L,c)u(L,\chi ) are of types \Bbb Z [A_￥ ]\Bbb Z [A_\infty ] or \Bbb Z [A_n]/(t)\Bbb Z [A_n]/(\tau ).     

Decompositions of nilpotent Lie algebras

It is proved that decompositions of nilpotent Lie algebras are global. In the complex case, nilpotency is also a necessary condition for every decomposition to be global. The results obtained are applied to the classification of complex homogeneous spaces of simply connected nilpotent Lie groups.Translated from Matematicheskie Zametki, Vol. 23, No. 1, pp. 27–30, January, 1978.In conclusion, the author would like to thank A. L. Onishchik for his interest in this research.     

The classification of two step nilpotent complex Lie algebras of dimension 8

A Lie algebra g is called two step nilpotent if g is not abelian and [g, g] lies in the center of g. Two step nilpotent Lie algebras are useful in the study of some geometric problems, such as commutative Riemannian manifolds, weakly symmetric Riemannian manifolds, homogeneous Einstein manifolds, etc. Moreover, the classification of two-step nilpotent Lie algebras has been an important problem in Lie theory. In this paper, we study two step nilpotent indecomposable Lie algebras of dimension 8 over the field of complex numbers. Based on the study of minimal systems of generators, we choose an appropriate basis and give a complete classification of two step nilpotent Lie algebras of dimension 8.     

Nilpotent Lie algebras with 2-dimensional commutator ideals

We classify all (finitely dimensional) nilpotent Lie k-algebras h with 2-dimensional commutator ideals h^′, extending a known result to the case where h^′ is non-central and k is an arbitrary field. It turns out that, while the structure of h depends on the field k if h^′ is central, it is independent of k if h^′ is non-central and is uniquely determined by the dimension of h. In the case where k is algebraically or real closed, we also list all nilpotent Lie k-algebras h with 2-dimensional central commutator ideals h^′ and dim_kh?11.     

Lie algebras with quadratic dimension equal to 2

The quadratic dimension of a Lie algebra is defined as the dimension of the linear space spanned by all its invariant non-degenerate symmetric bilinear forms. We prove that a quadratic Lie algebra with quadratic dimension equal to 2 is a local Lie algebra, this is to say, it admits a unique maximal ideal. We describe local quadratic Lie algebras using the notion of double extension and characterize those with quadratic dimension equal to 2 by the study of the centroid of such Lie algebras. We also give some necessary or sufficient conditions for a Lie algebra to have quadratic dimension equal to 2. Examples of local Lie algebras with quadratic dimension larger than 2 are given.