Nilpotent Lie Algebras Of Dimension Research Articles

Une propriété topologique de l'ensemble des algèbres de Lie caractéristiquement nilpotentes

In the variety of finite dimensional nilpotent Lie algebras over the field of complex numbers, the set of characteristically nilpotent Lie algebras is not closed. In this Note we show that it is also not open. To cite this article: J.M. Ancochea Bermúdez et al., C. R. Acad. Sci. Paris, Ser. I 337 (2003).

Rank-one Einstein solvmanifolds of dimension 7

We use a variational approach to determine explicitly all the 7-dimensional rank-one Einstein solvmanifolds. We prove that each one of the 34 nilpotent Lie algebras of dimension 6 admits a rank-one solvable extension which can be endowed with an Einstein left-invariant Riemannian metric. This also produces 34 Ricci soliton metrics on R 6 which are homogeneous.

Estimates on binomial sums of partition functions

Let p(n) denote the partition function and define where p(0)= 1. We prove that p(n,k) is unimodal and satisfies for fixed n≥ 1 and all 1≤k≤n. This result has an interesting application: the minimal dimension of a faithful module for a k-step nilpotent Lie algebra of dimension n is bounded by p(n,k) and hence by , independently of k. So far only the bound n n −1 was known. We will also prove that for n≥ 1 and .

Classification of nilpotent lie algebras of dimension nine whose maximum abelian ideal is of dimension seven

The classification of Nilpotent Lie Algebras has been partially achieved up to dimension eight but this specific problem remained unsolved, since it required an enormous amount of calculations. Nowadays, with the aid of modern computers, it can be solved by designing the proper algorithms. The aim of this paper is to provide the results of the classification of Nilpotent Lie Algebras of dimension nine, whose maximum abelian ideal is of dimension seven.

On a refinement of Ado's theorem

In this paper we study the minimal dimension $ \mu (g) $ of a faithful g-module for n-dimensional Lie algebras g. This is an interesting invariant of g which is difficult to compute. It is desirable to obtain good bounds for $ \mu (g) $ , especially for nilpotent Lie algebras. We will determine here $ \mu (g) $ for certain Lie algebras and prove upper bounds in general. For nilpotent Lie algebras of dimension n, the bound n n + 1 is known. We now obtain $ {\mu ({g})\le{\alpha \over \sqrt {n}}2^n} $ with some constant $ {\alpha \sim 2.76287} $ .

On the cohomology of a class of nilpotent Lie algebras

Let g denote a finite dimensional nilpotent Lie algebra over ℂ containing an Abelian ideal a of codimension 1, with z ∈ g/a. We give a combinatorial description of the Betti numbers of g in terms of the Jordan decomposition induced by ad(z)|a. As an application we prove that the filiform-nilpotent Lie algebras arising in the case t = 1 have unimodal Betti numbers.

Sur les composantes irréductibles de la varieté des lois d'algèbres de Lie nilpotentes

In this paper we determine all the components fo the variety of complex nilpotent Lie algebras of dimension 8. The technique is similar to that used for the smaller dimensions. But in this case big difficulties appear resulting from the complexity of the calculus. Thus we have developed a formal calculus for use in computers. The case of the dimension 8 is interesting because it is the first case where many nonfiliform components appear.

Blocks and projective modules for reduced universal enveloping algebras of a nilpotent restricted Lie algebra

In this note we re-derive and generalize a dimension formula for block ideals of reduced universal enveloping algebras of a finite dimensional nilpotent restricted Lie algebra due to V. M. Petrogradsky and provide related results on the corresponding projective indecomposable modules. In particular, the dimension and the composition length of every projective indecomposable module is always a p-power. The example at the end of the paper shows that even for algebraically closed fields our results are more complicated than the analoguous results for finite p-nilpotent groups (cf. [8, Proposition 2.4 (b)]). In the following F will always be an algebraically closed field of prime characteristic p. For other undefined notation and some fundamental facts from restricted Lie algebra theory we refer to [11] or [5]. According to a classical result of H. Zassenhaus [11, Corollary 1.4.4 and Proposition 1.4.6], for every simple module S of a finite dimensional nilpotent Lie algebra L, there exists a function co : L -~ F such that

On the Cohomology of Infinite Dimensional Nilpotent Lie Algebras

In the paper one- and two-dimensional cohomology is compared (or finite and infinite nilpotent Lie algebras, with coefficients in the adjoint representation. It turns out that, because the adjoint representation is not a highest weight representation in infinite dimension, the considered cohomology shows basic differences.

On the varieties of nilpotent Lie algebras of dimension 7 and 8

Let N n be the variety of n-dimensional complex nilpotent Lie algebras. We know that this algebraic variety is reducible for n≥11 and irreducible for n≤6. In this work we prove that N 7 is composed of two algebraic components and that N 8 is also reducible.

Some nilpotent Lie algebras of even dimension

For each even dimension greater than or equal to 8, an infinite family of 3-step nilpotent Lie algebras over ℂ is constructed. In dimension m, the family contains isomorphism classes parameterised locally by approximately m3/48 essential parameters.One particular case is studied further to get some global information about the variety of all nilpotent Lie algebras of dimension 8. Using the results obtained here, and some known facts, it is shown that there is a component consisting of algebras not having minimal possible central dimensions.

Constants of Lie algebra actions

Let R be an associative algebra over a field k and let L be a Lie algebra over k acting on R as derivations. We will be interested in studying the relationship between R and the ring of constants RL. Our situation is a special case of the study of rings of invariants of Hopf algebra actions and is also analogous to another special case of Hopf algebra actions, namely, the study of fixed rings of finite group actions. Many of the results in this paper are analogs of results on group actions and give rise to more general questions regarding Hopf algebra actions. In Section 1, we study actions on non-nilpotent algebras. We first show that any algebraic derivation has non-zero constants and then prove that any finite dimensional nilpotent Lie algebra of algebraic derivations must act with non-zero constants. It then follows, as a corollary, that any finite dimensional nilpotent restricted Lie algebra must act with non-zero constants. These are analogs of results on groups proved in [ 12, 5). In Section 2, we prove a necessary condition for the existence of nonzero invariants of certain Hopf algebra actions. We then use this result to prove the converse of our theorem on nilpotent restricted Lie algebras. We conclude the section with several questions on general Hopf algebra actions.

Sur les algèbres de Lie nilpotentes de dimension ⩽ 7

We consider the embedding problem of a nilpotent Lie algebra into a stratified one; we construct all nilpotent Lie algebras of dimension ⩽ 7 having a fixed Lie algebra of codimension 1, and we get among other results, a new classification of 6-dimensional nilpotent Lie algebras. This study has applications on the analysis of deformations of the internal space of 11 dimensional Kaluza-Klein theories (non abelian theories) and in 11-dimensional supergravity.

Sur les variétés d'Algèbres de Lie de dimension ⩽7

The varieties of Lie algebras of dimension m ⩽ 7 over an algebraically closed field of characteristic 0 are studied. The well-known classification of nilpotent Lie algebras of dimension ⩽6 and the density for m ⩽ 7 of the subspace consisting of laws which are decomposable (or splittable) are used. The irreductible components and the open orbits are determined.

Polynome de Hilbert-Samuel dans les algebres enveloppantes et les algebres de groupe

An analogue of the Hilbert-Samuel polynomial is considered with respect to the augmentation ideal of the enveloping algebra of a finite dimensional nilpotent Lie algebra and the group ring of a finitely generated, torsion-free nilpotent group. Then, the Hilbert series of finitely generated modules are rational.

Reflexive primes, localization and primary decomposition in maximal orders

If R is a maximal order and P a reflexive prime ideal of R, then the Goldie localization of R at P is shown to be the (partial) quotient ring of R with respect to the Ore set C(P) = {r E RIrx E P X x E P). This is accomplished by introducing new symbolic powers of the prime P which agree with Goldie's symbolic powers. As a consequence, whenever P is a reflexive prime ideal of R and p(n) the nth (Goldie) symbolic power of P, then an ideal B is reflexive if and only if B= n,= Pl(n') for uniquely determined reflexive primes Pi and integers ni > 0. More generally, each bounded essential right (left) ideal is shown to have a reduced primary decomposition and an explicit determination of the components is given in terms of the bound of the ideal. 1. In [67], Goldie gave a method for constructing a local ring Rp given an arbitrary two-sided Noetherian ring R and a prime ideal P, which for commutative R coincides with the usual commutative localization at P. Just as in the commutative case, Goldie's localization at P utilizes the set C(P) = {x E RI x + P is regular in R/P}. There are many instances where C(P) is an Ore set: R any commutative Noetherian ring, R an Asano order (Michler [69], Hajarnavis and Lenagan [71], Kuzmanovich [72]), R a hereditary Noetherian prime ring with P nonidempotent (Chatters and Ginn [72]), R the enveloping algebra of finite dimensional nilpotent Lie algebra (McConnell [68]) and R = AG, A a Noetherian prime ring of characteristic 0, G a finite group and P the augmentation ideal of R (Michler [72]). The object of the first section of this paper is to show that whenever R is a Noetherian maximal order and P a reflexive prime, C(P) is an Ore set of regular elements of R. Such rings include Asano orders and most classical maximal orders. To accomplish this, we first define a new symbolic power, denoted p(n), for a reflexive prime ideal of R. Actually, all of the results of this paper are valid for nonmaximal R and reflexive primes P satisfying P* P 0 P and PP* C R. Next, we establish the basic properties of these symbolic powers, ultimately showing that they are identical to the Hn of Goldie [67] (see also Michler [72] for another equivalent formulation). Although we use the Received by the editors February 19, 1975 and, in revised form, October 20, 1975. AMS (MOS) subject classifications (1970). Primary 16A08, 16A66; Secondary 16A 10, 16A46.