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Classification des algèbres de Lie nilpotentes complexes de dimension 7. (Classification of nilpotent complex Lie algebras of dimension 7)
Abstract
The authors give a complete list of the 7-dimensional complex nilpotent Lie algebras. This classification is obtained by using an invariant of nilpotent Lie algebras, called a characteristic sequence and defined by the maximum of the Segre symbols of the nilpotent linear maps ad x with x in the complement of the derived subalgebra. This invariant was introduced by them in an earlier paper
[C. R. Acad. Sci. Paris Ser. I Math. 302 (1986), no. 17, 611–613; in which they determined the nilpotent complex Lie algebras corresponding to the characteristic sequences (6, 1) and (5, 1, 1). The paper under review contains no proofs; for details the authors refer to another article [the authors, “Classification des algebres de Lie nilpotentes de dimension 7”, Univ. Louis Pasteur, Strasbourg, 1986; per bibl.].
... Since Safiullina's first attempt to classify all 7-dimensional nilpotent Lie algebras there have been a number of works in that direction, including this one. Various tactics have been implemented [1,14,18]. Second, in characteristic p, the Baker-Campbell-Hausdorff formula puts a group structure on a nilpotent Lie algebra. A project similar to this one has been carried out in all positive characteristics using linear methods with the help of computers [29]. ...
... I found no uniform way of doing this. If the first center is small, an inductive approach makes sense: L/Z contains a lot of information about L. From the multiplication for L/Z one knows the entire multiplication 1 The proof of the lemma turns out to be somewhat complicated. It is done by toying around with the inclusions and projections associated with the two decompositions, starting with one of the nonabelian Afs. ...
... As mentioned in the Introduction, this is one of three recent attempts to solve the classification problem in dimension 7. The others are due to Romdhani [18] and Ancochea-Bermudez and Goze [1]. Roger Carles has scrutinized these three works prior to publication, and has identified several mistakes in all of them. ...
All 7-dimensional nilpotent Lie algebras over ℂ are determined by elementary methods. A multiplication table is given for each isomorphism class. Distinguishing features are given, proving that the algebras are pairwise nonisomorphic. Moduli are given for the infinite families which are indexed by the value of a complex parameter.
... in particular, in [1,16,18,23,25,30]. There is no known classification of nilpotent algebras of dimensions greater than 7, even over the field of complex numbers C. ...
... The following lemma is almost obvious, and so we skip the proof. (1) = z(L). ...
We classify the non-degenerate two-step nilpotent Lie algebras of dimension 8 over the field of real numbers, using known results over complex numbers. We write explicit structure constants for these real Lie algebras.
... In this work, we establish a link between the dimension of the coadjoint orbit of the form α and cl(α) its class in Elie Cartan's sense. More precisely dim O(α) = 2 cl(α) 2 . Recall that the Cartan class of α corresponds to the number of independent Pfaffian forms needed to define α and its differential dα and it is equal to the codimension of the characteristic space [4,13,19]. ...
... This invariant was introduced in [2] in order to classify 7-dimensional nilpotent Lie algebras. A link between the notions of breath of nilpotent Lie algebra introduced in [22] and characteristic sequence is developed in [26]. ...
... In 1989 Romdhani obtained a classification of 7−dimensional complex nilpotent Lie algebras using only basic Linear algebra techniques such as Jordan forms of matrices and classification of bilinear forms [36]. Ancochea and Goze [6] and Seeley [38] are the other researchers who attacked the problem by following different approach, but they were later adjusted in [39] and [23]. In [11] De Graaf got the classification 6−dimensional complex nilpotent Lie algebras by using Gröbner bases and he compared it with the classifications of 6−dimensional complex nilpotent Lie algebras given before. ...
... Then the Leibniz identity [y 1 , [y 2 , y 1 ]] = [[y 1 , y 2 ], y 1 ]+[y 2 , [y 1 , y 1 ]] gives α 3 γ 2 = (α 4 +β 1 )γ 6 . This implies that α 4 + β 1 ≠ 0. ...
... They were introduced formally by Vergne [23,24] in the late 1960s, although Umlauf had already used them as an example in his thesis [21]. The distribution into isomorphism classes of n-dimensional filiform Lie algebras over the complex field is known for n ≤ 12 [6,13], whereas it is only known for nilpotent Lie algebras over the complex field of dimension n ≤ 7 [4]. More recently, some authors have dealt with the classification of n-dimensional nilpotent Lie algebras over finite fields F p = Z/pZ. ...
... Equation (6) can be used to fix an entry in F that does not appear in Eqs. (3)(4)(5). If a = 0 = A, then the variable f 46 can be isolated in Eq. (6) whenever ch(K) = 2, or the variable f 56 , otherwise. ...
Since the introduction of the concept of isotopism of algebras by Albert in 1942, a prolific literature on the subject has been developed for distinct types of algebras. Nevertheless, there barely exists any result on the problem of distributing Lie algebras into isotopism classes. The current paper is a first step to deal with such a problem. Specifically, we define a new series of isotopism invariants and we determine explicitly the distribution into isotopism classes of n-dimensional filiform Lie algebras, for . We
also deal with the distribution of such algebras into isomorphism classes, for which we confirm some known results and we prove that there exist p+8 isomorphism classes of seven-dimensional filiform Lie algebras over the finite field if .
... The classification of nilpotent Lie algebras over the complex numbers experimented an important advance based on the works of Ancochéa-Bermúdez and Goze [10] introducing an invariant more potent than the previously existing: the characteristic sequence or Goze's invariant (defined in Section 2.1). Those authors were able, by using the characteristic sequence as an invariant, to classify the nilpotent Lie algebras of dimension 7 [11] and the filiform Lie algebras of dimension 8 [12]. Later, by using that invariant, Gomez and Echarte [13] classify the filiform Lie algebras of dimension 9. Afterward, Castro et al. [14] develop an algorithm for symbolic language for finding the generic families of filiform Lie algebras in any dimension with the restrictions required to the parameters. ...
... If 11 ̸ = 0, 1 = 0, 2 ̸ = 8 , 2 ̸ = 0, 3 = 0, and 8 = 0, selecting 1 = 3 0 / 2 , the nullity of 2 9 + 4 is invariant. If 4 ̸ = − 2 9 , selecting 2 0 = 2 ( 2 9 + 4 )/ 2 2 ̸ = 0, the subfamily is 10 (0, 1, 0, 1, 0, 0, 0, 0); else with 1 = 3 0 / 2 the subfamily is 11 (0, 1, 0, 0, 0, 0, 0, 0). If 11 ̸ = 0, 1 = 0, 2 ̸ = 8 , 2 = 0, and 3 ̸ = 0, selecting 1 = 4 0 / 3 ̸ = 0, 0 = 3 / 8 ̸ = 0, and 1 = − 3 ( 4 − 9 8 )/ 3 8 , the subfamily is 12 (0, 0, 1, 0, 0, 0, 1, 0). ...
On the basis of the family of quasifiliform Lie algebra laws of dimension 9 of 16 parameters and 17 constraints, this paper is devoted to identify the invariants that completely
classify the algebras over the complex numbers except for isomorphism. It is proved that
the nullification of certain parameters or of parameter expressions divides the family into
subfamilies such that any couple of them is nonisomorphic and any quasifiliform Lie
algebra of dimension 9 is isomorphic to one of them. The iterative and exhaustive computation with Maple provides the classification, which divides the original family into
263 subfamilies, composed of 157 simple algebras, 77 families depending on 1 parameter,
24 families depending on 2 parameters, and 5 families depending on 3 parameters.
... To do it, he only used the elementary techniques of Linear Algebra, and excluded all algebras which decompose into a direct sum of lesser-dimensional factors. In 1989 Ancochea and Goze [10], by using a new invariant of filiform Lie algebras introduced by themselves, namely the characteristic sequence, got an incomplete classification of the nilpotent Lie algebras of dimension 7. It is convenient to note that Seeley [137] also classified the complex nilpotent Lie algebras of dimension 6 and 7, following a previous similar project in dimension 5 by Grunewald and O'Halloran [81], and by using the Iwasawa decomposition for GL 6 (C). Some years later, Goze and Remm [80] would rectify the classification given in [10] (another verification of these lists has been published in [106]). ...
... In 1989 Ancochea and Goze [10], by using a new invariant of filiform Lie algebras introduced by themselves, namely the characteristic sequence, got an incomplete classification of the nilpotent Lie algebras of dimension 7. It is convenient to note that Seeley [137] also classified the complex nilpotent Lie algebras of dimension 6 and 7, following a previous similar project in dimension 5 by Grunewald and O'Halloran [81], and by using the Iwasawa decomposition for GL 6 (C). Some years later, Goze and Remm [80] would rectify the classification given in [10] (another verification of these lists has been published in [106]). In 2007, de Graaf [53] retook the comparison of all the classifications of the nilpotent 6-dimensional Lie algebras. ...
The problem of Lie algebras’ classification, in their different varieties, has been dealt with by theory researchers since the early 20 th century. This problem has an intrinsically infinite nature since it can be inferred from the results obtained that there are features specific to each field and dimension. Despite the hundreds of attempts published, there are currently fields and dimensions in which only partial classifications of some families of algebras of low dimensions have been obtained. This article intends to bring some order to the achievements of this prolific line of research so far, in order to facilitate future research.
... It is known that over the complex numbers there are only finitely many isomorphism classes of nilpotent Lie algebras of dimension less than or equal to 6 [7,4,17] whereas in higher dimensions there are infinite families of pairwise nonisomorphic nilpotent Lie algebras [3,11]. In dimension 7, each infinite family can be parameterised by a single complex modulus, upon which the structure constants depend analytically [1,6,9,10,13]. An open problem is to determine exactly how many analytic parameters F n are needed to classify n-dimensional nilpotent Lie algebras. ...
... The closure of U is an algebraic component of dimension 55. There is in fact another algebraic component of dimension 55 containing a dense open subset of filiform Lie algebras [1,12]. (A nilpotent Lie algebra is filiform when there is an element whose centraliser is 2-dimensional.) ...
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 m ³ /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.
... The low-dimensional cohomology groups were used as invariants to study the physically motivated relations (contractions) between algebras (see, for instance, [3]). Also, the cohomology groups play a rigorous role in the classification problems of algebra structures on vector spaces (see [4], [5]). ...
Cohomology groups have a broad range of applications in algebraic and geometric classification problems of associative algebras. We focus on the second order cohomology groups of associative algebras. The main objectives of this study are precisely formulated two algorithms for describing the 2-cocycles and 2-coboundaries of an n-dimensional associative algebra (including unital and non-unital) in a matrix form. Using an existing classification result of low-dimensional associative algebras, we apply the algorithms to three-dimensional associative algebras.
... The low-dimensional cohomology groups were used as invariants to study the physically motivated relations (contractions) between algebras (see, for instance, [3]). Also, the cohonology groups play a rigorous role in the classification problems of algebra structures on vector spaces (see [4] and [5]) In the paper we make use a classification r e s ult o f twodimensional associative algebras given in [6] and a result on description of the derivations of all two-dimensional associative algebras from in [7]. The main purpose of this paper is to present some results on computing of low-order (co)homology groups of two-dimensional associative algebras. ...
In this paper, we calculate cohomology groups of low-dimensional complex associative algebras. The calculations are based on a classification result and description of derivations of low-dimensional associative algebras obtained earlier. For the first cohomology group, we give basic cocycles up to inner derivations. We also provide basic coboundaries for the second cohomology groups for low-dimensional associative algebras (including both unital and non unital).
... Classification theorems for nilpotent and solvable Lie groups with bi-invariant complex structures can be found in [62,63,125]. ...
It is well known that on any Lie group, a left-invariant Riemannian structure can be defined. For other left-invariant geometric structures, for example, complex, symplectic, or contact structures, there are difficult obstructions for their existence, which have still not been overcome, although a lot of works were devoted to them. In recent years, substantial progress in this direction has been made; in particular, classification theorems for low-dimensional groups have been obtained. This paper is a brief review of left-invariant complex, symplectic, pseudo-Kählerian, and contact structures on low-dimensional Lie groups and classification results for Lie groups of dimension 4, 5, and 6.
... It is important to find new invariants in the classification of Lie algebras. Goze and Ancochea obtained the classification of complex nilpotent Lie algebras of dimension 7 (see [1]) by introducing a new invariant which are called the characteristic sequence (see [8]). Later, using the same method, they also obtained the classification of complex filiform Lie algebras of dimension 8 (see [2]). ...
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.
... Definition 12. [1] ...
We classify, up to isomorphism, gradings by abelian groups on nilpotent
filiform Lie algebras of nonzero rank. In case of rank 0, we describe
conditions to obtain non trivial -gradings.
The way of communication coding using “multiplicative gammation” in algebras with strong filtration is proposed. This class of algebras was introduced earlier by the author for needs of Gröbner–Shirshov bases theory in a wide context. It includes semigroup algebras of ordered semigroups and universal enveloping algebras of Lie algebras, in particular, a polynomial algebra and a free associative algebra.
We study the coadjoint orbits of a Lie algebra in terms of Cartan class. In fact, the tangent space to a coadjoint orbit at the point corresponds to the characteristic space associated to the left invariant form and its dimension is the even part of the Cartan class of . We apply this remark to determine Lie algebras such that all the non singular orbits (non reduced to a point) have the same dimension, in particular when this dimension is 2 or 4. We determine also the Lie algebras of dimension 2n or 2n+1 having an orbit of dimension 2n.
We show how the one-mode pseudo-bosonic ladder operators provide concrete examples of nilpotent Lie algebras of dimension five. It is the first time that an algebraic-geometric structure of this kind is observed in the context of pseudo-bosonic operators. Indeed we don't find the well known Heisenberg algebras, which are involved in several quantum dynamical systems, but different Lie algebras which may be decomposed in the sum of two abelian Lie algebras in a prescribed way. We introduce the notion of semidirect sum (of Lie algebras) for this scope and find that it describes very well the behaviour of pseudo-bosonic operators in many quantum models.
A pair of sequences of nilpotent Lie algebras denoted by Nn, 7 and Nn, 16 are introduced. Here, n denotes the dimension of the algebras that are defined for n ≥ 6; the first terms in the sequences are denoted by 6.7 and 6.16, respectively, in the standard list of six-dimensional Lie algebras. For each of them, all possible solvable extensions are constructed so that Nn, 7 and Nn, 16 serve as the nilradical of the corresponding solvable algebras. The construction continues Winternitz’ and colleagues’ program of investigating solvable Lie algebras using special properties rather than trying to extend one dimension at a time.
In this paper we explain some tools of Differential Geometry. In detail we deal with Lie Theory, which is currently being investigated in Economics. Firstly we indicate the conditions demanded to production functions in such studies, and the definition of a particular type of technical progress: the Lie type. For this type, the three properties of Lie groups have to be verified by the technical progress. We also show the use of Lie operator for economical interpretations, and for quantifying the impact of the technical progress. This operator allows us to answer Solow-Stigler Controversy. Finally we introduce some applications of Lie Theory in Economics, and suggest new future research lines they can generate. In this way, our main aim is to show the actual and future applications of Lie Theory in Economics.
In this paper we describe some families of filiform Lie algebras by giving a method which allows to obtain them in any arbitrary dimension n starting from the triple (p,q,m), where m=n and p and q are, respectively, invariants z 1 and z 2 of those algebras. After obtaining the general law of complex filiform Lie algebras corresponding to triples (p,q,m), some concrete examples of this method are shown.
We collect information about the Klein quadric which is useful to determine the orbits of the group of all linear bijections of a four-dimensional vector space on the Grassmann manifolds of the exterior product. This information is used to classify nilpotent Lie algebras of small dimension, over arbitrary fields (including the characteristic 2 case). The invariants used are easy to read off from any set of structure constants.
We compute the dimension of the algebra of derivations of any n-dimensional nilpotent real or complex Lie algebra whose Goze's invariant (or characteristic sequence) is (n - 3, 1,...,1).
Lie's Third Theorem states that given a Lie algebra g of finite dimension, there exists a simply connected Lie group G whose associated Lie algebra is g. The classical proof of this result, which is not simple, is based on Ado's Theorem. According to it, every Lie algebra of finite dimension can be represented as a Lie subalgebra of the general linear group of matrices. We show in this paper a method to give a matrix representation of the simply connected group associated with a fixed nilpotent Lie algebra. Moreover, we give the representation of the Lie groups associated with filiform nilpotent Lie algebras whose derived is abelian.
We consider real nilpotent Lie algebras of positive rank. We fix a set A indexing the nonzero structure constants for a Lie algebra g with respect to a basis of eigenvectors for an R-split torus in the derivation algebra of g: We give criteria for when two Lie algebras with the same index set are isomorphic. We present a criterion for when there is a nilsoliton metric Lie algebra having a given index set, and we determine which nilsoliton metric Lie algebras have a given index set, up to isometric isomorphism and rescaling, in some common situations. We study the Nikolayevsky derivation, showing that it commutes with automorphisms that preserve certain inner products, and we find conditions on the Nikolayevsky derivation that insure that the isometry group of a metric Lie algebra is finite. We give examples showing that index sets and the Nikolayevsky derivation are useful invariants for nilpotent Lie algebras.
This paper, an approach to the classification of nilpotent Lie algebras g of any dimension with a Goze’s invariant (dimg — p, 1,…, 1), provides an explicit classification for p ≥ dimg — 3.
In this work the authors classify the filiform Lie algebras (i.e., Lie algebras that are nilpotent with an adjoint derivation of maximal order) of dimension m=8 over the field of complex numbers. These algebras, introduced by M. Vergne in her thesis ["Varietes des algebres de Lie nilpotentes'', These de 3 eme cycle, Univ. Paris, Paris, 1966; BullSig(110) 1967:299], form a Zariski-open set in the variety N m of nilpotent Lie algebra laws of C n . For each m≤6 there exists a filiform algebra which gives the only rigid law (i.e., the orbit is open under the natural action of the linear group GL m (C) ) in N m . In N 7 there exists a one-parameter family of filiform algebras but there is no rigid law.
For m=8 the authors enumerate six continuous families with one complex parameter and fourteen algebras, one of which is rigid in N 8 ; this is the most remarkable fact. The methods use perturbations, which are the analogue of deformations in the language of nonstandard analysis, and similarity invariants for a nilpotent matrix. Notice that we do not have for the moment an example of a Lie algebra which is nilpotent and rigid in the variety of all the Lie algebras in dimension m ; such an algebra cannot be filiform.
In this paper we study the existence of symplectic structures on the product
where M is a nilmanifold of dimension 5,7.
In this work large families of naturally graded nilpotent Lie algebras in arbitrary dimension and characteristic sequence (n,q,1) with satisfying the centralizer property are given. This centralizer property constitutes a generalization, for any nilpotent algebra, of the structural properties characterizing the Lie algebra Qn. By considering certain cohomological classes of the space , it is shown that, with few exceptions, the isomorphism classes of these algebras are given by central extensions of Qn by which preserve the nilindex and the natural graduation.
We give the classification, in any dimension, of nilpotent Lie algebras g with Goze’s invariant , being n = dim(g), The technique used to get our goal is to consider those algebras as central extensions.
We construct large families of characteristically nilpotent Lie algebras by analyzing the centralizers of the ideals in the central descending sequence of the Lie algebra Qn and deforming its extensions preserving the structure of these centralizers and the natural graduation. This provides characteristically nilpotent Lie algebras in any dimension and mixed characteristic sequences.
In this paper we consider the problem of classifying the (n−5)-filiform Lie algebras. This is the first index for which infinite parametrized families appear, as can be seen in dimension 7. Moreover we obtain large families of characteristic nilpotent Lie algebras with nilpotence index 5 and show that at least for dimension 10 there is a characteristic nilpotent Lie algebra with nilpotence index 4 which is the algebra of derivations of a nilpotent Lie algebra.
We give a complete classification up to isomorphisms of complex filiform Lie algebras of dimension m with m ≤ 11.
Apart from the own dimension n of a complex filiform Lie algebra, the first invariant known for these algebras was the lower central sequence of the algebra, obtained by Ancochea and Goze in 1989. Later, Echarte, Gómez and Núñez, in 1996, and ourselves, in the same year, obtained the invariants i and j, respectively. In this paper, we prove new properties of the invariant j and we use them to set some relations among these invariants and the dimension n. These results allow us to classify complex filiform Lie algebras according to some values of those invariants.
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 dimkh⩽11.
In this paper we give the list of all 7-dimensional nilpotent real Lie
algebras that admit a contact structure. Based on this list, we describe all
7-dimensional nilmanifolds that admit an invariant contact structure.
In this article we present the classification of the 3-filiform Leibniz algebras of maximum length, whose associated naturally graded algebras are Lie algebras. Our main tools are a previous existence result by Cabezas and Pastor [J.M. Cabezas and E. Pastor, Naturally graded p-filiform Lie algebras in arbitrary finite dimension, J. Lie Theory 15 (2005), pp. 379–391] and the construction of appropriate homogeneous bases in the connected gradation considered. This is a continuation of the work done in Ref. [J.M. Cabezas, L.M. Camacho, and I.M. Rodríguez, On filiform and 2-filiform Leibniz algebras of maximum length, J. Lie Theory 18 (2008), pp. 335–350].
Nilpotent Lie algebras with maximal subalgebras are considered. A bound on the dimension of these algebras as a function of their coclass is obtained. These algebras are then classified up to coclass two. The result is field dependent and several fields are considered.
A p-filiform Lie algebra g is a nilpotent Lie algebra for which Goze’s invariant is (n–p,1,…,1). These Lie algebras are well known for P ≥ n-4n = dim(g). In this paper we describe the p-filiform Lie algebras, for p = n-5 and we gjive their classification when the derived subalgebra is maximal.
In this paper we give the explicit classification of complex filiform Lie algebras of dimension 11. To do this, we use a method previously obtained by us in an earlier paper, which is based on the concept of isomorphism between Lie algebras. At present, this explicit classification is not known, although Gómez, Jiménez and Khakimdjanov gave a list of these algebras in a non-explicit way, but in terms of cocycles. We find that there exist 188 families of complex filiform Lie algebras of dimension 11.
The aim of this paper is the study of abelian Lie algebras as subalgebras of the nilpotent Lie algebra gn associated with Lie groups of upper-triangular square matrices whose main diagonal is formed by 1. We also give an obstruction to obtain the abelian Lie algebra of dimension one unit less than the corresponding to gn as a Lie subalgebra of gn. Moreover, we give a procedure to obtain abelian Lie subalgebras of gn up to the dimension which we think it is the maximum.
The work of Golod and Šafarevič on class field towers motivated the conjecture that b2 > b21/4 for nilpotent Lie algebras of dimension at least 3, where bi
denotes the i
th Betti number. Using a new lower bound for b
2 and a characterization of Lie algebras of the form g/Z(g), we prove this conjecture for 2-step algebras. We also give the Betti numbers of nilpotent Lie algebras of dimension at most 7 and use them to establish the conjecture for all nilpotent Lie algebras whose centres have codimension ≤ 7.
We study the left-invariant Riemannian metrics on a class of models of nilpotent Lie groups. In particular we prove that the Heisenberg groups are, up to local isomorphism, the only nilpotent non-decomposable Lie groups endowed with a homogeneous Riemannian naturally reductive space for every left invariant metric.
Using the Iwasawa composition and some facts about complete varieties, it is proved in a rather general setting that if the Lie algebra L_1 degenerates to L_0 (i.e. L_0 ls in the closure of the orbit of L_1 under changes of basis) then the same is true (after possibly adding a central abelian ideal to L_0) of their central quotients.
The present paper is devoted to the study of low dimensional Leibniz algebras over the field of p-adic numbers. The classification up to isomorphism of three-dimensional Lie algebras over the integer p-adic numbers is already known [8]. Here, we extend this classification to solvable Lie and non-Lie Leibniz algebras over
the field of p-adic numbers.
Key wordsLie algebra-Leibniz algebra-classification-p-adic field-solvability-nilpotency
For any complex 6-dimensional nilpotent Lie algebra
\mathfrakg,\mathfrak{g}, we compute the strain of all indecomposable 7-dimensional nilpotent Lie algebras which contain
\mathfrakg\mathfrak{g} by the adjoining a derivation method. We get a new determination of all 7-dimensional complex nilpotent Lie algebras, allowing
to check earlier results (some contain errors), along with a cross table intertwining nilpotent 6- and 7-dimensional Lie algebras.
KeywordsNilpotent Lie algebras-7-dimensional-Determination-Adjoining a derivation
Mathematics Subject Classification (2000)Primary 17B30
Let Nn 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 N7 is composed of two algebraic components and that N8 is also reducible.
We study left invariant contact forms and left invariant symplectic forms on Lie groups. In the case of filiform Lie groups we give a necessary and sufficient condition for the existence of a left invariant contact form and we prove the uniqueness of this contact form up to a nonzero scalar multiple. As an application we classify all symplectic structures on nilpotent Lie algebras of dimension ⩽6.
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.
In this work the authors classify the filiform Lie algebras (i.e., Lie algebras that are nilpotent with an adjoint derivation of maximal order) of dimension m=8 over the field of complex numbers. These algebras, introduced by M. Vergne in her thesis ["Varietes des algebres de Lie nilpotentes'', These de 3 eme cycle, Univ. Paris, Paris, 1966; BullSig(110) 1967:299], form a Zariski-open set in the variety N m of nilpotent Lie algebra laws of C n . For each m≤6 there exists a filiform algebra which gives the only rigid law (i.e., the orbit is open under the natural action of the linear group GL m (C) ) in N m . In N 7 there exists a one-parameter family of filiform algebras but there is no rigid law.
For m=8 the authors enumerate six continuous families with one complex parameter and fourteen algebras, one of which is rigid in N 8 ; this is the most remarkable fact. The methods use perturbations, which are the analogue of deformations in the language of nonstandard analysis, and similarity invariants for a nilpotent matrix. Notice that we do not have for the moment an example of a Lie algebra which is nilpotent and rigid in the variety of all the Lie algebras in dimension m ; such an algebra cannot be filiform.
Soient Γ un groupe localement compact, l'ensemble des classes de représentations unitaires irréductibles de Γ. Dans (4), Godement a défini une topologie sur que nous appellerons "topologie canonique." Cette topologie est facile à étudier lorsque Γ est abélien ou compact, mais fort mal connue en général. Dans un article récent (3), Fell a repris l'étude de la topologie canonique de , et réussi à la calculer lorsque Γ est le groupe spécial linéaire complexe à n variables. L'espace est alors "presque" séparé, et, par suite, "presque" localement compact.
Il semble nécessaire d'étudier complètement quelques cas particuliers avant d'entreprendre une théorie générale de .
Würzburg, Univ., Naturwiss. Fachbereich IV - Mathematik, Diss., 1976. (Nicht f. d. Austausch.).
A class of 3-Engel groups
- H Heiner~en
H. HEINEr~EN, A class of 3-Engel groups. J. Algebra 17, 41-345 (1971).
Gruppen deren Untergruppen subnormal vom Defekt zwei sind Anschrift des Autors: K. Mahdavi Department of Mathematics State University College of Arts and Sciences Potsdam *) Eine Neufassung ging am 30
- M Stadelmann
M. STADELMANN, Gruppen deren Untergruppen subnormal vom Defekt zwei sind. Arch. Math. 30, 364-371 (1978). Anschrift des Autors: K. Mahdavi Department of Mathematics State University College of Arts and Sciences Potsdam, New York 13676 USA Eingegangen am 2. 11. 1985") *) Eine Neufassung ging am 30. 11. 1985 ein.
Calcul duH 2 (g, g) sur IBMPC
- M Goze
- N Makklouf
Classification des algèbres de Lie nilpotentes de dimension 6
- V V Morozov
- V. V. Morozov