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Nonfiliform characteristically nilpotent Lie algebras
Abstract
We construct large families of characteristically nilpotent Lie algebras by considering deformations of the Lie algebra g_{m,m-1}^{4} of type Q_{n},and which arises as a central extension fo the filiform Lie algebra L_{n}. By studying the graded cohomology spaces we obtain that the sill algebras are isomorphic to the nilradicals of solvable, complete Lie algebra laws. For extremal cocycles these laws are also rigid. Considering supplementary cocycles we construcy, for dimensions n>8, nonfiliform characteristically nilpotent Lie algebras and show that for certain deformations these are compatible with central extensions.
... as a global classification of nilpotent Lie algebras is a hopeless problem [4], it seems more reasonable to analyze the graded structures and then use deformation theory to obtain concrete isomorphism classes satisfying certain properties. This has for example been done to obtain nilradicals of complete and rigid Lie algebras [3] ...
For sufficiently high dimensions, the naturally graded nonsplit nilpotent Lie algebras with linear characteristic sequence are classified.
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.
We determine the solvable complete Lie algebras whose nilradical is isomorphic to a filiform Lie algebra. Moreover we show that for any positive integer n there exists a solvable complete Lie algebras whose second cohomology group with values in the adjoint module has dimension at least n. 1 Complete Lie algebras Complete Lie algebras first appeared in 1951, in the context of ScheNkman’s theory of subinvariant Lie algebras [16]. In his famous derivation tower theorem, he proved that for centerless solvable Lie algebras the tower of derivation algebras Der n g = Der n−1 (Derg) was finite, and that the last term, which is also centerless, had only inner derivations. However, the formal definition of complete Lie algeba was not given until 1962. Their relation with the tower theorem has proven the importance of these algebras, which also play a role in rigidity theory of algebraic structures. In the last years, different authors have concentrated on classifications and structural properties of complete Lie algebras [3, 7, 13, 14], and various interesting results concerning these structures have been obtained. For our purpose it will be more convenient to define completeness of Lie algebras in terms of cohomology. Therefore we recall the elementary facts: Let g be a Lie algebra. A p-dimensional cochain of g (with values in g) is a p-linear alternating mapping of g p in g (p ∈ N ∗). A 0-cochain is a constant corresponding author:
The aim of this paper is to study the intercon-nectivity among Lie algebras that are characteristically nilpotent, derived and c-graded filiform, establishing some characterization theorems involving them. At the same time we offer an algorithm that will allow us to know, given a complex filiform Lie algebra, which of those properties hold.
We study a certain class of non-maximal rank contractions of the nilpotent Lie algebra and show that these contractions are completable Lie algebras. As a consequence a family of solvable complete Lie algebras of non-maximal rank is given in arbitrary dimension.
We study the structure of the rigid Lie algebras over an algebraically closed field of characteristic 0. They are algebraic algebras. When the radical is not nilpotent their dimension is egal to the dimension of the algebra of derivations. When the radical is nilpotent it belongs to one of the three cases: perfect; direct product of a perfect Lie algebra with the field; all the semi-simple derivations are inner.
Let L
n be the variety of n-dimensional complex Lie algebras (see Chapter 5). We denote by N
pn the subset of L
n constituted from the nilpotent Lie algebras whose nilindex is less than p. This subset is defined by
We prove the existence of a nonempty Zariski open subset consisting of
characteristically nilpotent Lie algebras in the variety of structures
of nilpotent Lie algebras of dimension >=12. We construct vast
families of characteristically nilpotent Lie algebras of arbitrary
dimension >=7.
Let L be a Lie algebra over a field of characteristic 0 and let D(L) be the derivation algebra of L , that is, the Lie algebra of all derivations of L . Then it is natural to ask the following questions: What is the structure of D(L)? What are the relations of the structures of D(L) and L ? It is the main purpose of this paper to present some results on D(L) as the answers to these questions in simple cases.
Concerning the questions above, we give an example showing that there exist non-isomorphic Lie algebras whose derivation algebras are isomorphic (Example 3 in § 5). Therefore the structure of a Lie algebra L is not completely determined by the structure of D(L) . However, there is still some intimate connection between the structure of D(L) and that of L .
In this paper we develop a method of construction of complex rigid solvable Lie algebras which is independent of coho-mological techniques or classification of Lie algebras. As an application, we classify all rigid solvable Lie algebras in dimension less or equal than eight, and we obtain partial results in dimension nine. Moreover, we give several examples of families of rigid Lie algebras in arbitrary dimension, some of them having its second cohomology group, in the Chevalley cohomology, non trivial.
In a recent paper Borei and Serre proved the theorem: If is a Lie algebra of characteristic 0 and has an automorphism of prime period without fixed points ≠ 0, then is nilpotent.1 In this note we give a proof valid also for characteristic p ≠ 0. By the same method we can prove several other similar results on automorphisms and derivations. Our method is based on decompositions of the Lie algebra which determine weakly closed sets of linear transformations. Such a set has, by definition, the closure property that if then there exists a γ(A, B) in the base field such that The main result we shall need is the generalized Engel theorem that if is a weakly closed set of nilpotent linear transformations in a finite-dimensional vector space, then the enveloping associative algebra of is nilpotent [3].
: The theories of deformations of associative algebras, Lie algebras, and of representations and homomorphisms of these all show a striking similarity to the theory of deformations of complex analytic manifolds. The common context of all these turns out to be the problem of deforming one single element x of degree one in a graded Lie algebra subject to the condition (x,x) = 0. The first part of this report gives general properties of graded Lie algebras over fields of characteristic not 2; the second part deals with the case of characteristic 2. The third and fourth parts give general theorems on deformability for the respective cases when the base field is the real or complex numbers (analytic methods), and general algebraically closed fields (algebraic methods). It is shown that deformability is to a large extent determined by certain cohomology groups and mappings of these. (Author)
An approach to perturbation theory is considered based on the formula exp (—iB) A exp (iB) = A exp (iB]), where [AB] = AB − BA is the commutator. The operators A constitute a linear space A and the operators B considered are such that B] take A into itself. The present discussion considers the case where A is finite dimensional with coordinates c1, ⋅⋅⋅, cn; i.e., A is isomorphic with the set of n‐dimensional vectors c. Under these circumstances sufficient conditions for the basic formula are given in terms of the ``analytic vectors'' of Nelson. The set, B, of B's available can be considered closed under the processes of taking linear combinations and forming the commutator. Thus B is a Lie algebra. Exponentiation leads to a Lie group of operators U, and A and A′ are said to be B equivalent if A′ = U−1AU. For A finite dimensional each B is associated with an n × n matrix, b, which specifies the operation B] relative to the vectors c. Effectively then, B is finite dimensional. The matrices b form a Lie algebra with a corresponding Lie group of matrices u such that A and A′ are B equivalent if and only if the corresponding two vectors c and c′ are u images; i.e., c′ = u c. Computationally, therefore, the set of A′ equivalent to a given A is obtained by considering the orbit of a given vector c under the Lie group; i.e., the set of u c. A neighborhood A′ of a given A consists of those operators A′ in the form A + δA where δA is arbitrary except for a restriction on the size of its coordinates, ci. Given A, a neighborhood A′ can be found for which one can obtain by a well‐known construction on the orbits a set of functionally complete and functionally independent invariants for B equivalence. Computationally global and rational invariants are desirable and these can be obtained in the form of similarity invariants, provided that a can be mapped onto a set of n × n matrices a in such a way that if c corresponds to a, then b c corresponds to a′ = [ab]. If the sets A and B are identical such a mapping is immediately available and if, in addition, the corresponding Lie algebra is semisimple, in general, given A, the A's in some neighborhood are each equivalent to an A″ which is a function of A. This corresponds to a case in which a very simple perturbation of A levels occurs. Two examples are discussed.
In this paper, we will discuss the properties of solvable complete Lie algebra, describe the structures of the root spaces of solvable complete Lie algebra, prove that solvable Lie algebras of maximal rank are com-plete, and construct some new complete Lie algebras from Kac-Moody algebras.
At head of title: The University of Michigan, College of Literature, Science and the Arts, Department of Mathematics. "ORA project 03000 under contract with: Mathematical Sciences Directorate, Air Force Office of Scientific Research, contract no. AF49(638)-104, Washington, D.C., administered through: Office of Research Administration, Ann Arbor. Technical note 03000-2-T." Issued also as thesis, Michigan, University, in microfilm form. Includes bibliographical references.
We review the known results about characteristically nilpotent complex Lie algebras, as well as we comment recent developements in the theory.
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