Supersolvable Lie Algebras Research Articles

Weak c-ideals of a Lie algebra

A subalgebra $B$ of a Lie algebra $L$ is called a weak c-ideal of $L$ if there is a subideal $C$ of $L$ such that $L=B+C$ and $B\cap C\leq B_{L} $ where $B_{L}$ is the largest ideal of $L$ contained in $B.$ This is analogous to the concept of weakly c-normal subgroups, which has been studied by a number of authors. We obtain some properties of weak c-ideals and use them to give some characterisations of solvable and supersolvable Lie algebras. We also note that one-dimensional weak c-ideals are c-ideals.

Abelian subalgebras and ideals of maximal dimension in supersolvable and nilpotent lie algebras

In this paper, we continue the study of abelian subalgebras and ideals of maximal dimension for finite-dimensional supersolvable and nilpotent Lie algebras. We show that supersolvable Lie algebras with an abelian subalgebra of codimension 3 contain an abelian ideal with the same dimension, provided that the characteristic of the underlying field is not two, and that the same is true for nilpotent Lie algebras with an abelian subalgebra of codimension 4, provided that the characteristic of the field is greater than five.

Extending structures, Galois groups and supersolvable associative algebras

Let A be a unital associative algebra over a field k. All unital associative algebras containing A as a subalgebra of a given codimension \(\mathfrak {c}\) are described and classified. For a fixed vector space V of dimension \(\mathfrak {c}\), two non-abelian cohomological type objects are explicitly constructed: \({\mathcal {A}}{{\mathcal {H}}}^{2}_{A} \, (V, \, A)\) will classify all such algebras up to an isomorphism that stabilizes A while \({\mathcal {A}}{{\mathcal {H}}}^{2} \, (V, \, A)\) provides the classification from Holder’s extension problem viewpoint. A new product, called the unified product, is introduced as a tool of our approach. The classical crossed product or the twisted tensor product of algebras are special cases of the unified product. Two main applications are given: the Galois group \(\mathrm{Gal} \, (B/A)\) of an extension \(A \subseteq B\) of associative algebras is explicitly described as a subgroup of a semidirect product of groups \(\mathrm{GL}_k (V) \rtimes \mathrm{Hom}_k (V, \, A)\), where the vector space V is a complement of A in B. The second application refers to supersolvable algebras introduced as the associative algebra counterpart of supersolvable Lie algebras. Several explicit examples are given for supersolvable algebras over an arbitrary base field, including those of characteristic two whose difficulty is illustrated.

The Ideal Index of Subalgebras of a Finite Dimensional Lie Algebra

D. A. Towers introduced the notion of ideal index of a maximal subalgebra of a Lie algebra, and used it to analyze the influence of maximal subalgebras on the structure of a finite dimensional Lie algebras. In this article, we generalize the ideal index from maximal subalgebras to all subalgebras, and obtain some new characterizations of solvable and supersolvable Lie algebras by the ideal indices of some certain subalgebras.

On abelian subalgebras and ideals of maximal dimension in supersolvable Lie algebras

In this paper, the main objective is to compare the abelian subalgebras and ideals of maximal dimension for finite-dimensional supersolvable Lie algebras. We characterise the maximal abelian subalgebras of solvable Lie algebras and study solvable Lie algebras containing an abelian subalgebra of codimension 2. Finally, we prove that nilpotent Lie algebras with an abelian subalgebra of codimension 3 contain an abelian ideal with the same dimension, provided that the characteristic of the underlying field is not 2. Throughout the paper, we also give several examples to clarify some results.

Characterizations for Supersolvable Lie Algebras

In (2009), Towers [10] presented the notion of c-ideality of a subalgebra of a Lie algebra, and gave some characterizations of solvable and supersolvable Lie algebras. In this article, we further investigate the influence of c-ideality of some subalgebras on the structure of Lie algebras. We also obtain some equivalent conditions for supersolvability of a finite dimensional Lie algebra.

The index complex of a maximal subalgebra of a Lie algebra

AbstractLet M be a maximal subalgebra of the Lie algebra L. A subalgebra C of L is said to be a completion for M if C is not contained in M but every proper subalgebra of C that is an ideal of L is contained in M. The set I(M) of all completions of M is called the index complex of M in L. We use this concept to investigate the influence of the maximal subalgebras on the structure of a Lie algebra, in particular, finding new characterizations of solvable and supersolvable Lie algebras.

C-Ideals of Lie Algebras

A subalgebra B of a Lie algebra L is called a c-ideal of L if there is an ideal C of L such that L = B + C and B ∩ C ≤ B L , where B L is the largest ideal of L contained in B. This is analogous to the concept of c-normal subgroup, which has been studied by a number of authors. We obtain some properties of c-ideals and use them to give some characterisations of solvable and supersolvable Lie algebras. We also classify those Lie algebras in which every one-dimensional subalgebra is a c-ideal.

Two generator subalgebras of Lie algebras

In [Thompson, J., 1968, Non-solvable finite groups all of whose local subgroups are solvable. Bulletin of the American Mathematical Society, 74, 383–437.], Thompson showed that a finite group G is solvable if and only if every two-generated subgroup is solvable (Corollary 2, p. 388). Recently, Grunevald et al. [Grunewald et al., 2000, Two-variable identities in groups and Lie algebras. Rossiiskaya Akademiya Nauk POMI, 272, 161–176; 2003. Journal of Mathematical Sciences (New York), 116, 2972–2981.] have shown that the analogue holds for finite-dimensional Lie algebras over infinite fields of characteristic greater than 5. It is a natural question to ask to what extent the two-generated subalgebras determine the structure of the algebra. It is to this question that this article is addressed. Here, we consider the classes of strongly-solvable and of supersolvable Lie algebras, and the property of triangulability.

On Auslander-Reiten Quivers of Enveloping Algebras of Restricted Lie Algebras

Let (L, [p]) be a finite dimensional restricted Lie algebra over an algebraically closed field F of characteristic p ≥ 3, X ∈ L* a linear form. In this article we study the Auslander-Reiten quivers of certain blocks of the reduced enveloping algebra u(L,x). In particular, it is shown that the enveloping algebras of supersolvable Lie algebras do not possess AR-components of Euclidean type.

Lie Algebras in which the Lattice formed by the Ideals is a Chain

The structure of the Lie algebras in which the ideals are totally ordered by set inclusion is studied. A complete classification is given of the supersolvable Lie algebras in this class over a field ofcharacteristic zero

Some finite varieties of Lie algebras

in the class whenever all n-generated subalgebras of L are in the class. Thus the classes JV of nilpotent Lie algebras and d of abelian Lie algebras are 2-recognizable. The problems are to find if a given class is n-recognizable and to find the smallest n. We consider the classes &‘JV of Lie algebras which have an abelian ideal with nilpotent quotient, JV&, of Lie algebras which have a nilpotent ideal with abelian quotient, % of supersolvable Lie algebras, and Y of solvable Lie algebras. The same notation is used for the corresponding classes of finite groups. Most notation is standard except where indicated.

Almost nilpotent Lie algebras

Throughout we shall consider only finite-dimensional Lie algebras over a field of characteristic zero. In [3] it was shown that the classes of solvable and of supersolvable Lie algebras of dimension greater than two are characterised by the structure of their subalgebra lattices. The same is true of the classes of simple and of semisimple Lie algebras of dimension greater than three. However, it is not true of the class of nilpotent Lie algebras. We seek here the smallest class containing all nilpotent Lie algebras which is so characterised.