Perfect Lie Algebras Research Articles

Some properties of the c-nilpotent multiplier of Lie algebras

In this article, we give a sufficient condition for the c-nilpotent multiplier of a Lie algebra to be finite dimensional. Also, we show that the c-nilpotent multipliers of perfect Lie algebras are isomorphic.

On Covers of Perfect Lie Algebras

A Lie algebra is said to be perfect when it coincides with its derived subalgebra. The paper is devoted to give a complete structure of covers of perfect Lie algebras. Also, similar to a result of Alperin and Gorenstein (1966) in group theory, it is shown that every automorphism of a finite dimensional perfect Lie algebra may be lifted to an automorphism of its cover. Moreover, we present the concepts of irreducible and primitive extensions of an arbitrary Lie algebra and give some equivalent conditions for a central extension to be irreducible or primitive. Finally, we study the connection between the primitive extensions and the stem covers of finite dimensional perfect Lie algebras.

A new matrix method for the Casimir operators of the Lie algebras and

A method is given to determine the Casimir operators of the perfect Lie algebras and the inhomogeneous Lie algebras in terms of polynomials associated with a parametrized (2N + 1) × (2N + 1)-matrix. For the inhomogeneous symplectic algebras this matrix is shown to be associated to a faithful representation. We further analyse the invariants for the extended Schrodinger algebra in (N + 1) dimensions, which arises naturally as a subalgebra of . The method is extended to other classes of Lie algebras, and some applications to the missing label problem are given.

Derivation algebras of centerless perfect Lie algebras are complete

It is proved that the derivation algebra of a centerless perfect Lie algebra of arbitrary dimension over any field of arbitrary characteristic is complete and that the holomorph of a centerless perfect Lie algebra is complete if and only if its outer derivation algebra is centerless.

The structure of the invariants of perfect Lie algebras

The structure of the invariants of perfect Lie algebras

The structure of the invariants of perfect Lie algebras II

The structure of the invariants of perfect Lie algebras II

The structure of the invariants of perfect Lie algebras

The structure of the invariants of perfect Lie algebras with nontrivial centre is analysed. It is shown that if the radical r of a semidirect sum s −→⊕Rr of a semisimple and a nilpotent Lie algebra has a one-dimensional centre, then the defining representation R contains a copy of the trivial representation D0 of s. Using this fact, a criterion can be deduced to eliminate variables and to characterize the Casimir operators of an algebra g by means of certain subrepresentations R � of R. For rank 1 Levi subalgebras s, all representations leading to perfect Lie algebras with a radical isomorphic to a Heisenberg Lie algebra are determined. We prove that the number of Casimir operators of such an algebra is fixed for any dimension, and moreover that they can be explicitly computed by means of determinantal formulae obtained from the brackets of the algebra. For rank n � 2 Levi part a stabilization result of this nature is also given

An extension based determinantal method to compute Casimir operators of Lie algebras

We present a method based on degree one extensions of Lie algebras by a derivation to compute the Casimir operator of perfect Lie algebras having only one invariant for the coadjoint representation and an Abelian radical. In particular, the Casimir operator of the special affine Lie algebras sa(n, R) results from the determinant of the commutator matrix of an extension. Examples are given for the case of non-Abelian radicals, and the corresponding generalization of the method for this case is formulated.

PLAIN REPRESENTATIONS OF LIE ALGEBRAS

In this paper we study representations of finite dimensional Lie algebras. In this case representations are not necessarily completely reducible. As the general problem is known to be of enormous complexity, we restrict ourselves to representations that behave particularly well on Levi subalgebras. We call such representations plain (Definition 1.1). Informally, we show that the theory of plain representations of a given Lie algebra L is equivalent to representation theory of finitely many finite dimensional associative algebras, also non-semisimple. The sense of this is to distinguish representations of Lie algebras that are of complexity comparable with that of representations of associative algebras. Non-plain representations are intrinsically much more complex than plain ones. We view our work as a step toward understanding this complexity phenomenon.We restrict ourselves also to perfect Lie algebras L, that is, such that L = [L, L]. In our main results we assume that L is perfect and [sfr ][lfr ]2-free (which means that L has no quotient isomorphic to [sfr ][lfr ]2). The ground field [ ] is always assumed to be algebraically closed and of characteristic 0.

Structure of perfect Lie algebras without center and outer derivations

On montre que certaines proprietes classiques des algebres de Lie semi-simples restent vraies pour les algebres de Lie g qui verifient [g, g] = g, Der(g) = ad(g), Z(g) = {0}, que nous appellerons les algebres de Lie sympathiques. On construit une algebre de Lie sympathique non semi-simple de dimension 25. Cette algebre de Lie est la plus petite algebre de Lie sympathique non senti-simple connue a ce jour. Si g est une algebre de Lie, on montre que g contient un plus grand ideal sympathique M, et qu'il existe un ideal resoluble de g note P(g) qui est le plus grand ideal resoluble I de g tel que I ∩ M = {0}. Et on montre l'existence d'une sous-algebre de Lie sympathique m telle que g = m? P(g), et g est sympathique si et seulement si P(g) = {0}. On etudie les ideaux I d'une algebre de Lie g tel que g/I est sympathique.

Lie algebras graded by finite root systems and the intersection matrix algebras of Slodowy

This paper is about toroidal Lie algebras, certain intersection matrix Lie algebras defined by Slodowy, and their relationship to one another and to certain Lie algebra analogues of Steinberg groups. The main result of the paper is the identification of the intersection matrix algebras arising from multiply-affinized Cartan matrices of types A, D and E with certain Steinberg Lie algebras and toroidal Lie algebras (Propositions 5.9 and 5.10). A major part of the paper studies and classifies Lie algebras graded by finite root systems. These become the princi- pal tool in our analysis of intersection matrix algebras. Each Lie algebra graded by a simply-laced finite root system of rank > 2 has attached to it an algebra which, according to the type and rank, is either commutative and associative, only associative, or alternative. All these possibilities occur in our description of inter- section matrix algebras. Let R be any associative algebra with identity, not necessarily finite dimen- sional, over a field k of characteristic 0. For each positive integer n the associative algebra M,(R) of n “ n matrices with coefficients in R forms a Lie algebra over k under the commutator product. We denote this Lie algebra by ol,(R). Let Eis be the (i, j) matrix unit of M.(R) and assume that n > 2. The subalgebra e.(R) of OI.(R) generated by the elements rEis, r ~ R, i 4= j, is an ideal of 91,(R) and is perfect, i.e. it is its own derived algebra. Now any perfect Lie algebra O has a universal central extension, also perfect, called a universal covering algebra (u.c.a.) of O [Ga], so in particular, e.(R) has a u.c.a, that we will denote by ~t,(R). We define 112,.(R) by the exact sequence (0.1) 0 ~ f2,,(R) ~ ~t.(R) ~ e,(R) ~ 0 * Dedicated to our teacher Maria J. Wonenburger ** Both authors gratefully acknowledge the support of the Natural Sciences and Engineering Research Council of Canada

The Genus of a Perfect Lie Algebra

The Genus of a Perfect Lie Algebra