Prime Lie Algebras Research Articles

On modular representations of finite-dimensional Lie superalgebras

In this paper, we studied representations of finite-dimensional Lie superalgebras over an algebraically closed field \begin{document} $\mathbb{F}$ \end{document} of characteristic p > 2. It was shown that simple modules of a finite-dimensional Lie superalgebra over \begin{document} $\mathbb{F}$ \end{document} are finite-dimensional, and there exists an upper bound on the dimensions of simple modules. Moreover, a finite-dimensional Lie superalgebra can be embedded into a finite-dimensional restricted Lie superalgebra. We gave a criterion on simplicity of modules over a finite-dimensional restricted Lie superalgebra \begin{document} ${\mathfrak{g}}$ \end{document} , and defined a restricted Lie super subalgebra, then obtained a bijection between the isomorphism classes of simple modules of \begin{document} ${\mathfrak{g}}$ \end{document} and those of this restricted subalgebra. These results are generalization of the corresponding ones in Lie algebras of prime characteristic.

On special Lie algebras having a faithful module with Krull dimension

For special Lie algebras we prove an analogue of Markov's theorem on -algebras having a faithful module with Krull dimension: the solubility of the prime radical. We give an example of a semiprime Lie algebra that has a faithful module with Krull dimension but cannot be represented as a subdirect product of finitely many prime Lie algebras. We prove a criterion for a semiprime Lie algebra to be representable as such a subdirect product.

Prime Lie algebras satisfying the standard Lie identity of degree 5

For every commutative differential algebra one can define the Lie algebra of special derivations. It is known for years that not every Lie algebra can be embedded into the Lie algebra of special derivations of some differential algebra. More precisely, the Lie algebra of special derivations of a commutative algebra always satisfies the standard Lie identity of degree 5. The problem of existence of such embedding is a long-standing problem (see [2,4,3]), which is closely related to the Lie algebra of vector fields on the affine line (see [2]). It was solved by Razmyslov in [2] for simple Lie algebras satisfying this identity (see also [3, Th. 16]). We extend this result to prime (and semiprime) Lie algebras over a field of zero characteristic satisfying the standard Lie identity of degree 5. As an application, we prove that for any semiprime Lie algebra the standard identity St5 implies all other identities of the Lie algebra of polynomial vector fields on the affine line.We also generalize some previous results about primeness of the Lie algebra of special derivations of a prime differential algebra to the case of non-unitary differential algebra.

A radical for graded Lie algebras

For an arbitrary group G and a G-graded Lie algebra L over a field of characteristic zero we show that the Kostrikin radical of L is graded and coincides with the graded Kostrikin radical of L. As an important tool for our proof we show that the graded Kostrikin radical is the intersection of all graded-strongly prime ideals of L. In particular, graded-nondegenerate Lie algebras are subdirect products of graded-strongly prime Lie algebras.

A characterization of the Kostrikin radical of a Lie algebra

In this paper we study if the Kostrikin radical of a Lie algebra is the intersection of all its strongly prime ideals, and prove that this result is true for Lie algebras over fields of characteristic zero, for Lie algebras arising from associative algebras over rings of scalars with no 2-torsion, for Artinian Lie algebras over arbitrary rings of scalars, and for some others. In all these cases, this implies that nondegenerate Lie algebras are subdirect products of strongly prime Lie algebras, providing a structure theory for Lie algebras without any restriction on their dimension.

Prime and semiprime lattice ordered lie algebras

The concepts of prime Lie algebras and semiprime Lie algebras are important in the study of Lie algebras. The purpose of this paper is to investigate generalizations of these concepts to lattice ordered Lie algebras over partially ordered fields. Some results concerning the properties of l-prime and l-semiprime lattice ordered Lie algebras are obtained. A necessary and sufficient condition for a lattice ordered Lie algebra to be an l-prime Lie l-algebra is presented.

An elemental characterization of strong primeness in Lie algebras

In this paper we prove that a Lie algebra L is strongly prime if and only if [ x , [ y , L ] ] ≠ 0 for every nonzero elements x , y ∈ L . As a consequence, we give an elementary proof, without the classification theorem of strongly prime Jordan algebras, of the fact that a linear Jordan algebra or Jordan pair T is strongly prime if and only if { x , T , y } ≠ 0 for every x , y ∈ T . Moreover, we prove that the Jordan algebras at nonzero Jordan elements of strongly prime Lie algebras are strongly prime.

δ-derivations of prime Lie algebras

δ-derivations of prime Lie algebras