The lie n-engel property in group rings

Let FGbe the group ring of a group Gover a field Fwhose characteristic is p≠ 2 Let ? denote the involution on FGwhich sends each group element to its inverse. Let (FG)⁺and (FG)^–denote, respectively, the sets of symmetric and skew elements with respect to ?.The conditions under which the group ring is Lie n-Engel for some nare known.We show that if either (FG)⁺or (FG)^–- is Lie n-Engel, and Gis devoid of 2-elements, then FGis Lie m-Engel for some m. Furthermore, we completely classify the remaining groups for which (FG)⁺is Lie n-Engel.     

Finite vertex algebras and nilpotence

I show that simple finite vertex algebras are commutative, and that the Lie conformal algebra structure underlying a reduced (= without nilpotent elements) finite vertex algebra is nilpotent.     

Quaternionic lie algebras

Lie algebras which are isomorphic to central quotients of quaternion division algebras are investigated.     

HNN-extensions of lie algebras

We define HNN-extensions of Lie algebras and study their properties. In particular, a sufficient condition for freeness of subalgebras is obtained. We also study differential HNN-extensions of associative rings. These constructions are used to give short proofs of Malcev's and Shirshov's theorems that an associative or Lie algebra of finite or countable dimension is embeddable into a two-generator algebra.

     

On primitive lie algebras

Sufficient conditions for Lie algebra primitiveness and examples of primitive Lie algebras and nonprimitive Lie algebras are given.     

The centralizer algebras of lie color algebras

In his 1977 paper, V.G. Kac classified the finite-dimensional simple complex Lie superalgebras. After Kac’s paper, M. Scheunert initiated the study of a generalization of Lie superalgebras - the Lie color algebras. We construct some new families of simple Lie color algebras. Following the work of A. Berele and A. Regev and A.N. Sergeev, who studied the general linear and sq(n)-series superalgebra cases, and the work of G. Benkart, C. Lee Shader, and A. Ram, who studied the orthosymplectic cases, we examine the centralizer algebras of some classical Lie superalgebras and their Lie color algebra counterparts acting on tensor space and derive Schur-Weyl duality results for these algebras and their centralizers.     

Underlying lie algebras of quadratic Novikov algebras

Novikov algebras were introduced in connection with the Poisson brackets of hydrodynamic type and the Hamiltonian operators in formal variational calculus. In this note we prove that the underlying Lie algebras of quadratic Novikov algebras are 2-step nilpotent. Moreover, we give the classification up to dimension 10.     

Rigidity of free algebras in varieties of lie algebras

In this paper the main result is the rigidity of varietally free Lie algebras within the varieties where they are free. In fact, these algebras are usually free in some larger varieties, such as various types of the commutator varieties, and this is demonstrated in this paper as well. A related paper is [2] where, among some other results, we claimed the rigidity of free metabelian algebras within the respective variety of Lie algebras.