Helena Albuquerque

Clifford algebras obtained by twisting of group algebras

Abstract We investigate the construction and properties of Clifford algebras by a similar manner as our previous construction of the octonions, namely as a twisting of group algebras of Z 2 n by a cocycle. Our approach is more general than the usual one based on generators and relations. We obtain, in particular, the periodicity properties and a new construction of spinors in terms of left and right multiplication in the Clifford algebra.

Quadratic Malcev superalgebras

A quadratic Malcev superalgebra is a Malcev superalgebra M=M0⊕M1 with a non-degenerate supersymmetric even invariant bilinear form B; B is called an invariant scalar product on M. In this paper, we obtain the inductive classifications of quadratic Malcev algebras and of Malcev superalgebras M=M0⊕M1 such that M0 is a reductive Malcev algebra and the action of the M0 on M1 is completely reducible.

Malcev superalgebras with trival nucleus

The nucleus of a Malcev superalgebra M measures how far it is from being a Lie superalgebraM being a Lie superalgebra if and only if its nucleus is the whole M. This paper is devoted to study Malcev superalgebras in the opposite direction, that is, with trivial nucleus. The odd part of any finite-dimensional Malcev superalgebra with trivial nucleus is shown to be contained in the solvable radical. For algebraically closed fields, any such superalgebra splits as the sum of its solvable radical and a semisimple Malcev algebra contained in the even part, which is a direct sum of copies of sl(2, F) and the seven-dimensional simple non-Lie Malcev algebra, obtained from the Cayley-Dickson algebra.

Quadratic Malcev Superalgebras with Reductive Even Part

It is our goal to give an inductive description of quadratic Malcev superalgebras with reductive even part. We use the notion of double extension of Malcev superalgebras presented by Albuquerque and Benayadi in [4] and transfer to Malcev superalgebras the concept of generalized double extension given in [6] for Lie superalgebras.

On the generators of Lie superalgebras

It is shown that any classical simple Lie superalgebra over an algebraically closed field of characteristic zero contains an element which generates the whole super-algebra.

Homogeneous Symmetric Antiassociative Quasialgebras

Our main purpose is to provide for homogeneous (even or odd) symmetric antiassociative quasialgebras a structure theory analogous to that for homogeneous symmetric associative superalgebras given in [5] and to present an inductive description of these classes of algebras.

Engel’s theorem for malcev superalgebras

A version of Engel’s theorem for Malcev superalgebras is proved in the spirit of theJacobson-Engel theorem for Lie algebras. Some consequences for the structure of Malcev superalgebras with trivial Lie nucleus are derived.

On the generators of semisimple Lie algebras

Abstract This article concerns the finite generation problem for semisimple Lie algebras. Having noticed a gap in the proof of a theorem of Ionescus, we show how the arguments can be modified in order to prove that result. We also present new results for some classes of semisimple Lie algebras which are most important in applications.

Alternative Twisted Tensor Products and Cayley Algebras

We introduce what we call alternative twisted tensor products for not necessarily associative algebras, as a common generalization of several different constructions: the Cayley–Dickson process, the Clifford process, and the twisted tensor product of two associative algebras, one of them being commutative. We show that some very basic facts concerning the Cayley–Dickson process (the equivalence between the two different formulations of it and the lifting of the involution) are particular cases of general results about alternative twisted tensor products of algebras. As a class of examples of alternative twisted tensor products, we introduce a tripling process for an algebra endowed with a strong involution, containing the Cayley–Dickson doubling as a subalgebra and sharing some of its basic properties.

On the nullity of lie algebras

Abstract This paper deals with the problem of the determination of the nullity of a finite dimensional Lie algebra L over a field of characteristic 0. If L is solvable, we get this nullity. In general, we obtain a natural number m such that m⩽nul(L)⩽m + 2.