Jordan Superalgebras of Type and the Wedderburn Principal Theorem (WPT)

An analogue of the Wedderburn Principal Theorem (WPT) is considered for finite-dimensional Jordan superalgebras 𝔍 with solvable radical 𝒩, 𝒩 ² = 0, and such that , where 𝔽 is a field of characteristic zero. It is proved that the WPT is valid under some restrictions over the irreducible -bimodules contained in 𝒩, and it is shown with counterexamples that these restrictions cannot be weakened.     

Finitely Generated PI-Superalgebras with Bounded Multiplicities of the Cocharacters

ABSTRACT

In this note, we characterize finitely generated superalgebras satisfying an ordinary polynomial identity whose multiplicities of the supercocharacters are bounded by a constant.     

Jordan Superhomomorphisms of Superalgebras

The aim of this article is to investigate Jordan superhomomorphisms of superalgebras (with and without superinvolution), using the theory of functional identities in superalgebras. As a consequence we slightly improve a result on Jordan homomorphisms of rings with involution.     

2-local superderivations on a superalgebra $${{\rm M}_{\rm n}({\mathbb C})}$$

The aim of this paper is to show that every 2-local superderivation on an associative superalgebra is a superderivation.      

On Lie supergroups and superbundles defined via the Baker-Campbell-Hausdorff formula

The Baker-Campbell-Hausdorff formula is extended to the class of Lie superalgebras and then is used to define a class of objects, which are called Lie supergroups. Such Lie supergroups turn out to be either Lie groups (then the Jacobi Z₂-identity is simultaneously the usual Jacobi identity) or some sets provided with a partially associative multiplication. Thus they are different objects from supergroups in the sense of Kostant, Berezin or A. Rogers. In particular no Grassman algebra is used in the paper. Examples of

1. 1) some matrix Lie superalgebras,

2. 2) certain superalgebras of the super-Poincaré algebra

3. 3) the special Lie superalgebras D(2, 1, ) and their superalgebras are presented.

Furthermore, cocycles of functions taking values in the Lie supergroups are defined and their partially associative multiplicative relations are considered. Some fibre bundles, whose transition functions determine such cocycles, are distinguished.     

Graded parafermions

A graded generalization of the Z_k parafermionic current algebra is constructed. This symmetry is realized in the osp(1|2)/U(1) coset conformal field theory. The structure of the parafermionic highest-weight modules is analyzed and the dimensions of the fields of the theory are determined. A free field realization of the graded parafermionic system is obtained and the structure constants of the current algebra are found. Although the theory is not unitary, it presents good reducibility properties.     

On Right Alternative Superalgebras

Simple right alternative superalgebras which have a simple algebra as even part and, as odd part, an irreducible bimodule over the even part are investigated. Under these conditions, superalgebras with one dimensional even part are classified, as well as superalgebras having M ₂(F) as even part and a unital irreducible bimodule over M ₂(F) of dimension less than or equal to 6 as odd part. It is shown that there is only a unique non alternative simple right alternative superalgebra of the first type and, for the second type, there is a infinite family depending on a single parameter.     

Triple products on  n

Triple products in  _n whose related algebra is  _n itself or  _n are classified up to isomorphism. This classification is obtained using the intimate relation between triple products and Lie (super)algebra structures.