Page 145 - Textos de Matemática Vol. 39

SIMPLE QUASIASSOCIATIVE ALGEBRAS
ANA PAULA SANTANA AND HELENA ALBUQUERQUE
Dedicated to Eduardo Marques de Sa´ on the occasion of his 60th birthday.
Abstract. In this paper we study simple quasiassociative algebras with ar- tinian null part, proving that they necessarily have semisimple null part. We classify the particular class of simple quasiassociative superalgebras, showing that each of these algebras is isomorphic either to a unique algebra of ma- trices Matn(∆), where ∆ is an division quasiassociative superalgebra, or to the quasiassociative superalgebra of matrices Matn,m(D), for some division algebra D and natural numbers n and m.
We also present a ‘quasi-LU’ decomposition of a square matrix. This exam- ple illustrates a setting we might call ‘Quasilinear’ Algebra, motivated by the notion of quasi-representations of quasiassociative algebras.
1. Introduction
Given a group G, the definition of a G-graded quasiassociative algebra was introduced by Albuquerque and Majid in [3]. This concept generalizes in a natural way the notion of a G−graded associative algebra.
Definition 1.1. [3] Given a group G, a field K and any invertible group cocycle φ : G × G × G → K∗, we say that a G-graded K-algebra A = ⊕g∈GAg is quasiassociative if it is unital and satisfies
( a b ) c = φ ( a¯ , ¯b , c¯ ) a ( b c ) ,
f o r a n y h o m o g e n e o u s a ∈ A a¯ , b ∈ A ¯b , c ∈ A c¯ ( a¯ , ¯b , c¯ ∈ G ) .
Werecallthatφ:G×G×G→K∗ isacocycleinGif,forallx,y,z,t∈G,
φ(x, e, y) = 1;
φ(x, y, z)φ(y, z, t) = φ(xy,z,t)φ(x,y,zt) . φ(x,yz,t)
Interesting examples of quasiassociative algebras are the deformed group algebras KF G which are defined as follows: the underlying vector space of KF G is the same as the underlying vector space of the group algebra KG, but the
2000 Mathematics Subject Classification. 17A70.
Key words and phrases. Nonassociative Algebras; Linear Algebra. The work was supported by CMUC-FCT.
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