Page 153 - Textos de Matemática Vol. 39

QUASIASSOCIATIVE ALGEBRAS 143
(2) Given a division associative algebra D and two natural numbers n and m (possibly equal), let Matn,m(D) be the set of (n + m) × (n + m) matrices over D, with the chess-board Z2-grading:
Matn,m(D)0 = a 0 : a ∈ Matn(D),b ∈ Matm(D) 0b
Matn,m(D)1 = 0 v :v∈Matn×m(D),w∈Matm×n(D), w0
with multiplication given by
a1 v1 · a2 v2 = a1a2 +v1w2 a1v2 +v1b2 .
w1 b1 w2 b2 w1a2 + b1w2 −w1v2 + b1b2
It is easy to check that Matn,m(D) is a quasiassociative superalgebra,
whose even part is isomorphic to M atn(D) × M atm(D). Then the following theorem holds.
Theorem 3.4. [1] Any nonassociative quasiassociative superalgebra with semi- simple artinian even part A = A0 ⊕ A1 is a finite direct sum of ideals
A = A 1 ⊕ · · · ⊕ A r ⊕ Aˆ 1 ⊕ · · · ⊕ Aˆ s ⊕ A˜ w h e r e :
For i = 1,...,r, Ai is isomorphic to Mat (∆i) for some n and some
ni i division nonassociative quasiassociative superalgebra ∆i;
ˆ
For j = 1,...,s, Aj is isomorphic to Mat (Dj) for some division
nj ,mj associative algebra Dj and natural numbers nj and mj;
A˜ is a trivial nonassociative quasiassociative superalgebra with semisimple even part.
Moreover, r, s, the ni’s and the pairs {nj,mj} are uniquely determined by A and so are (up to isomorphism) A˜, the division nonassociative quasiasso- ciative superalgebras ∆i’s and the division algebras Dj’s.
4. Simple quasiassociative algebras
The aim of this section is to study simple nonassociative quasiassociative algebras with artinian null part. The associative case was considered in [2] and [8].
Definition 4.1. We say that a quasiassociative algebra A = ⊕g∈GAg is simple if it has no nontrivial two-sided graded ideals.
Recall that I ⊆ A is a two-sided graded ideal if I = ⊕g∈GIg and Ig =I∩Ag.