Page 152 - Textos de Matemática Vol. 39

142 ANA PAULA SANTANA AND HELENA ALBUQUERQUE
Theorem 3.1. [1] Given a division associative algebra D, σ an automorphism ofDsuchthatthereisanelement0̸=a∈Dwithσ2 :d→ada−1 andwith σ(a)=−a, on the direct sum of two copies of D: ∆=D⊕Du (here u is just a marking device), define the multiplication
(de + d1u)(fe + f1u) = (defe + d1σ(f1)a) + (def1 + d1σ(fe))u.
Then with ∆0 = D and ∆1 = Du, this is easily seen to be a division quasiassociative superalgebra that is not associative.
These quasiassociative superalgebras exhaust, up to isomorphism, the di- vision quasiassociative superalgebras that are not associative with nonzero odd part.
We generalized this theorem for G-graded division quasiassociative alge- bras.
Theorem 3.2. [6] Let D be a division associative algebra, G a finite group
and φ : G×G×G → K∗ a cocycle. Suppose that, for each g,h,l ∈ G,
there are automorphisms ψg of D, and nonzero elements cg,h of D satisfying
ψgψh = cg,hψghc−1 , and cg,hcgh,l = φ(g, h, l)ψg(ch,l)cg,hl. In the direct sum g,h
of |G| copies of D, ∆ = g∈G Dug (here ug is just a marking device, with ue = 1), consider the multiplication defined by
d1(d2ug) = (d1d2)ug
(d1ug)d2 = (d1ψg(d2))ug
(d1ug)(d2uh) = (d1ψg(d2)cg,h)ugh
foranyd1,d2 ∈Dandg,h∈G.Then,with∆e =Dand∆g =Dug,∆is
a quasiassociative division algebra. Conversely, every quasiassociative division algebra can be obtained this way.
This classification of division quasiassociative algebras allowed us to de- termine the structure of quasiassociative algebras with simple artinian null part.
Theorem 3.3. [6] Any nonassociative quasiassociative algebra A with simple artinian null part is isomorphic to an algebra of matrices Matn(∆), for some integer n and nonassociative quasiassociative division algebra ∆. The integer n is uniquely determined by A and so is, up to isomorphism, the division algebra ∆.
In case G = Z2 it was possible to go a bit further and classify quasi- associative algebras with artinian semisimple even part. For this we need to consider the following two types of nonassociative quasiassociative superalge- bras of matrices with semisimple artinian even part:
(1) Given a division quasiassociative superalgebra ∆ and a natural number n, Matn(∆), with the usual product, is a quasiassociative superalgebra with even part Matn(∆0) which is a simple artinian algebra.