Page 154 - Textos de Matemática Vol. 39

144 ANA PAULA SANTANA AND HELENA ALBUQUERQUE
To study simple quasiassociative algebras it is useful to introduce the notion of radical.
Definition 4.2. Let A be a quasiassociative algebra. The radical of A, denoted by rad(A), is defined by rad(A) = ∩ {ann M: M simple graded left A-module}, where ann M denotes the annihilator of M in A.
It is easy to check that the radical of a quasiassociative algebra A is a proper two-sided graded ideal of A. So rad(A) = 0 if A is simple.
Theorem 4.3. Let A be a simple quasiassociative algebra with artinian null part Ae. Then Ae is a semisimple associative algebra.
Proof. Let J(Ae) denote the Jacobson radical of Ae. Given a simple graded A-module M = ⊕g∈GMg, each Mg is a simple Ae-module. So if a0 ∈ J(Ae), then a0Mg = 0,∀g. Thus J(Ae) ⊆ rad(A) = {0} and Ae is semisimple.
In case G = Z2, Theorem 3.4 gives a complete classification of nonasso- ciative quasiassociative superalgebras with semisimple artinian null part. So we only need to check which of these are simple to obtain the following result.
Theorem 4.4. Any simple nonassociative quasiassociative superalgebra A = A0 ⊕ A1, with A0 artinian, is isomorphic to one of the following alge-
bras:
Matn(∆), for some n and some division quasiassociative superalgebra ∆; Matn,m(D), for some division associative algebra D and natural numbers
n and m.
Moreover, the natural numbers {n, m} are uniquely determined by A and
so are (up to isomorphism) the division quasiassociative superalgebra ∆ and the division algebra D.
In the general case, the quasiassociative algebras with semisimple artinian null part are not classified. Nonetheless it is easy to see that the algebras Matn(∆), for some integer n and nonassociative quasiassociative division alge- bra ∆, exhibited in Theorem 3.3, are simple.
References
[1] Helena Albuquerque, Alberto Elduque e Jos´e P. Izquierdo, Z2 quasialgebras, Comm. in Algebra, 30 (2002), 2161-2174.
[2] Helena Albuquerque, Alberto Elduque e Jesus Laliena, Superalgebras with semisimple even part, Comm. in Algebra, 25 (1997), 1573-1587.
[3] Helena Albuquerque and Shahn Majid, Quasialgebra structure of the octonions, J. Al- gebra, 200 (1999), 188-224.
[4] Helena Albuquerque and Shahn Majid, Clifford Algebras obtained by twisting of Group Algebras, Journal of Pure and Applied Algebra, 171 (2002), 133-148.
[5] Helena Albuquerque and Ana Paula Santana, A note on quasiassociative algebras, The J.A. Pereira da Silva Birthday Schrift, Departamento de Matem´atica da Universidade de Coimbra, Coimbra, 2002.