Prime alternative algebras that are nearly commutative

S. V. Pchelintsev

Izvestiya. Mathematics , Tome 68 (2004) no. 1, pp. 181-204

Résumé

We prove that by deforming the multiplication in a prime commutative alternative algebra using a C-operation we obtain a prime non-commutative alternative algebra. Under certain restrictions on non-commutative algebras this relation between algebras is reversible. Isotopes are special cases of deformations. We introduce and study a linear space generated by the Bruck C-operations. We prove that the Bruck space is generated by operations of rank 1 and 2 and that “general” Bruck operations of rank 2 are independent in the following sense: a sum of $n$ operations of rank 2 cannot be written as a linear combination of $(n-1)$ operations of rank 2 and an arbitrary operation of rank 1. We describe infinite series of non-isomorphic prime non-commutative algebras of bounded degree that are deformations of a concrete prime commutative algebra.