On the classification of the real flexible division algebras

Erik Darpö¹

¹ Matematiska Institutionen Uppsala Universitet Box 480 S-75106 Uppsala, Sweden

Colloquium Mathematicum, Tome 105 (2006) no. 1, pp. 1-17

Résumé

We investigate the class of finite-dimensional real flexible division algebras.We classify the commutative division algebras, completing an approach by Althoen and Kugler. We solve the isomorphism problem for scalar isotopes of quadratic division algebras, and classify the generalised pseudo-octonion algebras. In view of earlier results by Benkart, Britten and Osborn and Cuenca Mira et al., this reduces the problem of classifying the real flexible division algebras to the normal form problem of the action of the group $\mathcal{G}_2$ by conjugation on the set of positive definite symmetric linear endomorphisms of $\mathbb R^7$. A method leading to the solution of this problem is demonstrated.In addition, the automorphism groups of the real flexible division algebras are described.

Détail

DOI : 10.4064/cm105-1-1

Keywords: investigate class finite dimensional real flexible division algebras classify commutative division algebras completing approach althoen kugler solve isomorphism problem scalar isotopes quadratic division algebras classify generalised pseudo octonion algebras view earlier results benkart britten osborn cuenca nbsp mira nbsp nbsp reduces problem classifying real flexible division algebras normal form problem action group mathcal conjugation set positive definite symmetric linear endomorphisms mathbb method leading solution problem demonstrated addition automorphism groups real flexible division algebras described