On the classification of the real flexible division algebras

Tom 105 / 2006

Erik Darpö Colloquium Mathematicum 105 (2006), 1-17 MSC: 15A21, 17A20, 17A35, 17A36, 17A45. DOI: 10.4064/cm105-1-1


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.


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