Inversion and quasigroup identities in division algebras

The present article is concerned with division algebras that are structurally close to alternative algebras, in the sense that they satisfy some identity or other algebraic property that holds for all alternative division algebras. Motivated by Belousov's ideas on quasigroups, we explore a new approach to the classification of division algebras. By a detailed study of the representations of the Lie group of autotopies of real division algebras we show that, if the group of autotopies has a sufficiently rich structure then the algebra is isotopic to an alternative division algebra. On the other hand, it is straightforward to check that required conditions hold for large classes of real division algebras, including many defined by identites expressable in a quasigroup. Some of the algebras that appear in our results are characterized by the existence of a well-behaved inversion map. We give an irredundant classification of these algebras in dimension 4, and partial results in the 8-dimensional case.