Real division algebras with large automorphism group

Abstract

We continue the study of the 8-dimensional real division algebras whose derivation algebra is large (of type G 2, A 2, 2 A 1, or A 1) begun by Benkart and Osborn in the early 1980s. For some of the families of real algebras that they constructed, we find necessary and sufficient conditions for them to be division algebras, determine when two division algebras in such a family are isomorphic or determine the automorphism group of such an algebra. We use one of these families to prove that every 2-dimensional real division algebra embeds in a 4-dimensional and in an 8-dimensional real division algebra. A new family, F 6 , of non-isomorphic real 8-dimensional algebras, parametrized by the Euclidean space R 6, is constructed and studied in detail. The division algebras in F 6 correspond to a non-empty open subset of the parameter space. We also introduce an interesting 2-parameter subfamily F 2⊂ F 6 , which contains the generalized pseudo-octonion algebras. We obtain necessary and sufficient conditions for (A,μ)∈ F 2 to be a division algebra. In the generic case, the algebras in F 6 are 1-generated and have SO(3) as the automorphism group. We also determine all (1-, 2-, and 4-dimensional) subalgebras of the division algebras in F 6 . We show that there exist 8-dimensional (and 4-dimensional) real division algebras having a 1-dimensional subalgebra not contained in any 2-dimensional subalgebra. We also construct 8-dimensional real division algebras having a 2-dimensional subalgebra not contained in any 4-dimensional subalgebra.