Corestricted group actions and eight-dimensional absolute valued algebras

Abstract

Abstract.A condition for when two eight-dimensional absolute valued algebras are isomorphic was given in [4]. We use this condition to deduce a description (in the sense of Dieterich, [9]) of the category of such algebras, and show how previous descriptions of some full subcategories fit in this description. Led by the structure of these examples, we aim at systematically constructing new subcategories whose classification is manageable. To this end we propose, in greater generality, the definition of sharp stabilizers for group actions, and use these to obtain conditions for when certain subcategories of groupoids are full. This we apply to the category of eight-dimensional absolute valued algebras and obtain a class of subcategories, for which we simplify, and partially solve, the classification problem.

Classification of the finite-dimensional real division composition algebras having a non-Abelian derivation algebra

“…Any permutation of a Cayley triple is a Cayley triple. Given a Cayley triple (u, v, z), the element u generates the two-dimensional subalgebra 1, u ≃ C, and {u, v} generates the four-dimensional subalgebra 1, u, v, uv ≃ H. 2 It is known that the group G 2 acts transitively on the set of all Cayley triples, and that for any two such triples there is a unique φ ∈ G 2 mapping one to the other. Thus by fixing a Cayley triple (u, v, z) once and for all, the elements of G 2 correspond bijectively to the set of all Cayley triples via φ → (φ(u), φ(v), φ(z)).…”

An approach to finite-dimensional real division composition algebras through reflections

“…an exhaustive list of pairwise non-isomorphic objects. The following isomorphism condition, formulated in terms of isotopes of O, is due to [4].…”

“…From this isomorphism condition we derived in [3] a description (in the sense of Dieterich) of A 8 . To define a description, we first recall that for each set X and group G acting on X, the group action category G X is the category with object set X, and, for each x, y ∈ X, morphism set G X(x, y) = (g, x, y) g ∈ G, gx = y .…”

“…For such algebras, the technicalities associated with the triality phenomenon disappear. One motivation to our idea above is that this occurs, as we will see, for several other values of m and n. This was investigated for m = 1 in [3].…”

“…Conditions for when two eight-dimensional absolute valued algebras are isomorphic were obtained in [4], and this was formalized as a description (in the sense of Dieterich) in [3]. These results are reviewed in Section 2.…”