Classification of the finite dimensional absolute valued algebras having a non-zero central idempotent or a one-sided unity (Q966113)

Absolute valued finite dimensional algebras having a non-zero central idempotent or a one-sided unit are classified up to isomorphism. As it is well known finite dimensional absolute valued algebras over the reals have dimension \(1\), \(2\), \(4\) or \(8\). The classification problem for dimensions \(\leq 4\) can be considered as closed after the studies of M. Ramírez and others. The complexity in dimension eight is much higher and this work is a contribution to this problem. A. Rochdi showed that the problem of classifying \(8\)-dimensional absolute valued algebras having a nonzero central idempotent or a one-sided unit is equivalent to the description of the orbits of the action of \(G_2\) on \(O(7)\) (considering \(G_2\) as a subgroup of \(O(7)\) and acting by conjugation). This problem is solved in the present paper (sections 5 and 6). The authors recover Rochdi's result from a much more general situation using composition \(k\)-algebras with an LR-bijective idempotent. They relate these with the pairs of isometrical unit-fixing linear operators on a unital composition algebra in terms of a certain equivalence of categories. Also, as a product of this research, the paper reveals a connection between certain representations of a suitable quiver, and anticommutative quadruples. As the authors mention, this establishes a link between a class of modules over the path algebra of the quiver (which is associative) and an interesting class of nonassociative algebras.