Alternative algebras having scalar involutions (Q1082425)

This paper classifies all alternative algebras, including infinite- dimensional ones, which admit an involution \(a\to a'\) such that \(a+a'\) is a scalar multiple of the identity for all a. If the form \((ab'+ba')\) is nondegenerate, then these algebras are called composition algebras, are known to be semisimple of dimension 1,2,4 or 8 and their structure is well known. The present paper considers the case when the form could be degenerate. Let B be a maximal nonsingular subalgebra of A and R be the radical of the form. Then A is the direct sum of B and R, R is the union of all nilpotent ideals of A, B has dimension \(2^ n\) where \(0\leq n\leq 3\) and \(R^{4-n}=0\). The structure of B is well known and R is both anti- commutative and anti-associative. By considering the possible structures for B the authors obtain the structures for A.