Composition algebras with large derivation algebras
An algebra \(A\) over a field \(F\) is called a composition algebra if \(A\) is equipped with a strictly nondegenerate quadratic form \(q:A\rightarrow F\) verifying \(q(xy)=q(x)q(y)\). The structure of unital composition algebras is well known: they all have dimension \(1,\;2,\;4,\) or \(8\), and are either the field \(F\), if its characteristic is not \(2\), \(F\oplus F\), the quadratic separable field extensions \(K\) of \(F\), the generalized quaternion algebras over \(F\), and the Cayley-Dickson algebras over \(F\). The authors call these algebras Hurwitz algebras. It is known that any finite dimensional composition algebra can be obtained from a Hurwitz algebra \(C\), with norm \(q\), by means of a new multiplication \(x\ast y=\phi (x)\psi(y)\), where \(\phi\) and \(\psi\) are orthogonal transformations relative to \(q\). Let \(x\mapsto \bar x\) be a canonical involution in a Hurwitz algebra \(C\). The authors call an algebra \(A\) a standard composition algebra associated to \(C\) if the product in \(A\) is given by one of the terms: \(xy, \;\bar x y,\;x \bar y,\;\bar x \bar y\), where juxtaposition denotes the multiplication in \(C\). In dimension 2, any composition algebra over an algebraically closed field is standard [\textit{H. Petersson}, Abh. Math. Semin. Univ. Hamb. 35, 215-222 (1971; Zbl 0217.06701)]. One of the main results of the paper states that the same is true in dimensions 4 and 8 for composition algebras with the largest possible derivation algebras. More precisely, the authors determine all the possibilities for the Lie algebra \(\text{Der } A\) of derivations for composition algebras \(A\) of dimension 4 and all the possibilities with maximal toral rank (equal to 2) in dimension 8, and then classify the composition algebras with the largest derivation algebras. In \(\dim A=4\) case \(\dim \text{Der } A\leq 3\), and \(A\) is standard if and only if \(\text{Der } A\) is a central simple 3-dimensional algebra; and in \(\dim A=8\) case \(A\) is standard if and only if \(\text{Der } A\) is a central simple Lie algebra of type \(G_2\). In dimension 8 case the composition algebras \(A\) with \(\text{Der } A\) a central simple Lie algebra of type \(A_2\) are also classified: they are isomorphic to certain algebras introduced by S. Okubo in connection with the \(SU(3)\) particle physics or to certain composition algebras related to the so-called color algebra.
scientific article; zbMATH DE number 1014003
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