On automorphisms and derivations of Cayley-Dickson algebras
It is well-known that the generalized Cayley-Dickson algebra \(A_n^{\Gamma}\) (where \(A_0^{\Gamma}=F\), the base field of characteristic \(\neq 2\), \(A^{\Gamma}_{i+1}=A_i^{\Gamma}\otimes A_i^{\Gamma}\) with involution, defined by \(\overline{(a,b)}=(\bar a,-b)\) and multiplication \((a,b)(c,d)=(ac+\gamma_{i+1}\bar db,b\bar c+da)\); \(\Gamma\) is the sequence of structure constants, a collection of nonzero elements of F and \(\gamma_{i+1}\in \Gamma)\) is not associative for \(n\geq 3\), but does satisfy the flexible law. The authors prove that if \(\{x,y,z\}=0\) for certain \(x,z\in A_n^{\Gamma}\) and all \(y\in A_n^{\Gamma}\) \((n\geq 3)\), where \(z\) satisfies special conditions, then the vector-space, spanned by \([1,x,z]\) is a commutative algebra of dimension \(\leq 2\). Then it is possible to give a proof of a theorem on derivations of Cayley-Dickson algebras, originating from Schafer. This result leads to the calculation of the automorphism groups of Cayley-Dickson algebras: \[ \Aut_F(A^{\Gamma}_{n+1})\cong \Aut_F(A_n^{\Gamma})\oplus G, \] where \(G\cong \mathbb Z/2\) or \(G\cong S_3\), depending on the values of the structure constants. Finally the authors prove for \(n\geq 4\): \(A_n^{\Gamma}\cong A_n^{\Gamma '}\) if and only if \(\gamma_n\lambda_n^{-1}\) is a square in \(F\) and \(A^{\Gamma}_{n-1}\cong A^{\Gamma '}_{n-1}\). Here the characteristic of \(F\neq 2\) and 3, and \(\gamma_i\) and \(\lambda_j\) are the structure constants in \(A_n^{\Gamma}\) resp. \(A_n^{\Gamma '}\). The isomorphisms of Cayley-Dickson algebras of order \(\leq 3\) are already well-known.
scientific article; zbMATH DE number 4146218