Fused four-dimensional real division algebras
The authors consider two modifications of the Cayley-Dickson process. -- Let \({\mathcal A}= (\mathbb{R}^2, \cdot )\) be any two-dimensional real algebra and \(\varphi\) a nonsingular linear transformation of \(\mathbb{R}^2\) to \(\mathbb{R}^2\). Defining multiplication on the vector space \(\mathbb{R}^2 \oplus \mathbb{R}^2\) by \[ (a, b) (c, d)= (ac- b\varphi (d), ad+ b\varphi (c)) \] one gets a four-dimensional algebra, the \({\mathcal A}\)-based \(\varphi\)-algebra, which contains an isomorphic copy of \({\mathcal A}\) as a subalgebra. -- Taking any two-dimensional real algebras \({\mathcal A}= (\mathbb{R}^2, \cdot )\) and \({\mathcal B}= (\mathbb{R}^2, *)\) for the second modification and defining multiplication on \(\mathbb{R}^2 \oplus \mathbb{R}^2\) by \[ (a, b) (c, d)= (ac- b* d, ad+ b*c) \] the authors construct a four-dimensional real algebra, the \({\mathcal A}\)-based fused algebra \({\mathcal A} \oplus {\mathcal B}\), which again contains an isomorphic copy of \({\mathcal A}\) as a subalgebra. -- Since fused algebras are a generalization of \(\varphi\)-algebras only the former are investigated. Using a criterion for the positive definiteness of a certain quaternary quadratic form over \(\mathbb{R}\) the authors succeed in proving: \({\mathcal A} \oplus {\mathcal B}\) is a division algebra iff \({\mathcal A}\) and \({\mathcal B}\) are division algebras and \[ (\alpha \delta- \beta \gamma) (\pi\tau- \rho\sigma) <0 \] holds for any basis \(\{a, b\}\) of \(\mathbb{R}^2\) with \(a^2= \alpha a+ \beta b\), \(ab= \gamma a+ \delta b\) \((\alpha, \beta, \gamma, \delta\in \mathbb{R}\}\) in \({\mathcal A}\) and \[ a*a= \pi a+ \rho b, \quad a*b= \sigma a+\tau b \qquad (\pi, \rho, \sigma, \tau\in \mathbb{R}) \] in \({\mathcal B}\). As a corollary one has: an \({\mathcal A}\)-based \(\varphi\)-algebra is a division algebra iff \({\mathcal A}\) is a division algebra and \(\text{det } \varphi<0\). -- Examples of division algebras are given showing that the class of fused algebras is larger than the class of \(\varphi\)- algebras, that there are fused algebras which are neither quadratic nor \(\mathbb{C}\)-associative nor rotational scaled quaternion algebras and that there are fused rotational scaled quaternion algebras besides the quaternion algebra. -- Finally the quaternion algebra is proven to be the only quadratic fused division algebra and the only \(\mathbb{C}\)-associative fused division algebra.
scientific article; zbMATH DE number 703240