Universal factorization equalities over generalized quaternion algebras (Q2743174)

A generalized quaternion algebra \(({u,v\over F})\) over an arbitrary field \(F\) of characteristic not two is studied [cf. \textit{T. Y. Lam}, The algebraic theory of quadratic forms (1973; Zbl 0259.10019); \textit{R. S. Pierce}, Associative algebras (1982; Zbl 0497.16001); \textit{B. Farb} and \textit{R. K. Dennis}, Noncommutative algebra (1993; Zbl 0797.16001)]. Due to the definition, a generalized quaternion \(a=a_0 1+a_1i+a_2j+ a_3k\in ({u,v\over F})\) is presented in the basis \(1,i,j,k\) with following properties: \(ij=-ji=k\), \(i^2=u\), \(j^2=v\) \((u,v,\in F)\). In the paper, some fundamental equalities, namely, so called universal factorization equalities for an element \(a=({u,v \over F})\) and its matrix representation \(\varphi(a)\) are established, for example, as follows: NEWLINE\[NEWLINEP\left (\begin{matrix} a & & & \\ & a\\ & & a\\ & & & a \end{matrix} \right)P^{-1}= \varphi(a)NEWLINE\]NEWLINE where \(P(\varphi(a))\) are \(4\times 4\)-matrices with elements expressed in terms of quantities \(a_0,a_1,a_2,a_3\); \(u,v\) \((1,i,j,k;u,v)\). As a consequence, various operation properties of the generalized quaternions are obtained. Some applications and possible extensions of the equalities under consideration are also discussed.