On a noncommutative algebraic geometry (Q2791817)

Let \(\mathbb{H}\) denote the field of quaternions, and consider \(\mathbb{H}\)-valued \(C^\infty\) quaternionic functions \(f\) on an open set \(U\) of \(\mathbb{H}\). For \(q\in\mathbb{H}\), write \(q= z_1+ z_2{\mathbf j}\) where \(z_1,z_2\in\mathbb{C}\), and writing \(f=f_1+f_2{\mathbf j}\) and NEWLINE\[NEWLINEDf(q)= {1\over 2}\Biggl({\delta\over\delta\overline{z_1}}+{\mathbf j}{\delta\over\delta\overline{z_2}}\Biggr)\;f(q),NEWLINE\]NEWLINE \(f\) is termed hyperholomorphic if \(Df=0\). The author characterizes those functions which are hyperholomorphic on \(U\) and whose inverses are hyperholomorphic on \(U\).NEWLINENEWLINE Section 4 of the paper describes a subspace of these functions which has ``good'' properties of addition and multiplication. In Section 5, Hamilton 4-manifolds for \(\mathbb{H}\) (analogous to Riemann surfaces for \(\mathbb{C}\)) are defined, examples of such manifolds are presented, and the description of a class of four-dimensional manifolds is begun.NEWLINENEWLINEFor the entire collection see [Zbl 1337.41001].