The algebraic structure of quaternionic analysis (Q657674)

The aim of the paper is to present in detail a new interpretation of the notion of regularity, in the sense of Fueter, of quaternionic functions. It is based on a new decomposition of the real differential of a quaternionic function which, in turn, is a consequence of a general notion of quaternionic linearity. First, the author explains in the extensive Section 1 ``Introduction'' the main ideas of the work. The contents of Section 2 ``Orthogonal automorphisms of \(\mathbb{H}\)'' mostly can be found in the literature or is known as folklore but the author presents it from a certain original point of view. The title of Section 3 ``Quaternionic linearity'' is a bit misleading since it is not about the usual linearity which is classically defined as follows: let \(X\), \(Y\) be linear spaces over \(\mathbb{H}\) and let \(\ast\) denote the multiplications by the scalars for both spaces; an additive map \(\Lambda:X\to Y\) is said to be quaternionic linear when \[ \Lambda(q\ast x)=q\ast\Lambda(x)\tag{3.1} \] for every \(x\in X\) and \(q\in\mathbb{H}\). The author defines a ``relaxed'' version, called \(\varphi\)-linearity, or linearity with respect to \(\varphi\), if (3.1) is replaced by \[ \Lambda(q\ast x)=\varphi(q)\ast\Lambda(x)\tag{3.2} \] where \(\varphi:\mathbb{H}\to\mathbb{H}\) is now any given map. The associated class of maps is denoted by \(\mathrm{Lin}_{\varphi}^{\mathbb{H}}(X;Y)\). It turns out that the definition makes sense for a very restricted class of \(\varphi\)'s. Proposition 3.1. {Assume that \(\mathrm{Lin}_{\varphi}^{\mathbb{H}}(X;Y)\neq\{0\}\). Then either \(\varphi\) is an automorphism of \(\mathbb{H}\) or \(\bar{\varphi}\) is, depending on whether \(X\) and \(Y\) have or have not the same chirality.} (About the later one reads on p. 578: A vector space over \(\mathbb{H}\) may be a left space or a right space: the choice between the two possibilities is named here the chirality of the space itself. The term chirality context is used also.) Other properties of this linearity are studied as well. Next Section 4 ``Canonical decomposition of real linear maps and regularity'' relates the above \(\varphi\)-linearities with regularity. First of all, it reminds that every real linear map \(\Lambda:X\to Y\) between complex spaces may be uniquely decomposed into the sum \[ \Lambda=L_{\Lambda}+A_{\Lambda}\tag{4.1} \] of a complex linear and a complex anti-linear part, respectively. Straightforward computations lead to \[ L_{\Lambda}(x)=\frac{\Lambda(x)-i\Lambda(ix)}{2},\qquad A_{\Lambda}(x)=\frac{\Lambda(x)+i\Lambda(ix)}{2}. \] If \(\Lambda\) is the real differential of a function \(f:X\to Y\), the vanishing of one or the other component in an open subset \(\mathcal{U}\subset X\) decides the holomorphy type of \(f\) in \(\mathcal{U}\): by expressing it in a complex basis of \(X\), one finds the classical Cauchy-Riemann equations of complex analysis. The quaternionic generalization is presented in a series of statements. Proposition 4.1. {Assume that the automorphisms \(\omega_0\), \(\omega_1\), \(\omega_2\), \(\omega_3\) are quaternionic orthogonal. Then every \(\Lambda\in\mathrm{Lin}^{\mathbb{R}}(X;Y)\) can be uniquely decomposed as \[ \Lambda=\Lambda_0+\Lambda_1+\Lambda_2+\Lambda_3 \] where \(\Lambda_{h}\in\mathrm{Lin}_{\varphi_{h}}^{\mathbb{H}}(X;Y)\) for every \(h\), and \[ \varphi_h=\omega_h\quad or\quad \varphi_h=\bar{\omega}_h \] depending on whether \(X\) and \(Y\) have or have not the same chirality. Moreover, for every \(h\) one has \[ \Lambda_h(x)=\frac{1}{4}\sum_{k=0}^{3}\bar{\varphi}_h(e_k)\ast\Lambda(e_k\ast x)\tag{4.2} \] with \(e_0\), \(e_1\), \(e_2\), \(e_3\) any real orthonormal basis of \(\mathbb{H}\).} Corollary 4.2. {If \(\omega_0=\mathrm{id}\) in Proposition 4.1, then the two components \[ \mathcal{L}_{\Lambda}=\Lambda_0\quad \text{and}\quad \mathcal{R}_{\Lambda}=\Lambda_1+\Lambda_2+\Lambda_3 \] do not change when the imaginary orthogonal automorphisms \(\omega_1\), \(\omega_2\), \(\omega_3\) vary.} Definition 4.3. {A map \(\Lambda\) is regular when \(\mathcal{L}_{\Lambda}=0\).} Corollary 4.4. {A map \(\Lambda\in\mathrm{Lin}^{\mathbb{R}}(\mathbb{H})\) is regular if and only if \[ (\Lambda|\varphi_0)=0\tag{4.5} \] where either \(\varphi_0=\mathrm{id}\) or \(\varphi_0=\mathrm{cj}\), depending on the chirality context.} When all this is specified to the Fueter operator, the regularity in the sense of Definition 4.3 becomes the usual Fueter regularity and the Cauchy-Riemann equations for them arise. The last two sections 5 ``Quaternionic size of regular maps and complex linearity'' and 6 ``Some nonlinear consequences'' explain more peculiarities of the relation between linearity and regularity.