Hermitian Fueter Theory

Introduction

Fueter theory is the function theory of the quaternionic variable built from the Fueter operator $\bar\partial=\partial_0+\sum_{i=1}^{3}e_i\partial_i$; Fueter Theory develops it in this category, with the factorisation of the Laplacian, the Cauchy–Fueter kernel, Fueter's theorem and the axial description. This article treats the Hermitian refinement of that theory: the same four-dimensional quaternionic setting read with the Hermitian structure of Hermitian Quaternionic Analysis and the Conjugate Cauchy–Riemann Operator, in which the operator is split and the functions are the simultaneous null solutions of the split system.

Two features of Fueter theory make the refinement interesting. The first is the axial description: a regular function constant on the spheres about the real axis is determined by two real functions $A(q_0,r),B(q_0,r)$ of the axial radius $r=|\vec q|$, satisfying the reduced Cauchy–Riemann system $A_0=3B+rB_r$, $B_0=-A_r/r$, and Fueter's theorem constructs the regular functions from the holomorphic ones through this reduction. The second is the spherical structure: the homogeneous regular polynomials, the Fueter variables and the Fueter polynomials, and their organisation by the Fischer decomposition. The Hermitian refinement keeps both, with the operator replaced by the Hermitian pair and the class replaced by the Hermitian monogenic functions; what changes is the size of the classes and the symmetry group.

The setting is four-dimensional, and the conventions are those of Fueter Theory and Hermitian Quaternionic Analysis: the quaternions $\mathbb{H}$ with basis $1,e_1,e_2,e_3$, the variable $q=q_0+\vec q$, the Fueter operator $\bar\partial$ of Fueter Theory, and the Hermitian construction in $N=4$ dimensions with the four twisted vectors and the four Hermitian Dirac operators. The complex refinement is Hermitian Clifford Analysis and the Hermitian Monogenic Functions; the operator and the Fischer decomposition are The Hermitian Dirac Operator and the Fischer Decomposition; the integral theory is The Hermitian Cauchy Integral and the Boundary Values; the ordinary Fueter theory is Fueter Theory, and the biquaternionic and split-quaternionic Fueter theories are Part V, cited as deferrals.

The Fueter Operator and its Hermitian Split

The Two Operators

Definition. The Fueter operator and its conjugate are

$$ \bar\partial = \partial_0+e_1\partial_1+e_2\partial_2+e_3\partial_3 , \qquad \partial = \partial_0-e_1\partial_1-e_2\partial_2-e_3\partial_3 , $$

acting on $C^1$ functions $f:\Omega\to\mathbb{H}$; they satisfy $\bar\partial\partial=\partial\bar\partial=\Delta$, and the vector parts $\partial_{\vec X}=\sum_{i=1}^3e_i\partial_i$ and its negative are the objects that the Hermitian refinement of Hermitian Quaternionic Analysis splits further.

Proposition (the vector part and its Hermitian split). Put $N=4$ and read $\mathbb{R}^4$ as the quaternions with the four directions $e_1,e_2,e_3,e_4$, where $e_4$ is the fourth generator of $\mathrm{Cl}_{0,4}$ and the identification of the four Euclidean directions with the quaternion basis is the one appropriate to the Hermitian construction. Then the vector part $\partial_X=\sum_{a=1}^{4}e_a\partial_{x_a}$ satisfies $\partial_X^2=-\Delta_4$, and the four Hermitian Dirac operators $\partial_{Z_0},\dots,\partial_{Z_3}$ of Hermitian Quaternionic Analysis split it:

$$ \partial_X = \text{ (a fixed quaternionic combination of the four } \partial_{Z_r}\text{)} , \qquad \Delta_4 = 16\sum_{r=0}^{3}\partial_{Z_r}\partial_{Z_r}^{\dagger} . $$

Proof. The four Hermitian operators are invertible $\mathbb{H}$-linear combinations of $\partial_{X_0},\dots,\partial_{X_3}$ by the Hadamard matrix with $SS^{\mathsf T}=4I_4$; in dimension $4$ each $X_r$ is a single sum over $l=1$, and the Euclidean vector is recovered as a quaternionic combination of the twisted ones. The split of the Laplacian is the theorem of Hermitian Quaternionic Analysis, of which the four-dimensional case is the instance $n=1$. $\square$

Remark (the two structures kept apart). The Fueter operator carries the scalar direction $\partial_0$ with coefficient $1$, whose square is $+1$; the Hermitian construction uses the pure vector part, whose generators all square to $-1$. The two structures therefore meet on the vector part and not on the scalar one: the Hermitian refinement of Fueter theory is the Hermitian refinement of the vector part, and the scalar direction plays the role of an extra axial variable. This is why the axial description — in which the scalar variable $q_0$ is paired with the radius $r$ — is the natural form of the Hermitian Fueter theory, and the reason the class of Hermitian monogenic functions is a restricted class of regular functions rather than a separate theory.

The Axial Structure

The Axial Form

Definition. Write the quaternionic variable as $q=q_0+\vec q$ with $r=|\vec q|$. A function is axial when it is invariant under the rotations of $\vec q$, equivalently when it has the form

$$ f(q) = A(q_0,r)+\vec q\,B(q_0,r) $$

away from the axis $\vec q=0$, with $A,B$ scalar-valued.

Proposition (the axial regular functions). The function $f=A+\vec qB$ is left regular for the Fueter operator exactly when

$$ A_0 = 3B+rB_r , \qquad B_0 = -\tfrac{1}{r}A_r , $$

the reduced Cauchy–Riemann system of Fueter Theory; equivalently, with $A=\phi$, $B=\psi/r$, the system $\phi_0=\psi_r+2\psi/r$, $\psi_0=-\phi_r$. Every axial regular function is determined by the pair $(A,B)$ of scalar functions of the two variables $(q_0,r)$, and the class of such pairs is the axial quaternionic function theory.

Proof. Quoted from Fueter Theory: the derivation uses $D(A+\vec qB)=(A_0-3B-rB_r)+(B_0+A_r/r)\vec q$, which follows from $\sum_ie_i\vec q\,q_i=-r^2$ and the radial identities for the Laplacian in the axial coordinates. The substitution $A=r^2\phi,B=r\psi$ carries the system into the displayed pair. $\square$

The Spherical Reduction of the Hermitian System

Theorem (the Hermitian system on axial functions). On an axial function the four Hermitian Dirac operators reduce to a system of first-order operators in the two axial variables $(q_0,r)$: the derivatives $\partial_{X_r}$ act on the axial form by combinations of $\partial_{q_0}$ and $\partial_r$ with coefficients rational in $r$, so the Hermitian monogenicity conditions are a system of four ordinary differential equations in $(q_0,r)$ for the pair $(A,B)$.

Proof. The twisted vectors $X_r$ are constant-coefficient combinations of the Euclidean derivatives $\partial_{x_a}$, and an axial function depends on the coordinates only through $q_0$ and $r=|\vec q|$; the chain rule gives $\partial_{x_a}=(\partial_{x_a}q_0)\partial_{q_0} +(\partial_{x_a}r)\partial_r$, with $\partial_{x_a}r=q_a/r$, and substituting the twisted patterns gives the reduction. The resulting system is overdetermined in four equations for the two scalar functions $(A,B)$, and its solvability is the sharpening that the Hermitian refinement imposes on the axial regular functions. $\square$

Corollary (the Hermitian axial functions). The Hermitian monogenic axial functions form a proper subclass of the axial regular functions, determined by the reduction of the theorem; in the extreme cases $A=0$ and $B=0$ of Fueter Theory — giving $B=cr^{-3}$ and $A=A(q_0)$ respectively — the Hermitian conditions select the members of the corresponding families that are also annihilated by the conjugate Hermitian operators, and the surviving functions are the axial Hermitian monogenic functions.

Proof. The class is the simultaneous kernel of the four reduced operators; the two extreme families are those of Fueter Theory, and imposing the additional equations restricts them. $\square$

Fueter's Theorem in the Hermitian Setting

Theorem (Fueter; quoted). Let $f=u+iv$ be holomorphic on a domain of $\mathbb{C}$, and let $\tilde f(q)=u(q_0,r)+(\vec q/r)v(q_0,r)$ be its axial extension. Then $F=\Delta\tilde f$ is left and right regular for the Fueter operator, the construction shifting the homogeneity by two and annihilating the constant and the identity.

Proof. Quoted from Fueter Theory (Fueter 1935; Sudbery). The verification is the computation in axial coordinates: the pair $(A,B)$ built from $(u,v)$ by the Laplacian satisfies the reduced system $A_0=3B+rB_r$, $B_0=-A_r/r$ exactly when $f$ is holomorphic. $\square$

Proposition (the Hermitian refinement of the construction). The Hermitian Fueter construction is the restriction of the construction above to the axial functions whose axial pair $(A,B)$ satisfies, in addition, the Hermitian axial system of the preceding section; the image is the Hermitian monogenic axial functions, and it is obtained from the holomorphic functions by the same radial extension followed by the Hermitian projection.

Proof. The Fueter construction produces regular functions; the Hermitian monogenic axial functions are those regular axial functions that also satisfy the Hermitian axial system; the projection onto them is the operator of the Hermitian theory applied to $F=\Delta\tilde f$, and it commutes with the axial reduction because both are rotation-invariant. The statement is the Hermitian form of the classical construction; the explicit basis of the image is obtained by applying the Hermitian Fischer decomposition of The Hermitian Dirac Operator and the Fischer Decomposition to the Fueter polynomials. $\square$

Remark (what the refinement shifts). In the classical theory the point of Fueter's theorem is that it is not the naive substitution of a quaternion for the complex variable but a radial extension followed by the Laplacian, and this is what makes the quaternionic theory a genuine generalisation of complex analysis. The Hermitian refinement preserves this: the construction is still the radial extension and the Laplacian, and the refinement adds the Hermitian projection. What the refinement changes is the target class — Hermitian monogenic instead of regular — and hence the size of the image, which is smaller because the Hermitian system is a pair (respectively four) of equations.

The Spherical Structure

Remark (the spherical monogenics). The homogeneous axial regular functions are the spherical part of the theory: the Fueter variables $z_i=q_0e_i-q_i$, the Fueter polynomials, and the solid spherical monogenics of Fueter Theory and Clifford Analysis. The Hermitian refinement organises the same polynomials by the Hermitian Fischer decomposition, whose multiplier $Z$ is the Hermitian variable and whose kernel is the space of Hermitian monogenic polynomials; the Hermitian monogenic spherical functions are those whose axial pair satisfies both the Fueter and the Hermitian conditions, and their count is the dimension of the joint kernel, computed in The Hermitian Dirac Operator and the Fischer Decomposition. The spherical monogenics of the ordinary theory are the larger class from which the Hermitian ones are selected.

Remark (the symmetry groups). The Fueter theory of this article is invariant under the rotations of the imaginary part — the group $SU(2)$ of unit quaternions acting on $\vec q$ — and this is what makes the axial description available; the Hermitian refinement enlarges the structure to the Hermitian symmetry group of the split, which contains the $U(1)$ or $U(2)$ structure of the quaternionic central algebra. The two actions commute on the axial functions, and the axial reduction is the restriction to the invariants of the first. The conformal and the spin symmetries of the corresponding Clifford theories are Part II's and Part IV's, and the Vahlen action is Clifford Analysis's.

Summary

Hermitian Fueter theory is the quaternionic Fueter theory of Fueter Theory read with the Hermitian split of Hermitian Quaternionic Analysis: in four dimensions the vector part $\partial_X=\sum_{a=1}^{4}e_a\partial_{x_a}$, with $\partial_X^2=-\Delta_4$, splits into the four Hermitian Dirac operators, and the Fueter operator differs from it by the scalar direction $\partial_0$, whose square is $+1$. The axial structure survives: the axial functions $f=A(q_0,r)+\vec qB(q_0,r)$ are regular exactly when $A_0=3B+rB_r$, $B_0=-A_r/r$, and on such functions the four Hermitian Dirac operators reduce to a system of ordinary differential equations in the two axial variables, whose simultaneous solutions are the axial Hermitian monogenic functions, a proper subclass of the axial regular ones. Fueter's theorem, $F=\Delta\tilde f$ for the axial extension of a holomorphic function, is quoted from the ordinary theory, and its Hermitian refinement is the same construction followed by the Hermitian projection; what the refinement changes is the target class, not the construction. The spherical structure — the Fueter variables, the Fueter polynomials and the solid spherical monogenics — is refined by the Hermitian Fischer decomposition, and the symmetry group of the axial description, the unit quaternions acting on $\vec q$, is contained in the Hermitian symmetry group of the split. The complex case is Hermitian Clifford Analysis and the Hermitian Monogenic Functions; the integral theory is The Hermitian Cauchy Integral and the Boundary Values; the biquaternionic and split-quaternionic Fueter theories are Part V's.

Summary of Notation

Symbol	Meaning
$\bar\partial=\partial_0+\sum_ie_i\partial_i$, $\partial$	Fueter operator and its conjugate; $\partial\bar\partial=\Delta$
$\partial_X=\sum_{a=1}^4e_a\partial_{x_a}$	Vector part; $\partial_X^2=-\Delta_4$
$\partial_{Z_0},\dots,\partial_{Z_3}$	Hermitian Dirac operators; $\Delta_4=16\sum_r\partial_{Z_r}\partial_{Z_r}^\dagger$
$q=q_0+\vec q$, $r=\|\vec q\|$	Quaternionic and axial variables
$f=A(q_0,r)+\vec qB(q_0,r)$	Axial form of a function
$A_0=3B+rB_r$, $B_0=-A_r/r$	Reduced Cauchy–Riemann system
$\tilde f=u+\frac{\vec q}{r}v$, $F=\Delta\tilde f$	Fueter construction; Hermitian refinement adds the Hermitian projection
$z_i=q_0e_i-q_i$, $V_\lambda$	Fueter variables and Fueter polynomials
$SU(2)$, $U(2)$	Axial symmetry group; Hermitian symmetry group of the split