Slice Dirac operator over octonions

The slice Dirac operator over octonions is a slice counterpart of the Dirac operator over quaternions. It involves a new theory of stem functions, which is the extension from the commutative $ O(1) $ case to the non-commutative $ O(3) $ case. For functions in the kernel of the slice Dirac operator over octonions, we establish the representation formula, the Cauchy integral formula (and, more in general, the Cauchy-Pompeiu formula), and the Taylor as well as the Laurent series expansion formulas.


Introduction
The purpose of this article is to initiate the study of the slice Dirac operator over octonions. The Dirac operator for quaternions has its root in mathematical physics, quantum mechanics, special relativity, and engineering (see [1,2,22]) and it plays a key role in the Atiyah-Singer index theorem (ref. [5]). It may be called Dirac operator since it factorizes the 4-dimensional Laplacian. However, we note that in the literature (1.1) is often called generalized Cauchy-Riemann operator or Cauchy-Fueter operator, see e.g. [6,23,32], even though it has been originally introduced in a paper by Moisil, see [24]. Based on the Dirac operator for quaternions in (1.1), we shall introduce what we call the slice Dirac operator over octonions, using the slice technique. This technique was used by Gentili and Struppa for quaternions in [15,16] and for octonions in [17] based on Cullen's approach [11]. This technique makes it possible to extend some properties of holomorphic functions in one complex variable to the high dimensional and non-commutative case of quaternions. It has found significant applications especially in operator theory [3,9,10], differential geometry [14], geometric function theory [26,27] and it can be generalized to other higher dimensional settings like Clifford algebras [7,8] and real alternative algebras [18,19,20,28].
The heart of the slice technique comes from the slice structure of quaternions H, namely the fact that H can be expressed as union of complex half planes as where S denotes the set of imaginary unit in H, and C + I is the upper half plane {x + yI : x ∈ R, y ≥ 0}.
From this decomposition, it is then natural to say that quaternions have a book structure since C + I plays the role of a page in a book for any I ∈ S. The real axis R plays the role of the edge of the book in which all the pages of the book intersect, i.e., C + I ∩ C + J = R for any I = J.
The book structure for quaternions plays the same role as the sheaf or fiber bundle structure in differential geometry.
It is remarkable that the topology in the book structure is no longer the Euclidean one. Indeed, the distance compatible with the topology is given by the Euclidean one in a plane, otherwise the distance between any two points from distinct half planes is measured through the path of light via the real axis.
Following the Fueter's construction [13], see also [31], when one considers an open set O in the upper half complex plane C + minus the real line and a holomorphic function f (x + ιy) = F 1 (x, y) + ιF 2 (x, y), one may define a function defined over the quaternions using the book structure. In fact, if we consider q = x + Iy, y > 0, for some suitable I, we may set f (q) = f (x + Iy) = F 1 (x, y) + IF 2 (x, y). Note that q = x−Iy, y > 0 and so, by definition, f (q) = f (x−Iy) = F 1 (x, y)−IF 2 (x, y). Note also that the pair (F 1 , F 2 ) satisfies the Cauchy-Riemann system and thus f (x + Iy) is in the kernel of the Cauchy-Riemann operator ∂ x +I∂ y . If one is willing to extend the definition to the points of the real line, there is a problem since if q ∈ R then q = x + I0 and the imaginary unit I is no longer unique.
To solve this problem, one may consider a weaker notion of book structure and observe that H = in other words, we may consider H as the union of complex planes. Following a slight modification of the Fueter's construction, see [25], let O an open set symmetric with respect to the real axis (possibly intersecting the real axis) and a holomorphic function f (x + ιy) = F 1 (x, y) + ιF 2 (x, y). If F 1 , F 2 are an even-odd pair in the second variable, namely if they satisfy we may define a function on an open set in H (suitably constructed using O) Note that these conditions immediately imply that so that f is well defined. Moreover, the fact that F 2 is odd in the second variable imply that F 2 (x, 0) = 0, thus f is well defined also at real points. This second approach is the one that we will generalize to the octonionic case.
To this end, we set and we consider where O(1) is identified with the group of matrices 1 0 0 1 , Then we have 2) can be rewritten as Thus, following [29], we impose that and any F satisfying this condition is called stem function.
We can regard at this construction as the commutative stem function theory since F is invariant under the commutative group O(1).
As we shall see, the significant property of the slice regular function in H (noncommutative counterpart of holomorphic functions, i.e. holomorphic maps depending on the parameter I ∈ S) is given by the representation formula, which demonstrates that any slice regular function is completely determined by its evaluation at any two distinct half planes, or pages in this description.
In order to extend the slice theory for the Cauchy-Riemann operator over quaternions into the slice theory for the slice Dirac operator over octonions, we need to introduce a modified theory of stem functions. It turns out that the corresponding notion of stem function is invariant under the non-commutative group O(3). It will result in a new form of the representation formula, expressed in term of a quaternionic matrix.
We point out that the non-commutative and non-associative setting of octonions has found significant applications in the universal model of M -theory, in which the universe is given by the product of the 4-dimensional Minkowski space with a G 2manifold of very small scale. Here the exceptional Lie group G 2 is an automorphism group of octonions (ref. [4,21]).
We conclude this introduction with a remark about our definition of intrinsic and stem functions. Rinehart [29] studied the intrinsic functions as self-mappings of an associative algebra. In contrast, our intrinsic functions have distinct dimensions for their definition and target domains, and are constructed in the non-associative setting; see also [12,31]. Fueter [13] initiated the study of stem functions for complex-valued functions in his construction of radially holomorphic functions on the space of quaternions; see [18] for its recent development. However, their considerations are all restricted to the commutative O(1) setting. In this paper we initiate the study in the non-commutative O(3) setting. It is interesting to note that the procedure we followed in this paper may lead to further generalizations to higher dimensional algebras.
The structure of this paper is as follow: in section 2, we recall some important properties of octonionic algebra O. In section 3, we introduce the book structure in the octonionic algebra in terms of quaternionic subspaces and the stem function for the non-commutative group O(3); we also provide the representation formula which can be written via a quaternionic matrix. In section 4 we introduce the slice Dirac operator and a splitting property for slice Dirac functions. Section 5 contains the Cauchy-Pompeiu integral formula for slice functions and the Cauchy integral formula for slice Dirac-regular functions. Finally, in section 6 we give the expansion of slice Dirac-regular as Taylor series as well as Laurent series.

The algebra of octonions
The algebra of octonions O is a real, alternative, non-commutative and non-associative division algebra (see for example [30]). It is isomorphic to R 8 as a real inner product vector space and it can be equipped with the standard orthogonal basis : e 0 = 1, e 1 , . . . , e 7 .
The octonionic algebra O also can be generated from the quaternions algebra H Every x ∈ O can be written as We can introduce its conjugate e k x k , and then set The modulus is multiplicative, i.e.
In the sequel, given x ∈ O we introduce a left multiplication operator L In general, for any x, y ∈ O, L x L y = L xy , but equality may hold when suitable assumptions hold: [30]). The subalgebra generated by any two elements of an alternative algebra is associative. In particular, for all r ∈ R, and for all It is also useful to recall the following well known result:

Stem function in the octonionic setting
Let O be the algebra of octonions. The set of its imaginary units is a sphere of dimension six The set of all such row vectors I is denoted by N . For any I := (1, I, J, K) ∈ N , we consider the algebra of quaternions generated by it, i.e., We can endow the octonionic algebra with a structure that we still call book struc- as we prove in the following result: Proposition 3.1. The octonionic algebra has the structure: Proof. Any x ∈ O can be written as the sum of its real part x 0 and its imaginary part Im(x) = 7 k=1 e k x k . Therefore, it can be further expressed as x = x 0 + Iy with x 0 , y ∈ R and I = Im(x)/|Im(x)|. We have that thus I ∈ S 6 . Now we can choose J, K ∈ S 6 such that I := (1, I, J, K) ∈ N .
We note that, in general, any x ∈ O belongs to more than one quaternionic space, as the following example shows.
It is easy to see that Let O(4) be the group of orthogonal transformations of R 4 , and let O(3) be its subgroup keeping the real axis invariant. Therefore, any g ∈ O(3) can be regarded as matrix in the form where P is an orthogonal transformations of R 3 . The transformation g : R 4 −→ R 4 can be naturally extended to be a map g : We also recall that in the quaternionic case, the stem function is complex intrinsic i.e. it is invariant under the commutative group O(1). In other words, f (z) = f (z) wherez denotes the complex conjugation. In our setting, the stem function is intrinsic under the non-commutative group O(3) and this set is evidently nonempty since it contains, e.g., With the book structure, we can define a slice function by lifting a stem function.
Here we denote by x T the transpose of the row vector

If
[Ω] is a domain, then Ω is an axially symmetric domain.
For any x = (x 0 , x 1 , x 2 , x 3 ) ∈ R 4 , we consider the three involutions Let I ∈ N be fixed arbitrarily. By virtue of the identification of H I with R 4 , the map α can be identified with the map To ease the notation we still write α instead of α I . The same convention is adopted in the sequel for β, γ, F , V, V α , V, V, P α , P α , P α , A α , and B α which also depend on I ∈ N .
It is easy to check that Ω ⊂ Ω.
for any q ∈ Ω with q = Ix T for some I ∈ N . We say that f is a (left) slice function (induced by F ).
Since, in general, any element in O may belong to more than one H I we need to prove the following: Proof. Assume that q ∈ O can be written in two ways as We divide the proof into various cases. Case 1: Assume that which yields I ′ F (x ′ T ) = IF (x T ) and the assertion follows. Case 2: If H I = H I ′ , we claim that H I and H I ′ intersect at C I for some I ∈ S 6 . Indeed, since q = Ix T = I ′ x ′ T , there exists y 0 , y 1 ∈ R and I ∈ S 6 such that q = y 0 + Iy 1 ∈ C I and the claim follows. Therefore, we can choose J 1 , J ′ 1 ∈ S 6 respectively such that In conclusion, Definition 3.6 is well-posed. Now we provide the representation formula of slice functions in terms of a quaternion matrix: Theorem 3.9. Let f be a slice function on an axially symmetric set Ω in O. Let q ∈ O and let q = Ix T , for I ∈ N and x ∈ R 4 . Then for any p := I ′ x T with I ′ ∈ N the following formula holds: Proof. Since Ω is an axially symmetric set, we have α(q), β(q), γ(q) ∈ Ω for any q ∈ Ω. By definition, Thanks to Artin Theorem, see Theorem 2.2, we get By the definition of slice functions, for any I ′ = (1, I ′ , J ′ , K ′ ) ∈ N we then have This representation is very useful to prove further properties of slice functions. Moreover, notice that 2M I is an orthogonal matrix with elements in H I , i.e.
The following result shows that the slice function f (Ix T ) is a linear function of I. Proof. By construction, M I F (q) is independent of I ′ . Theorem 3.9 shows that holds for any I, which implies that M I F (q) is independent of I. (In alternative, one can prove the assertion noting that (3.5) shows that and so M I F (q) is independent of I). Moreover, the linearity in I ′ is immediate.
Remark 3.12. Also the representation formula for quaternionic slice regular functions can be written in matrix form. In fact, for any I, J ∈ S where S is the set of imaginary unit of quaternions H, and for any x, y ∈ R, the representation formula can be written as

Slice Dirac operator
In this section, we introduce the slice Dirac operator in O and establish the corresponding splitting lemma. We begin by recalling the Dirac operator (1.1) introduced in Section 1: , and its conjugate operator For any fixed I = (1, I, J, K) ∈ N , we define the slice Dirac operator in O as In the sequel, the restriction f | H I of a function f to H I shall be denoted by f I : We now introduce a main definition: Let Ω be an axially symmetric domain in O and let f ∈ S(Ω) ∩ then f is called a (left) slice Dirac-regular function in Ω.
We denote the set of slice Dirac-regular functions on the axially symmetric set Ω by SR(Ω). Proof. Let f be slice Dirac-regular and let q ∈ Ω ∩ H I , q = Ix T . Then using (4.2) we have:    Example 4.5. We further generalize the above example, by constructing a function
It is direct to verify that (3.1) holds so that F is an O−stem function. This stem function F induces a slice function f : O → O defined by for any I = (1, I, J, K) ∈ N . Since S, h satisfy equations (4.5), it is easy to verify that F is a solution of equations (4.2). This means that f is a slice Dirac-regular function on O.
An explicit example for F is given by , but many others can be easily written.
The restriction of a slice Dirac-regular function to a quaternionic space H I satisfy the following splitting property: f (q) = G 1 (q) + e 4 G 2 (q), ∀ q ∈ Ω I .
Proof. Since f is an octonion-valued function, there exists an element e 4 ∈ S 6 with e 4 ⊥ H I such that where G 1 , G 2 ∈ H I . Hence, also using Lemma 2.1, it follows that which implies D I G 1 = D I G 2 = 0, and the assertion follows.
Remark 4.7. We note that, in principle, one could have written and the condition of being slice Dirac regular would translate into obtaining that G 2 is right regular.

Cauchy integral formula
In this section, we present the Cauchy integral theory for slice Dirac operator. denote the unit exterior normal to the boundary ∂Ω I at ξ. We consider the Cauchy kernel in H I defined by and we finally let be the Lebesgue volume element in R 4 , and dS the induced surface element.
Theorem 5.1. Let f : Ω −→ O be a slice function on a bounded axially symmetric set Ω ⊂ O. Suppose that f I ∈ C 1 (Ω I ) and ∂Ω I is piecewise smooth for some given I ∈ N . Then for all q ∈ Ω I , we have Proof. The classical divergence theorem shows that for any real-valued function µ ∈ C 1 (Ω I ) ∩ C(Ω I ). Thus for the octonion-valued function f I ∈ C 1 (Ω) ∩ C(Ω), we have By Lemma 4.6, there exist e 4 ∈ S 6 with e 4 ⊥ H I such that f I = G 1 + e 4 G 2 with G 1 , G 2 which are H I -valued. Hence for any map V : Ω I → H I such that V ∈ C 1 (Ω I ) ∩ C(Ω I ), we have where we have used associativity in H I . Similarly, we have The equalities (5.4) and (5.5) hold, in particular, when V is the Cauchy kernel in (5.1). We now fix q ∈ Ω I and note that where D ξ denotes the Dirac operator with respect to the variable ξ. Indeed,

Straightforward calculations show that
Take a sufficient small ε such that the ball B ε (q) centered at q and with radius ε is contained in Ω I . From (5.4), (5.5), and Lemma 4.6 we have (5.7) Hence we can calculate this integral as follows: (5.8) :=I ∂Ω I − I ε By Equation (5.6) and Artin Theorem 2.2, we can evaluate the limit of I ε : Let ε → 0 in (5.8), we get Corollary 5.2. Let f be a slice Dirac-regular function on a bounded axially symmetric set Ω. Suppose that f I ∈ C 1 (Ω I ) and ∂Ω I is piecewise smooth for some given I ∈ N . then Proof. If q ∈ Ω I , then (5.10) follows from Theorem 5.1 since D ξ f I (ξ) = 0. For q / ∈ Ω I , the integral at the left hand side of (5.11) is a proper integral so that after limit process, (5.7) becomes Since f is slice Dirac-regular in Ω, the left hand side vanishes and we obtain (5.11).
Using the representation formula, we can introduce another kernel which allows to write a Cauchy formula of more general validity.
Theorem 5.4. Let f be a slice function on a bounded axially symmetric set Ω, suppose that f I ∈ C 1 (Ω I ) and ∂Ω I is piecewise smooth for some given I ∈ N . Then for any p ∈ Ω, there exists I ′ ∈ N such that p = I ′ x T with x ∈ R 4 and where q = Ix T ∈ Ω I and V is as in (5.12).
Proof. By the representation formula in Theorem 3.9, for any q = Ix T ∈ Ω I and any I ′ ∈ N we have Since Ω is an axially symmetric set, it follows that α(q), β(q), γ(q) ∈ Ω I for any q ∈ Ω I . Theorem 5.1 gives Substituting (5.15) into (5.14) and moving the integral out, we finally get and (5.12) allows to conclude.
Note that the monomials f (q) = q n are not Dirac-regular. Their Dirac-regular counterparts are the homogeneous left and right Dirac-regular polynomials P α , defined by where q = x 0 + ix 1 + jx 2 + kx 3 . Here the sum runs over all n! a! different orderings of α 1 1 ′ s, α 2 2 ′ s and α 3 3 ′ s and i β l ∈ {i, j, k} for any l = 1, 2, · · · , n.
The polynomials P α are homogeneous of degree n, while V α is homogeneous of degree −n − 3 (see [6]). Let U n be the right quaternionic vector space of homogeneous Dirac-regular functions of degree n ∈ N. Then, the polynomials P α (α ∈ N 3 ) are Dirac-regular and form a basis for U n . Theorem 6.1. Let f be a slice Dirac-regular function in the unit ball B ⊂ O centered at the origin and let f ∈ C 1 (B). For any q ∈ B, there exist I ∈ N such that q ∈ H I , and where the power series is uniformly convergent over B I .
Proof. Let q ∈ B, then there exists I ∈ N such that q ∈ H I . Moreover, there exists a closed ball B ρ with ρ < 1 such that q ∈ B ρ . By Lemma 4.6, we can pick e 4 ∈ S 6 with e 4 ⊥ H I , and write where G 1 and G 2 are H I −valued Dirac-regular and conjugate Dirac-regular, respectively. The integral formula (5.10) gives By Theorem 28 in [32], we can expand V (ξ − q) in power series for any |q| < |ξ| and the right hand side converges uniformly in any region {(ξ, q) : |q| ≤ r|ξ|} with r < 1. Since q ∈ B ρ and ξ ∈ ∂B, we have |q| ≤ r|ξ| with r < 1. Using the rightmost expression in (6.3), we get (6.4) Using the first expression in (6.3) and repeating the procedure, we have (6.5) Substituting (6.4) and (6.5) into (6.2) we obtain Differentiating both sides of the integral formula (6.2), we have In particular, letting q → 0 we conclude that For any q = x 0 + Ix 1 + Jx 2 + Kx 3 ∈ H I , by direct calculation we have Note that F 0 does not satisfy the compatibility conditions (3.1). Therefore, not all P α can be extended to a slice Dirac-regular function on the whole O.
Remark 6.3. We still do not know if the series in (6.1) is convergent uniformly on the whole unit ball B, besides on the subsets B I . Our proof on B I depends on the explicit formula of the kernel V and the associativity of quaternions. This technique obviously fails in the setting of octonions and to consider the uniform convergence over B, one should follow a different approach. In fact, for any f ∈ C 1 (B), one needs the estimate and it is problematic to show that |f (Ix T ) − f (I ′ x T )| is small enough.
By the representation formula, we have f (I ′ x T ) = I ′ (M I F (q)).