• PDF / 343,786 Bytes
• 30 Pages / 429.408 x 636.768 pts Page_size
• 12 Downloads / 254 Views

DOWNLOAD

REPORT

---

SLICE DIRAC OPERATOR OVER OCTONIONS∗

BY

Ming Jin and Guangbin Ren Department of Mathematics, University of Science and Technology of China Hefei 230026, China e-mail: [email protected], [email protected]

AND

Irene Sabadini Dipartimento di Matematica, Politecnico di Milano Via Bonardi, 9, 20133 Milano, Italy e-mail: [email protected]

ABSTRACT

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.

∗ This work was supported by the NNSF of China (11771412). © The authors 2020. This article is published with open access at link.springer.com. Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (https://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution and reproduction in any medium, provided the appropriate credit is given to the original authors and the source, and a link is provided to the Creative Commons license, indicating if changes were made.

Received February 12, 2019 and in revised form September 23, 2019

1

2

M. JIN, G.-B. REN AND I. SABADINI

Isr. J. Math.

1. Introduction The purpose of this article is to initiate the study of the slice Dirac operator over octonions. The Dirac operator for quaternions, ∂ ∂ ∂ ∂ +i +j +k , ∂x0 ∂x1 ∂x2 ∂x3 has its root in mathematical physics, quantum mechanics, special relativity, and engineering (see [1, 2, 21]) and it plays a key role in the Atiyah–Singer index theorem (see [5]). It may be called the Dirac operator since it factorizes the 4-dimensional Laplacian. However, we note that in the literature (1.1) is often called the generalized Cauchy–Riemann operator or Cauchy-Fueter operator, see e.g., [6, 23, 32], even though it was 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 expre