Slice Dirac operator over octonions

Jin, Ming; Ren, Guangbin; Sabadini, Irene

doi:10.1007/s11856-020-2067-z

Cited by 16 publications

(10 citation statements)

References 28 publications

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“…Apart from this regularity concept there is also the concept of sliceregularity in these algebras. See, for instance, [11] and the very recent paper [17] which is based on a different geometric splitting. However, in this paper we restrict ourselves to focus entirely on the following definition.…”

Section: Algebraic Relations Between the Cm-division Values Of Octonimentioning

confidence: 99%

Function Theories in Cayley-Dickson Algebras and Number Theory

Kraußhar

2021

Milan J. Math.

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In the recent years a lot of effort has been made to extend the theory of hyperholomorphic functions from the setting of associative Clifford algebras to non-associative Cayley-Dickson algebras, starting with the octonions.An important question is whether there appear really essentially different features in the treatment with Cayley-Dickson algebras that cannot be handled in the Clifford analysis setting. Here we give one concrete example: Cayley-Dickson algebras admit the construction of direct analogues of so-called CM-lattices, in particular, lattices that are closed under multiplication.Canonical examples are lattices with components from the algebraic number fields $$\mathbb{Q}{[\sqrt{m1}, \ldots \sqrt{mk}]}$$ Q [ m 1 , … mk ] . Note that the multiplication of two non-integer lattice paravectors does not give anymore a lattice paravector in the Clifford algebra. In this paper we exploit the tools of octonionic function theory to set up an algebraic relation between different octonionic generalized elliptic functions which give rise to octonionic elliptic curves. We present explicit formulas for the trace of the octonionic CM-division values.

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Section: Algebraic Relations Between the Cm-division Values Of Octonimentioning

confidence: 99%

Function Theories in Cayley-Dickson Algebras and Number Theory

Kraußhar

2021

Milan J. Math.

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“…In [28], a class of slice regular functions is defined on the n-dimensional quadratic cone of octonions by slice functions and corresponding stem functions, following the slice analysis on real alternative * -algebras, see [23]. This technique plays a fundamental role in slice analysis in one variable and was used recently in the octonionic case in a new way, connecting slice analysis with quaternionic analysis, see [27]. In this section, we will generalize the slice functions defined in [28].…”

Section: Slice Functionsmentioning

confidence: 99%

Weak Slice Regular Functions on the n-Dimensional Quadratic Cone of Octonions

et al. 2021

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In the literature on slice analysis in the hypercomplex setting, there are two main approaches to define slice regular functions in one variable: one consists in requiring that the restriction to any complex plane is holomorphic (with the same complex structure of the complex plane), the second one makes use of stem and slice functions. So far, in the setting of several hypercomplex variables, only the second approach has been considered, i.e. the one based on stem functions. In this paper, we use instead the first definition on the so-called n-dimensional quadratic cone of octonions. These two approaches yield the same class of slice regular functions on axially symmetric slice-domains, however, they are different on other types of domains. We call this new class of functions weak slice regular. We show that there exist weak slice regular functions which are not slice regular in the second approach. Moreover, we study various properties of these functions, including a Taylor expansion.

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“…It would be interesting to investigate possible extensions of the results in [26,28] according to this new definitions. We will not pursue this here.…”

Section: Slice Monogenic Functions Of An Octonionic Variablementioning

confidence: 99%

“…Later, other variations of the notion of slice hyperholomorphicity were introduced; see [10,18,25,26]; however all of them have in common the fact that the domain of the functions can be expressed as the union of complex planes. In the particular case of Clifford algebras, this means that one cannot consider a fully Clifford variable as input of a function.…”

Section: Introductionmentioning

confidence: 99%

Slice monogenic functions of a Clifford variable via the 𝑆-functional calculus

Colombo

Kimsey

Pinton

et al. 2021

Proc. Amer. Math. Soc. Ser. B

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In this paper we define a new function theory of slice monogenic functions of a Clifford variable using the S S -functional calculus for Clifford numbers. Previous attempts of such a function theory were obstructed by the fact that Clifford algebras, of sufficiently high order, have zero divisors. The fact that Clifford algebras have zero divisors does not pose any difficulty whatsoever with respect to our approach. The new class of functions introduced in this paper will be called the class of slice monogenic Clifford functions to stress the fact that they are defined on open sets of the Clifford algebra R n \mathbb {R}_n . The methodology can be generalized, for example, to handle the case of noncommuting matrix variables.

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Slice Dirac operator over octonions

Cited by 16 publications

References 28 publications

Function Theories in Cayley-Dickson Algebras and Number Theory

Function Theories in Cayley-Dickson Algebras and Number Theory

Weak Slice Regular Functions on the n-Dimensional Quadratic Cone of Octonions

Slice monogenic functions of a Clifford variable via the 𝑆-functional calculus

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