Division algebras of slice functions

Ghiloni, Riccardo; Perotti, Alessandro; Stoppato, Caterina

doi:10.1017/prm.2019.13

Cited by 25 publications

(31 citation statements)

References 21 publications

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“…Thanks to the latter property of quasi-open maps we obtain the Maximum Modulus Principle for slice regular functions defined on product domains, see [12,Theorems 7.1 and 7.2] for the case of slice domains and [1,Theorems 4.2] for a partial result in the case of product domains. See also [23] for a different approach. [21, §2] for the definition of the reciprocal function).…”

Section: The Fibers Of a Slice Regular Functionmentioning

confidence: 99%

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On a Class of Orientation-Preserving Maps of $$\pmb {\mathbb {R}}^4$$

Ghiloni¹,

Perotti²

2020

J Geom Anal

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The purpose of this paper is to present several new, sometimes surprising, results concerning a class of hyperholomorphic functions over quaternions, the so-called slice regular functions. The concept of slice regular function is a generalization of the one of holomorphic function in one complex variable. The results we present here show that such a generalization is multifaceted and highly non-trivial. We study the behavior of the Jacobian matrix J f of a slice regular function f proving in particular that det(J f ) ≥ 0, i.e. f is orientationpreserving. We give a complete characterization of the fibers of f making use of a new notion we introduce here, the one of wing of f . We investigate the singular set N f of f , i.e. the set in which J f is singular. The singular set N f turns out to be equal to the branch set of f , i.e. the set of points y such that f is not a homeomorphism locally at y. We establish the quasi-openness properties of f . As a consequence we deduce the validity of the Maximum Modulus Principle for f in its full generality. Our results are sharp as we show by explicit examples.2010 MSC: Primary 30G35; Secondary 30C15, 32A30, 57R45.Keywords: Quaternionic hyperholomorphic functions, Orientation-preserving maps, Singular and branch sets of differentiable maps, Quasi-openness, Maximum Modulus Principle. than two. During the last century several generalizations were introduced mainly over quaternions H, octonions O and Clifford algebras R m , see [5,27]. These generalized theories share many analytic and differential behaviors with the theory of holomorphic functions of a complex variable. However they do not include the classical theory of polynomials with noncommutative coefficients on one side, see [29].In 2006 Gentili and Struppa [13,14] remedied to this 'algebraic' lack introducing a new theory, the one of slice regular functions over quaternions. Such a theory was extended to octonions and Clifford algebras in [8,15,16]. In paper [20] we gave a unified and generalized approach valid over all real alternative * -algebras, based on the concept of stem function. The theory has developed rapidly, see e.g. [12,23] and references therein. It has proven also its effectiveness in applications to quaternionic functional calculus and mathematical foundation of quaternionic quantum mechanics (see e.g. [9,17,18]), classification of orthogonal complex structures in R 4 (see e.g. [3,4,10]), and operator semigroup theory in noncommutative setting (see e.g. [7,24,25]).The stem function approach over quaternions reads as follows, see [20]. Let H be the real division algebra of quaternions and let S H = {J ∈ H | J 2 = −1} be the 2-sphere of its imaginary units. For each J ∈ S H , we denote by C J = Span(1, J) ≃ C the subalgebra of H generated by J. Then we have the 'slice' decompositionGiven a (non-empty) subset D of C, we define the circularization Ω D of D as follows:

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Section: The Fibers Of a Slice Regular Functionmentioning

confidence: 99%

“…The theory has developed rapidly, see e.g. [12,23] and references therein. It has proven also its effectiveness in applications to quaternionic functional calculus and mathematical foundation of quaternionic quantum mechanics (see e.g.…”

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confidence: 99%

On a Class of Orientation-Preserving Maps of $$\pmb {\mathbb {R}}^4$$

Ghiloni¹,

Perotti²

2020

J Geom Anal

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“…In this paper we prove some new relations between the Cauchy-Riemann operator, the spherical Dirac operator, the Laplacian operator and the class of slice-regular functions on a Clifford algebra. Slice-regular functions constitute a recent but rapidly expanding function theory in several hypercomplex settings, including quaternions and real Clifford algebras (see [11,7,12,10,16,17]). This class of functions was introduced by Gentili and Struppa [11] in 2006-2007 for functions of one quaternionic variable.…”

Section: Introductionmentioning

confidence: 99%

Slice Regularity and Harmonicity on Clifford Algebras

Perotti

2019

Trends in Mathematics

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We present some new relations between the Cauchy-Riemann operator on the real Clifford algebra Rn of signature (0, n) and slice-regular functions on Rn. The class of slice-regular functions, which comprises all polynomials with coefficients on one side, is the base of a recent function theory in several hypercomplex settings, including quaternions and Clifford algebras. In this paper we present formulas, relating the Cauchy-Riemann operator, the spherical Dirac operator, the differential operator characterizing slice regularity, and the spherical derivative of a slice function. The computation of the Laplacian of the spherical derivative of a slice regular function gives a result which implies, in particular, the Fueter-Sce Theorem. In the two four-dimensional cases represented by the paravectors of R3 and by the space of quaternions, these results are related to zonal harmonics on the three-dimensional sphere and to the Poisson kernel of the unit ball of R 4 .

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“…This can be done by allowing poles and working with slice semi-regular functions. For a self-contained introduction to the topic of semi-regular function we refer to [12,13]. In particular, given any f ∈ R(Ω), we de ne the symmetrized function of f as the slice preserving function de ned by f s := f * f c = f , f * and the regular inverse as f −* := (f s ) − f c .…”

Section: De Nition 21mentioning

confidence: 99%

“…The aim of this paper is to exploit the powerfulness of Sylvester operators in the context of slice regularity, thus providing a series of outcomes, some of which quite unexpected, that deal with properties of commutation of semi-regular functions on axially symmetric domains contained in the skew algebra of quaternions. For a self-contained introduction to the subject of slice-regular and semi-regular functions see [9,11,13], while a systematic study of Sylvester operators in this setting can be found in [3].…”

Section: Introductionmentioning

confidence: 99%

Applications of the Sylvester operator in the space of slice semi-regular functions

Altavilla

Fabritiis

2020

Concrete Operators

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In this paper we apply the results obtained in [3] to establish some outcomes of the study of the behaviour of a class of linear operators, which include the Sylvester ones, acting on slice semi-regular functions. We first present a detailed study of the kernel of the linear operator ℒf,g (when not trivial), showing that it has dimension 2 if exactly one between f and g is a zero divisor, and it has dimension 3 if both f and g are zero divisors. Afterwards, we deepen the analysis of the behaviour of the -product, giving a complete classification of the cases when the functions fv, gv and fvgv are linearly dependent and obtaining, as a by-product, a necessary and sufficient condition on the functions f and g in order their *-product is slice-preserving. At last, we give an Embry-type result which classifies the functions f and g such that for any function h commuting with f + g and f * g, we have that h commutes with f and g, too.

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Division algebras of slice functions

Cited by 25 publications

References 21 publications

On a Class of Orientation-Preserving Maps of $$\pmb {\mathbb {R}}^4$$

On a Class of Orientation-Preserving Maps of $$\pmb {\mathbb {R}}^4$$

Slice Regularity and Harmonicity on Clifford Algebras

Applications of the Sylvester operator in the space of slice semi-regular functions

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