The algebra of slice functions

Ghiloni, Riccardo; Perotti, Alessandro; Stoppato, Caterina

doi:10.1090/tran/6816

Cited by 56 publications

(69 citation statements)

References 24 publications

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“…Let us recall some basic material concerning slice functions over H, see [20] and also [19,21,22] for a full account presented via the stem function approach, including generalizations.…”

Section: Preliminariesmentioning

confidence: 99%

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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“…Let us recall some basic material concerning slice functions over H, see [20] and also [19,21,22] for a full account presented via the stem function approach, including generalizations.…”

Section: Preliminariesmentioning

confidence: 99%

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

Ghiloni¹,

Perotti²

2020

J Geom Anal

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“…A different approach has been used in [10,Proposition 3.2]. A complete treatment has been given in [22,Section 2] in the case of slice functions, which subsumes the case of slice monogenic functions. In order to make our presentation self-contained, we provide here a detailed proof of Proposition 2.14.…”

Section: Preliminariesmentioning

confidence: 99%

Growth and distortion theorems for slice monogenic functions

Ren

Wang

2017

Pacific J. Math.

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Abstract. The sharp growth and distortion theorems are established for slice monogenic extensions of univalent functions on the unit disc D ⊂ C in the setting of Clifford algebras, based on a new convex combination identity. The analogous results are also valid in the quaternionic setting for slice regular functions and we can even prove the Koebe type one-quarter theorem in this case. Our growth and distortion theorems for slice regular (slice monogenic) extensions to higher dimensions of univalent holomorphic functions hold without extra geometric assumptions, in contrast to the setting of several complex variables in which the growth and distortion theorems fail in general and hold only for some subclasses with the starlike or convex assumption.

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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%

“…This approach to slice regularity can be pursued on an ample class of real algebras. Here we recall the basic definitions and refer to [12,16] for details and other references. Let A be a real alternative algebra with unity e. The real multiples of e in A are identified with the real numbers.…”

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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The algebra of slice functions

Cited by 56 publications

References 24 publications

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

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

Growth and distortion theorems for slice monogenic functions

Slice Regularity and Harmonicity on Clifford Algebras

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