In this chapter an introduction is given to Clifford analysis and the underlying Clifford algebras. The functions under consideration are defined on Euclidean space and take values in the universal real or complex Clifford algebra, the structure and properties of which are also recalled in detail. The function theory is centered around the notion of a monogenic function, which is a null solution of a generalized Cauchy–Riemann operator, which is rotation invariant and factorizes the Laplace operator. In this way, Clifford analysis may be considered as both a generalization to higher dimension of the theory of holomorphic functions in the complex plane and a refinement of classical harmonic analysis. A notion of monogenicity may also be associated with the vectorial part of the Cauchy–Riemann operator, which is called the Dirac operator; some attention is paid to the intimate relation between both notions. Since a product of monogenic functions is, in general, no longer monogenic, it is crucial to possess some tools for generating monogenic functions: such tools are provided by Fueter's theorem on one hand and the Cauchy–Kovalevskaya extension theorem on the other hand. A corner stone in this function theory is the Cauchy integral formula for representation of a monogenic function in the interior of its domain of monogenicity. Starting from this representation formula and related integral formulae, it is possible to consider integral transforms such as Cauchy, Hilbert, and Radon transforms, which are important both within the theoretical framework and in view of possible applications.

This work provides an overview of the algebraic properties of primitive idempotents, which are fundamental in defining spinor spaces within the Clifford algebra framework. In addition to the key concepts, we also present novel results. In particular, we show that the primitive idempotent can be expressed as a polynomial in a specific special bivector. More generally, we demonstrate that every endomorphism on the spinor space can be represented as a polynomial in this special bivector. We also establish that the primitive idempotent, interpreted as a zero projection, represents a special case of this broader polynomial framework. By combining established insights with new contributions, this article offers a fresh perspective on these fundamental structures.

As an analogue of the massless field equations in Euclidean space, we consider the so-called generalized Cauchy–Riemann equations introduced by E. Stein and G. Weiss. In the spin 1/2 case these equations reduce to the Dirac equation for spin 1/2 fields, which was thoroughly and intensively studied in Clifford analysis. For general spin it was recently shown that, in dimension 4, homogenous solutions form irreducible Spin modules. The next step then is to describe the corresponding Fischer decomposition, i.e. an irreducible decomposition of the space of spinor fields, which is well-known for spin 1/2 and for spin 1. The main aim of the present paper is to describe, still in dimension 4, the Fischer decomposition for spin 3/2.

There are two well-known ways of describing elements of the rotation group SO(m). First, according to the Cartan-Dieudonne theorem, every rotation matrix can be written as an even number of reflections. And second, they can also be expressed as the exponential of some anti-symmetric matrix. In this paper, we study similar descriptions of a group of rotations SO0 in the superspace setting. This group can be seen as the action of the functor of points of the orthosymplectic supergroup OSp(m vertical bar 2n) on a Grassmann algebra. While still being connected, the group SO0 is thus no longer compact. As a consequence, it cannot be fully described by just one action of the exponential map on its Lie algebra. Instead, we obtain an Iwasawa-type decomposition for this group in terms of three exponentials acting on three direct summands of the corresponding Lie algebra of supermatrices. At the same time, SO0 strictly contains the group generated by super-vector reflections. Therefore, its Lie algebra is isomorphic to a certain extension of the algebra of superbivectors. This means that the Spin group in this setting has to be seen as the group generated by the exponentials of the so-called extended superbivectors in order to cover SO0. We also study the actions of this Spin group on supervectors and provide a proper subset of it that is a double cover of SO0. Finally, we show that every fractional Fourier transform in n bosonic dimensions can be seen as an element of this spin group. (C) 2020 Elsevier B.V. All rights reserved.

As is the case for the theory of holomorphic functions in the complex plane, the Cauchy Integral Formula has proven to be a corner stone of Clifford analysis, the monogenic function theory in higher dimensional euclidean space. In recent years, several new branches of Clifford analysis have emerged. Similarly as hermitian Clifford analysis in euclidean space R^{2n} of even dimension emerged as a refinement of euclidean Clifford analysis by introducing a complex structure on R^{2n}, quaternionic Clifford analysis arose as a further refinement by introducing a so--called hypercomplex structure Q, i.e.\ three complex structures (I, J, K) which submit to the quaternionic multiplication rules, on R^{4p}, the dimension now being a fourfold. Two, respectively four, differential operators lead to first order systems invariant under the action of the respective symmetry groups U(n) and Sp(p). Their simultaneous null solutions are called hermitian monogenic and quaternionic monogenic functions respectively. In this contribution we further elaborate on the Caychy Integral Formula for hermitian and quaternionic monogenic functions. Moreover we establish Caychy integral formulae for osp(4|2)--monogenic functions, the newest branch of Clifford analysis refining quaternionic monogenicity by taking the underlying symplectic symmetry fully into account.

The Clifford-Cauchy integral formula has proven to be a corner stone of the monogenic function theory, as is the case for the traditional Cauchy formula in the theory of holomorphic functions in the complex plane. In the recent years, several new branches of Clifford analysis have emerged. Similarly as hermitian Clifford analysis was introduced in Euclidean space $$\mathbb {R}^{2n}$$ of even dimension as a refinement of Euclidean Clifford analysis by the introduction of a complex structure on $$\mathbb {R}^{2n}$$ , quaternionic Clifford analysis arose as a further refinement by the introduction of a so-called hypercomplex structure $$\mathbb {Q}$$ , i.e. three complex structures ( $$\mathbb {I}$$ , $$\mathbb {J}$$ , $$\mathbb {K}$$ ) which submit to the quaternionic multiplication rules, on Euclidean space $$\mathbb {R}^{4p}$$ , the dimension now being a fourfold. Two, respectively four differential operators are constructed, leading to invariant systems under the action of the respective symmetry groups U(n) and Sp(p). Their simultaneous null solutions are respectively called hermitian and quaternionic monogenic functions. The basics of hermitian monogenicity have been studied in e.g. Brackx et al. (Compl Anal Oper Theory 1(3):341–365, 2007; Complex Var Elliptic Equ 52(10–11):1063–1079, 2007; Appl Clifford Algebras 18(3–4):451–487, 2008). Quaternionic monogenicity has been developed in, amongst others, Peña-Peña (Complex Anal Oper Theory 1:97–113, 2007), Eelbode (Complex Var Elliptic Equ 53(10):975–987, 2008), Damiano et al. (Adv Geom 11:169–189, 2011), and Brackx et al. (Adv Appl Clifford Alg 24(4):955–980, 2014; Ann Glob Anal Geom 46:409–430, 2014). In this contribution, we give an overview of the ways in which a Cauchy integral representation formula has been established within each of these frameworks.

In [4] we studied the group invariance of the inner product of supervectors as introduced in the framework of Clifford analysis in superspace. The fundamental group SO0 leaving invariant such an inner product turns out to be an extension of SO(m)×Sp(2n) and gives rise to the definition of the spin group in superspace through the exponential of the so-called extended superbivectors, where the spin group can be seen as a double covering of SO0 by means of the representation h(s)[x]=sxs‾. In the present paper, we study the invariance of the Dirac operator in superspace under the classical H and L actions of the spin group on superfunctions. In addition, we consider the Hermitian Clifford setting in superspace, where we study the group invariance of the Hermitian inner product of supervectors introduced in [3]. The group of complex supermatrices leaving this inner product invariant constitutes an extension of U(m)×U(n) and is isomorphic to the subset SO0J of SO0 of elements that commute with the complex structure J. The realization of SO0J within the spin group is studied together with the invariance under its actions of the super Hermitian Dirac system. It is interesting to note that the spin element leading to the complex structure can be expressed in terms of the n-dimensional Fourier transform.

The massless field equations for lower integer and half-integer values of spin in Minkowski space are fundamental equations in mathematical physics. Their counterpart in Euclidean spacetime is a system of elliptic equations, which was already studied from the viewpoint of function theory in the framework of so-called Hodge systems for differential forms of various degrees. In dimension 4 it is possible to substitute spinor calculus for the usual tensor notation. In the present paper we concentrate on the case of the massless field equation for spin 1 in dimension 4, and we treat, in a spinor formalism, a fundamental concept of its function theory: the Fischer decomposition of polynomial spinor fields, for which we give simple and independent proofs.

In this paper we first recall the proper algebraic framework, i.e. the radial algebra, needed to extend Hermitian Clifford analysis to the superspace setting. The fundamental objects for this extension then are introduced by means of an abstract complex structure on the Hermitian radial algebra. This leads to a natural representation of this Hermitian radial algebra on superspace.

We recall the algebra of endomorphisms on the so-called radial algebra of abstract vector variables which generalizes both polynomial and Clifford algebras, and the defining building blocks of a function theory in this abstract framework. These building blocks are given by the abstract versions of the Dirac operator and the directional derivatives. Together with other fundamental endomorphisms, they lead to an algebraic structure which can be considered as the abstract equivalent of orthogonal Clifford analysis. Following a similar approach, we present the axiomatic definitions of the Hermitian radial algebra, leading to the abstract equivalent of Hermitian Clifford analysis.

Quaternionic Clifford analysis is a recent new branch of Clifford analysis, a higher dimensional function theory which refines harmonic analysis and generalizes to higher dimension the theory of holomorphic functions in the complex plane. So-called quaternionic monogenic functions satisfy a system of first order linear differential equations expressed in terms of four interrelated Dirac operators. The conceptual significance of quaternionic Clifford analysis is unraveled by showing that quaternionic monogenicity can be characterized by means of generalized gradients in the sense of Stein and Weiss. At the same time, connections between quaternionic monogenic functions and other branches of Clifford analysis, viz Hermitian monogenic and standard or Euclidean monogenic functions are established as well.

In this paper, we investigate the existence of a mixed Fischer decomposition in a generalized Clifford analysis setting, where a system of Dirac equations is considered, associated to different orthogonal bases ( or structural sets) in Euclidean space.

We prove the Fischer decomposition for the space of spinor-valued polynomials, defined on Euclidean space of four-fold dimension, in terms of irreducible modules for the symplectic group, consisting of so-called osp(4|2)-monogenics.

In this paper, we show that a higher order Borel–Pompeiu (Cauchy–Pompeiu) formula, associated with an arbitrary orthogonal basis (called structural set) of a Euclidean space, can be extended to the framework of generalized Clifford analysis. Furthermore, in lower dimensional cases, as well as for combinations of standard structural sets, explicit expressions of the kernel functions are derived. Copyright © 2015 John Wiley & Sons, Ltd.

Spaces of spinor-valued homogeneous polynomials, and in particular spaces of spinor-valued spherical harmonics, are decomposed in terms of irreducible representations of the symplectic group Sp(p). These Fischer decompositions involve spaces of homogeneous, so-called osp(4|2)-monogenic polynomials, the Lie super algebra osp(4|2) being the Howe dual partner to the symplectic group Sp(p). In order to obtain Sp(p)-irreducibility, this new concept of osp(4|2)-monogenicity has to be introduced as a refinement of quaternionic monogenicity; it is defined by means of the four quaternionic Dirac operators, a scalar Euler operator E underlying the notion of symplectic harmonicity and a multiplicative Clifford algebra operator P underlying the decomposition of spinor space into symplectic cells. These operators E and P, and their Hermitian conjugates, arise naturally when constructing the Howe dual pair osp(4|2)xSp(p), the action of which will make the Fischer decomposition multiplicity free. Copyright (c) 2016 John Wiley & Sons, Ltd.

The XXXth International Colloquium on Group Theoretical Methods in Physics (ICGTMP), also known as the Group30 conference, took place in Ghent (Belgium) from Monday 14 to Friday 18 July 2014. The conference was organised by Ghent University (Department of Applied Mathematics, Computer Science and Statistics, and Department of Mathematical Analysis). The website http://www.group30.ugent.be is still available.

We establish a higher order Borel-Pompeiu formula for monogenic functions associated to an arbitrary orthogonal basis (called structural set) of Euclidean space, or combinations of such structural sets.

We consider a nonstandard elliptic eigenvalue problem of second order on a two-component domain consisting of two intervals with a contact point. The interaction between the two domains is expressed through a coupling condition of nonlocal type, more specifically, in integral form. The problem under consideration is first stated in its variational form and next interpreted as a second-order differential eigenvalue problem. The aim is to set up a finite element method for this problem. The error analysis involved is shown to be affected by the nonlocal condition, which requires a suitable modification of the vector Lagrange interpolant on the overall finite element mesh. Nevertheless, we arrive at optimal error estimates. In the last section, an illustrative numerical example is given, which confirms the theoretical results.

In the framework of Clifford analysis, a chain of harmonic and monogenic potentials in the upper half of (m+1)-dimensional Euclidean space was recently constructed, including a higher dimensional analogue of the logarithmic function in the complex plane, and their distributional boundary values were computed. In this paper we determine these potentials in lower half-space, and investigate whether they can be extended through the boundary R^m. This is a stepping stone to the representation of a doubly infinite sequence of distributions in R^m, consisting of positive and negative integer powers of the Dirac and the Hilbert-Dirac operators, as the jump across R^m of monogenic functions in the upper and lower half-spaces, in this way providing a sequence of interesting examples of Clifford hyperfunctions.