Franciscus Sommen

Analysis of Dirac Systems and Computational Algebra

* Preface * Background Material * Computational Algebraic Analysis for Systems of Linear Constant Coefficients Differential Equations * The Cauchy-Fueter Systems and its Variations * Special First Order Systems in Clifford Analysis * Some First Order Linear Operators in Physics * Open Problems and Avenues for Further Research * References * Index

Lie groups as spin groups

It is shown that every Lie algebra can be represented as a bivector algebra; hence every Lie group can be represented as a spin group. Thus, the computational power of geometric algebra is available to simplify the analysis and applications of Lie groups and Lie algebras. The spin version of the general linear group is thoroughly analyzed, and an invariant method for constructing real spin representations of other classical groups is developed. Moreover, it is demonstrated that every linear transformation can be represented as a monomial of vectors in geometric algebra.

Integral theorems for functions and differential forms in C[m]

Introduction Differential Forms Differential Forms with Coefficients in 2 x 2 Matrices Hyperholomorphic Functions and Differential Forms in Cm Hyperholomorphic Cauchys Integral Theorems Hyperholomorphic Moreras Theorems Hyperholomorphic Cauchys Intergral Representations Hyperholomorphic D-Problem Complex Hodge-Dolbeault System, the ?-Problem and the Koppelman Formula Relation Between Hyperholomorphic Theory and Clifford Analysis

Special functions in Clifford analysis and axial symmetry

Abstract In our previous paper (Rend. Circ. Mat. Palermo 6 (1984), 259–269, we proved a general Laurent expansion for monogenic functions in symmetric domains of R m + 1, depending on the kind of symmetry involved. In this paper we consider axial symmetric domains and we apply our results in order to introduce new special monogenic functions. We investigated axial exponential functions, generalized powerfunctions and generalized Hermite polynomials.

Fundaments of Hermitean Clifford analysis part II: splitting of h -monogenic equations

Hermitean Clifford analysis focuses on h-monogenic functions taking values in a complex Clifford algebra or in a complex spinor space, where h-monogenicity is expressed by means of two complex and mutually adjoint Dirac operators, which are invariant under the action of a Clifford realization of the unitary group. In part 1 of the article the fundamental elements of the Hermitean setting have been introduced in a natural way, i.e., by introducing a complex structure on the underlying vector space, eventually extended to the whole complex Clifford algebra . The two Hermitean Dirac operators are then shown to originate as generalized gradients when projecting the gradient on invariant subspaces. In this part of the article, the aim is to further unravel the conceptual meaning of h-monogenicity, by studying possible splittings of the corresponding first-order system into independent parts without changing the properties of the solutions. In this way further connections with holomorphic functions of several complex variables are established. As an illustration, we give a full characterization of h-monogenic functions for the case n = 2. During the final redaction of this article, we received the sad news that our friend, colleague and co-author Jarolím Bureš died on 1 October 2006.

The Two-Dimensional Clifford-Fourier Transform

Recently several generalizations to higher dimension of the Fourier transform using Clifford algebra have been introduced, including the Clifford-Fourier transform by the authors, defined as an operator exponential with a Clifford algebra-valued kernel.In this paper an overview is given of all these generalizations and an in depth study of the two-dimensional Clifford-Fourier transform of the authors is presented. In this special two-dimensional case a closed form for the integral kernel may be obtained, leading to further properties, both in the L1 and in the L2 context. Furthermore, based on this Clifford-Fourier transform Clifford-Gabor filters are introduced.

Spherical harmonics and integration in superspace

The study of spherical harmonics in superspace, introduced in (De Bie and Sommen 2007 J. Phys. A: Math. Theor. 40 7193) is further elaborated. A detailed description of spherical harmonics of degree k is given in terms of bosonic and fermionic pieces, which also determines the irreducible pieces under the action of SO(m) x Sp(2n). In the second part of the paper, this decomposition is used to describe all possible integrations over the supersphere. It is then shown that only one possibility yields the orthogonality of spherical harmonics of different degrees. This is the so-called Pizzetti-integral of which it was shown in (De Bie and Sommen 2007 J. Phys. A: Math. Theor. 40 7193) that it leads to the Berezin integral.

On a Generalization of Fueter’s Theorem

In this paper we discuss a generalization of Fueters theorem which states that whenever f(xo,x) is holomorphic in x 0 +x, then it satisfies DOf = 0, D = O +t0 +j81 2 + kD,, 3 being the Fueter operator.

Symmetric Analogues of Rarita-Schwinger Equations

In this paper a generalization of the classicalRarita–Schwinger equations for spin 3/2 fields to the case of spin fieldswith values in irreducible representation spaces with weight k+1/2 isgiven. It corresponds to the study of serie of first orderconformal invariant operators, which are constructed from twisted Diracoperators. The representation character of polynomial solutions of the equations onflat space and their relations are described in details.

Some connections between clifford analysis and complex analysis

In the paper we construct kernels and mono-genic and holomorphic in in order to extend to several dimersions respectively the tranformation given by and the Fourier-Borel transformation T belonging to a space of analytic functionals. This leads toconnections between the theory of holomorphic functions of several variables and the theory of monogenic functions. These relationships are used to study the absolute convergence of the multiple Taylor series for monogenic functions.