Hermitean Clifford analysis focusses on h-monogenic functions taking values in a complex Clifford algebra or in a complex spinor space. Monogenicity is expressed here by means of two complex mutually adjoint Dirac operators, which are invariant under the action of a Clifford realisation of the unitary group. In this contribution we present a deeper insight in the transition from the orthogonal setting to the Hermitean one. Starting from the orthogonal Clifford setting, by simply introducing a so-called complex structure J ∈ SO(2n; R), the fundamental elements of the Hermitean setting arise in a quite natural way. Indeed, the corresponding projection operators 1/2 (1 ± iJ) project the initial basis (eα, α = 1, . . . , 2n) onto the Witt basis and moreover give rise to a direct sum decomposition of C C 2n into two components, where the SO(2n; R)-elements leaving those two subspaces invariant, commute with the complex structure J. They generate a subgroup which is doubly covered by a subgroup of Spin(2n; R), denoted SpinJ (2n; R), being isomorphic with the unitary group U(n; C). Finally the two Hermitean Dirac operators are shown to originate as generalized gradients when projecting the gradient on the invariant subspaces mentioned, which actually implies their invariance under the action of SpinJ (2n; R). The eventual goal is to extend the complex structure J to the whole Clifford algebra C C 2n , in order to conceptually unravel the true meaning of Hermitean monogenicity and its connections to orthogonal monogenicity.
Clifford analysis offers a higher dimensional function theory studying the null solutions of the rotation invariant, vector valued, first order Dirac operator ∂. In the more recent branch Hermitean Clifford analysis, this rotational invariance has been broken by introducing a complex structure J on Euclidean space and a corresponding second Dirac operator ∂ J , leading to the system of equations ∂f = 0 = ∂ J f expressing so-called Hermitean monogenicity. The invariance of this system is reduced to the unitary group. In this paper we show that this choice of equations is fully justified. Indeed, constructing the Howe dual for the action of the unitary group on the space of all spinor valued polynomials, the generators of the resulting Lie superalgebra reveal the natural set of equations to be considered in this context, which exactly coincide with the chosen ones.
Abstract. In this paper we construct the main ingredients of a discrete function theory in higher dimensions by means of a new "skew" type of Weyl relations. We will show that this new type overcomes the difficulties of working with standard Weyl relations in the discrete case. A Fischer decomposition, Euler operator, monogenic projection, and basic homogeneous powers will be constructed.
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