Fundaments of Hermitean Clifford Analysis Part I: Complex Structure

Abstract.

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;${\mathbb{R}}$), 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 (e_α, α = 1, . . . , 2n) onto the Witt basis and moreover give rise to a direct sum decomposition of into two components, where the SO(2n;${\mathbb{R}}$)-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;${\mathbb{R}}$), denoted Spin _J (2n;${\mathbb{R}}$), being isomorphic with the unitary group U(n;${\mathbb{C}}$). 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 Spin _J (2n;${\mathbb{R}}$). The eventual goal is to extend the complex structure J to the whole Clifford algebra , in order to conceptually unravel the true meaning of Hermitean monogenicity and its connections to orthogonal monogenicity.