Left Regular Polynomials in Even Dimensions, and Tensor Products of Clifford Algebras

Abstract

In [4] Fueter describes a method for characterizing homogeneous polynomial solutions to the quaternionic analogue of the Cauchy-Riemann equations. In [3] Delanghe demonstrates that this method may be generalized to characterize homogeneous polynomial solutions to a generalized Cauchy-Riemann equation defined over a Clifford algebra. The function theory associated with this particular equation has been extensively pursued in recent years by a number of authors (eg [2,5,7,8,9,10]). However, the author shows in [7] that the methods used in [3,4] may be extended to characterize homogeneous polynomial solutions to homogeneous, first order, constant coefficient differential equations defined over arbitrary, finite dimensional, associative algebras with identity. For this reason it would appear desirable to find another method to characterize these solutions, to generalized Cauchy-Riemann equations over Clifford algebras, which is more closely aligned to the properties of Clifford algebras.

In this paper another approach is adopted to characterize homogeneous polynomial solutions, in even dimensions, of the equations considered by Delanghe in [3]. It is demonstrated that these polynomials form an orthogonal basis with respect to the inner product introduced, by Sommen [8] for solutions to the homogeneous Dirac equation in Rⁿ. Moreover, the method used here to introduce the inner product makes use of a natural automorphism within the Clifford algebra and bypasses the inversion transform employed in [8]. We also show that these basis elements contain all homogeneous polynomial solutions to the split Dirac equations described by Sommen in [9]. We conclude by giving a local characterization of the solutions to these split Dirac equations in terms of canonical isomorphisms between Clifford algebras and tensor products of lower dimensional Clifford algebras.