Triple Monogenic Functions and Higher Spin Dirac Operators

Abstract

In the Clifford analysis context a specific type of solutions for the higher spin Dirac operators Q k,l (k ≥ l ∈ N) is studied; these higher spin Dirac operators can be seen as generalizations of the classical Rarita-Schwinger operator. To that end subspaces of the space of triple monogenic polynomials are introduced and their algebraic structure is investigated. Also a dimensional analysis is carried out.

The Polynomial Null Solutions of a Higher Spin Dirac Operator in Two Vector Variables

“…By construction, M s h,k,l is the space of h-homogeneous type A solutions for Q k,l . Also in [5] we proved that this space decomposes into Spin(m)-irreducibles as…”

“…the action of Spin(m), and in [5] the decomposition of this space in terms of irreducible Spin(m)-modules was determined. Define M s h,k,l to be the space…”

Higher Spin Dirac Operators Between Spaces of Simplicial Monogenics in Two Vector Variables

“…This vector space is highly reducible with respect to the action of Spin(m), and in [4] we have determined the decomposition of this space in terms of irreducible Spin(m)-modules, making use of the fact that each vector space S p,q,r can be seen as a highest weight vector for the algebra gl 3 , with positive root vectors…”

“…The decomposition of this space into irreducible spaces for Spin(m) was also determined in [4], using branching rules from gl 3 to gl 2 . Using the so-called raising and lowering operators u, ∂ x and v, ∂ x (E u − E v ) − u, ∂ x v, ∂ u , which were studied in the much broader setting of transvector algebras and weight bases for Lie algebras in e.g.…”

The Monogenic Fischer Decomposition: Two Vector Variables

“…[8,22]) , the study of higher spin (Stein-Weiss) and RaritaSchwinger operators (see e.g. [3,4,5,6,7,13,30]). The paper is organized as follows.…”