Polynomial Null-solutions Research Articles

GENERALISED SYMMETRIES AND BASES FOR DUNKL MONOGENICS

We introduce a family of commuting generalised symmetries of the Dunkl–Dirac operator inspired by the Maxwell construction in harmonic analysis. As an application, we use these generalised symmetries to construct bases of the polynomial null-solutions of the Dunkl–Dirac operator. These polynomial spaces form representation spaces of the Dunkl–Dirac symmetry algebra. For the ℤ2d case, the results are compared with previous investigations.

Polynomial null solutions to bosonic Laplacians, bosonic Bergman and Hardy spaces

AbstractA bosonic Laplacian, which is a generalization of Laplacian, is constructed as a second-order conformally invariant differential operator acting on functions taking values in irreducible representations of the special orthogonal group, hence of the spin group. In this paper, we firstly introduce some properties for homogeneous polynomial null solutions to bosonic Laplacians, which give us some important results, such as an orthogonal decomposition of the space of polynomials in terms of homogeneous polynomial null solutions to bosonic Laplacians, etc. This work helps us to introduce Bergman spaces related to bosonic Laplacians, named as bosonic Bergman spaces, in higher spin spaces. Reproducing kernels for bosonic Bergman spaces in the unit ball and a description of bosonic Bergman projection are given as well. At the end, we investigate bosonic Hardy spaces, which are considered as generalizations of harmonic Hardy spaces. Analogs of some well-known results for harmonic Hardy spaces are provided here. For instance, connections to certain complex Borel measure spaces, growth estimates for functions in the bosonic Hardy spaces, etc.

Construction of Special Solutions for the Maxwell Equations

In this paper, we consider homogeneous polynomial solutions for the classical Maxwell equations in a time-space domain with time variable $$t<t_0$$, where $$t_0$$ is an arbitrary constant. The technique is motivated by the study of the so-called generalized Maxwell operators, which are constructed as conformally invariant differential operators in the framework of Clifford analysis. We firstly show that solutions of the classical Maxwell equations are equivalent to null solutions of the generalized Maxwell operator, then a direct decomposition and a partially orthogonal polynomial basis for the space of homogeneous polynomial null solutions of the generalized Maxwell operator are provided. At the end, a basis of homogeneous polynomial solutions for the source-free Maxwell equations on a bounded time-space domain is given.

The Higher Rank q-Deformed Bannai-Ito and Askey-Wilson Algebra

The $q$-deformed Bannai-Ito algebra was recently constructed in the threefold tensor product of the quantum superalgebra $\mathfrak{osp}_q(1\vert 2)$. It turned out to be isomorphic to the Askey-Wilson algebra. In the present paper these results will be extended to higher rank. The rank $n-2$ $q$-Bannai-Ito algebra $\mathcal{A}_n^q$, which by the established isomorphism also yields a higher rank version of the Askey-Wilson algebra, is constructed in the $n$-fold tensor product of $\mathfrak{osp}_q(1\vert 2)$. An explicit realization in terms of $q$-shift operators and reflections is proposed, which will be called the $\mathbb{Z}_2^n$ $q$-Dirac-Dunkl model. The algebra $\mathcal{A}_n^q$ is shown to arise as the symmetry algebra of the constructed $\mathbb{Z}_2^n$ $q$-Dirac-Dunkl operator and to act irreducibly on modules of its polynomial null-solutions. An explicit basis for these modules is obtained using a $q$-deformed $\mathbf{CK}$-extension and Fischer decomposition.

The [formula omitted] Dirac–Dunkl operator and a higher rank Bannai–Ito algebra

The kernel of the Z2n Dirac–Dunkl operator is examined. The symmetry algebra An of the associated Dirac–Dunkl equation on Sn−1 is determined and is seen to correspond to a higher rank generalization of the Bannai–Ito algebra. A basis for the polynomial null-solutions of the Dirac–Dunkl operator is constructed. The basis elements are joint eigenfunctions of a maximal commutative subalgebra of An and are given explicitly in terms of Jacobi polynomials. The symmetry algebra is shown to act irreducibly on this basis via raising/lowering operators. A scalar realization of An is proposed.

Reproducing Kernels for Polynomial Null-Solutions of Dirac Operators

It is well known that the reproducing kernel of the space of spherical harmonics of fixed homogeneity is given by a Gegenbauer polynomial. By going over to complex variables and restricting to suitable bihomogeneous subspaces, one obtains a reproducing kernel expressed as a Jacobi polynomial, which leads to Koornwinder’s celebrated result on the addition formula. In the present paper, the space of Hermitian monogenics, which is the space of polynomial bihomogeneous null-solutions of a set of two complex conjugated Dirac operators, is considered. The reproducing kernel for this space is obtained and expressed in terms of sums of Jacobi polynomials. This is achieved through use of the underlying Lie superalgebra \(\mathfrak {sl}(1|2)\), combined with the equivalence between the \(L^2\) inner product on the unit sphere and the Fischer inner product. The latter also leads to a new proof in the standard Dirac case related to the Lie superalgebra \(\mathfrak {osp}(1|2)\).

Algebraic Approach to Slice Monogenic Functions

In recent years, the study of slice monogenic functions has attracted more and more attention in the literature. In this paper, an extension of the well-known Dirac operator is defined which allows to establish the Lie superalgebra structure behind the theory of slice monogenic functions. Subsequently, an inner product is defined corresponding to this slice Dirac operator and its polynomial null-solutions are determined. Finally, analogues of the Hermite polynomials and Hermite functions are constructed in this context and their properties are studied.

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

The polynomial null solutions are studied of the higher spin Dirac operator Q k,l acting on functions taking values in an irreducible representation space for Spin(m) with highest weight $${(k + \frac{1}{2},l+\frac{1}{2},\frac{1}{2},\ldots,\frac{1}{2})}$$ , with $${k,l \in \mathbb {N}}$$ and k = l.

A toy model for higher spin Dirac operators

This paper deals with the higher spin Dirac operator Q2,1 acting on functions taking values in an irreducible representation space for so(m) with highest weight \( (\tfrac{5} {2},\tfrac{3} {2},\tfrac{1} {2},...,\tfrac{1} {2}) \). This operator acts as a toy model for generalizations of the classical Rarita—Schwinger equations in Clifford analysis. Polynomial null solutions for this operator are studied in particular.

Powers of the Parabolic Dirac Operator

The following paper deals with the characterization of the polynomial null solutions for a homogeneous and a non-homogeneous parabolic Dirac operator, via Cauchy-Riemann-like systems. For that we will use the factorizations of the Laplace and the heat operator, respectively. We will present a Fischer decomposition for the case of homogeneous polynomials. At the end of the paper two examples are presented.

On Homogeneous Polynomial Solutions of the Moisil-Théodoresco System in ℝ3

The space of homogeneous polynomial solutions of degree k of the Moisil-Theodoresco system in ℝ3 is isomorphic to the real vector space MT+ (ℝ3; ℝ0,3 +; k) of homogeneous ℝ0,3 +-valued polynomial null-solutions of degree k of the Cauchy-Riemann operator D x in ℝ3. Hereby ℝ0,3 + is the even subalgebra of the Clifford algebra ℝ0,3. A structure theorem is proved for the elements W k ∈ MT+ (ℝ3; ℝ0,3 +; k), based essentially on conjugate harmonicity, and a general procedure is elaborated for constructing bases of MT+ (ℝ3; ℝ0,3 +; k).

Spherical harmonics and integration in superspace

In this paper, the classical theory of spherical harmonics in is extended to superspace using techniques from Clifford analysis. After defining a super-Laplace operator and studying some basic properties of polynomial null-solutions of this operator, a new type of integration over the supersphere is introduced by exploiting the formal equivalence with an old result of Pizzetti. This integral is then used to prove orthogonality of spherical harmonics of different degree, Green-like theorems and also an extension of the important Funk–Hecke theorem to superspace. Finally, this integration over the supersphere is used to define an integral over the whole superspace, and it is proven that this is equivalent with the Berezin integral, thus providing a more sound definition of the Berezin integral.

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