The Symplectic Fueter–Sce Theorem

Eelbode, David; Hohloch, Sonja; Muarem, Güner

doi:10.1007/s00006-020-01077-5

Cited by 5 publications

(6 citation statements)

References 27 publications

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“…In order to deal with the fractional laplacian he uses the techniques of Fourier multipliers in the space of distributions. We refer to [25,37,38,40,41,43,47] for several generalizations.…”

mentioning

confidence: 99%

Generalized Appell polynomials and Fueter–Bargmann transforms in the polyanalytic setting

Martino

Diki

2022

Anal. Appl.

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This paper deals with some special integral transforms in the setting of quaternionic valued slice polyanalytic functions. In particular, using the polyanalytic Fueter mappings, it is possible to construct a new family of polynomials which are called the generalized Appell polynomials. Furthermore, the range of the polyanalytic Fueter mappings on two different polyanalytic Fock spaces is characterized. Finally, we study the polyanalytic Fueter–Bargmann transforms.

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“…In order to deal with the fractional laplacian he uses the techniques of Fourier multipliers in the space of distributions. We refer to [25,37,38,40,41,43,47] for several generalizations.…”

mentioning

confidence: 99%

Generalized Appell polynomials and Fueter–Bargmann transforms in the polyanalytic setting

Martino

Diki

2022

Anal. Appl.

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“…In order to deal with the fractional Laplacian, he uses the techniques of Fourier multipliers in the space of distributions. We refer to [21, 32, 33, 39, 42, 43, 49] for several generalizations.…”

Section: Preliminary Resultsmentioning

confidence: 99%

The Fueter-Sce mapping and the Clifford–Appell polynomials

Martino¹,

Diki²,

Adán³

2023

Proceedings of the Edinburgh Mathematical Society

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The Fueter-Sce theorem provides a procedure to obtain axially monogenic functions, which are in the kernel of generalized Cauchy–Riemann operator in ${\mathbb{R}}^{n+1}$ . This result is obtained by using two operators. The first one is the slice operator, which extends holomorphic functions of one complex variable to slice monogenic functions in $ \mathbb{R}^{n+1}$ . The second one is a suitable power of the Laplace operator in n + 1 variables. Another way to get axially monogenic functions is the generalized Cauchy–Kovalevskaya (CK) extension. This characterizes axial monogenic functions by their restriction to the real line. In this paper, using the connection between the Fueter-Sce map and the generalized CK-extension, we explicitly compute the actions $\Delta_{\mathbb{R}^{n+1}}^{\frac{n-1}{2}} x^k$ , where $x \in \mathbb{R}^{n+1}$ . The expressions obtained is related to a well-known class of Clifford–Appell polynomials. These are the building blocks to write a Taylor series for axially monogenic functions. By using the connections between the Fueter-Sce map and the generalized CK extension, we characterize the range and the kernel of the Fueter-Sce map. Furthermore, we focus on studying the Clifford–Appell–Fock space and the Clifford–Appell–Hardy space. Finally, using the polyanalytic Fueter-Sce theorems, we obtain a new family of polyanalytic monogenic polynomials, which extends to higher dimensions the Clifford–Appell polynomials.

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“…Recall that these are the symplectic spinor-valued polynomial functions f ∈ P( 2n , ) ⊗ S( n ) such that D s f = D s f = 0. The explicit calculation of the Cartan product (and more generally the tensor product) will be done in [6]. Moreover, as an application we will prove a Fischer decomposition for the Howe dual pair we obtained in this paper.…”

Section: A Unitary Howe Duality Associated With D S and D Smentioning

confidence: 89%

“…In order to describe these systematically, we will need the notion of Howe dualities and corresponding Fischer decompositions. This will be done in full detail in our upcoming paper [6]. In the following section we will reveal the algebraic structures required for this approach.…”

Section: 5mentioning

confidence: 99%

Unitary Howe dualities in fermionic and bosonic algebras and related Dirac operators

Muarem

2023

SciPost Phys. Proc.

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In this paper we use the canonical complex structure \mathbb{J}𝕁 on \mathbb{R}^{2n}ℝ2n to introduce a twist of the symplectic Dirac operator. This can be interpreted as the bosonic analogue of the Dirac operators on a Hermitian manifold. Moreover, we prove that the algebra of these Dirac operators is isomorphic to the Lie algebra \mathfrak{su}(1,2)𝔰𝔲(1,2) which leads to the Howe dual pair (U(n),\mathfrak{su}(1,2))(U(n),𝔰𝔲(1,2)).

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The Symplectic Fueter–Sce Theorem

Cited by 5 publications

References 27 publications

Generalized Appell polynomials and Fueter–Bargmann transforms in the polyanalytic setting

Generalized Appell polynomials and Fueter–Bargmann transforms in the polyanalytic setting

The Fueter-Sce mapping and the Clifford–Appell polynomials

Unitary Howe dualities in fermionic and bosonic algebras and related Dirac operators

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