四元数分析中的准正函数
Quasi Regular Functions in Quaternionic Analysis
AI总结：
本文研究了四元数分析中新的准正函数，探讨了其再生核结构及与共形不变代数的应用。
摘要：
我们研究了一类在四元数分析中自然出现的新函数，我们称之为“准正函数”。与著名的四元数正函数类似，这些函数提供了四元数共形群的表示。然而，与正函数不同，准正函数不具有不变的酉结构，而是具有伪酉等价结构。这些函数的再生核有特别简单的形式：(Z-W)^{-1}。我们描述了准正函数的K型基底，并推导了再生核展开式。我们还显示了由准正函数形成的不可约表示限制到庞加莱群时有三个不可约组成部分。我们的兴趣源于将其应用于研究四元数函数的共形不变代数。我们还引入了某些 intertwining 运算符在四元数函数空间张量积之间的分解。这种分解是通过由准正函数构造的费米子福克空间获得的。
英文摘要：
We study a new class of functions that arise naturally in quaternionic analysis, we call them "quasi regular functions". Like the well-known quaternionic regular functions, these functions provide representations of the quaternionic conformal group. However, unlike the regular functions, the quasi regular ones do not admit an invariant unitary structure but rather a pseudounitary equivalent. The reproducing kernels of these functions have an especially simple form: (Z-W)^{-1}. We describe the K-type bases of quasi regular functions and derive the reproducing kernel expansions. We also show that the restrictions of the irreducible representations formed from the quasi regular functions to the Poincare group have three irreducible components. Our interest in the quasi regular functions arises from an application to the study of conformal-invariant algebras of quaternionic functions. We also introduce a factorization of certain intertwining operators between tensor products of spaces of quaternionic functions. This factorization is obtained using fermionic Fock spaces constructed from the quasi regular functions.
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