Paragrassmann Algebras as Quantum Spaces, Part I: Reproducing Kernels

TL;DR

This paper explores paragrassmann algebras as quantum spaces, establishing their structure as reproducing kernel Hilbert spaces and highlighting their unique properties distinct from classical function spaces.

Contribution

It introduces a sesquilinear form on paragrassmann algebras, constructs their reproducing kernels, and analyzes their properties as non-commutative quantum spaces.

Findings

The Segal-Bargmann space within paragrassmann algebras has a reproducing kernel.

These algebras are not isomorphic to classical function algebras, requiring careful interpretation of evaluation.

The reproducing kernel exhibits most standard properties, demonstrating the space's structure.

Abstract

Paragrassmann algebras are given a sesquilinear form for which one subalgebra becomes a Hilbert space known as the Segal-Bargmann space. This Hilbert space as well as the ambient space of the paragrassmann algebra itself are shown to have reproducing kernels. These algebras are not isomorphic to algebras of functions so some care must be taken in defining what "evaluation at a point" corresponds to in this context. The reproducing kernel in the Segal-Bargmann space is shown to have most, though not all, of the standard properties. These quantum spaces provide non-trivial examples of spaces which have a reproducing kernel but which are not spaces of functions.