Deformed Heisenberg algebra with reflection

TL;DR

This paper uncovers the universal structure of a deformed Heisenberg algebra with reflection, revealing its finite and infinite-dimensional representations, connections to parafermions, superalgebras, and generalized statistics.

Contribution

It demonstrates the algebra's universal representations, linking deformed Heisenberg algebra with reflection to parafermionic, superalgebra, and generalized statistics frameworks.

Findings

Finite-dimensional representations are deformed parafermionic with Z_2-grading.

All representations yield irreducible osp(1|2) superalgebra representations.

Normalized form relates to guon algebra and generalized statistics.

Abstract

A universality of deformed Heisenberg algebra involving the reflection operator is revealed. It is shown that in addition to the well-known infinite-dimensional representations related to parabosons, the algebra has also finite-dimensional representations of the parafermionic nature. We demonstrate that finite-dimensional representations are representations of deformed parafermionic algebra with internal Z_2-grading structure. On the other hand, any finite- or infinite-dimensional representation of the algebra supply us with irreducible representation of osp(1|2) superalgebra. We show that the normalized form of deformed Heisenberg algebra with reflection has the structure of guon algebra related to the generalized statistics.