Quantum Harmonic Oscillator Algebra and Link Invariants

TL;DR

This paper explores the q-deformation of the harmonic oscillator algebra as a Ribbon Hopf algebra, constructs associated link invariants, and relates them to classical polynomials like Alexander-Conway, expanding the mathematical framework of quantum invariants.

Contribution

It introduces a new q-deformed harmonic oscillator algebra as a Ribbon Hopf algebra and connects it to link invariants and multicolored braid group representations.

Findings

Link invariant matches inverse Alexander-Conway polynomial

Constructed braid group representations relate to multivariable Alexander polynomial

Interpreted R-matrix as baxterization of semicyclic $SU(2)_q$ representations

Abstract

The $q$--deformation $U_q (h_4)$ of the harmonic oscillator algebra is defined and proved to be a Ribbon Hopf algebra.Associated with this Hopf algebra we define an infinite dimensional braid group representation on the Hilbert space of the harmonic oscillator, and an extended Yang--Baxter system in the sense of Turaev. The corresponding link invariant is computed in some particular cases and coincides with the inverse of the Alexander--Conway polynomial. The $R$ matrix of $U_q (h_4)$ can be interpreted as defining a baxterization of the intertwiners for semicyclic representations of $SU(2)_q$ at $q=e^{2 \pi i/N}$ in the $N \rightarrow \infty$ limit.Finally we define new multicolored braid group representations and study their relation to the multivariable Alexander--Conway polynomial.