Star product for deformed oscillator algebra $\mathsf{Aq}(2,\nu)$

TL;DR

This paper introduces a generalized star product for the deformed oscillator algebra $ ext{Aq}(2, u)$, extending the Moyal product with additional parameters and complex integration techniques, enriching the algebraic structure.

Contribution

It presents a novel star product formulation for the deformed oscillator algebra incorporating homotopy-like parameters and complex integration over Riemann surfaces.

Findings

Generalized star product with additional parameters

Integration over Riemann surfaces using Pochhammer formula

Extension of Moyal star product to deformed oscillator algebra

Abstract

An analogue of the Moyal star product is presented for the deformed oscillator algebra. It contains several homotopy-like additional integration parameters in the multiplication kernel generalizing the differential Moyal star-product formula $\exp[i\epsilon_{\alpha\beta}\partial^\alpha \partial^\beta]$. Using Pochhammer formula, integration over these parameters is carried over a Riemann surface associated with the expression of the type $z^x (1-z)^y$ where $x$ and $y$ are arbitrary real numbers.