q-Oscillators, q-Epsilon Tensor, q-Groups

TL;DR

This paper constructs a $q$-deformed epsilon tensor from $q$-oscillators invariant under $U(n)$, revealing connections to quantum groups $SL_q(n)$ and quantum hyperplanes, thus linking $q$-oscillator invariants to quantum group structures.

Contribution

It introduces a $q$-deformed epsilon tensor from $q$-oscillators and demonstrates its invariance leads to the quantum group $SL_q(n)$, establishing a new link between $q$-oscillators and quantum groups.

Findings

Constructed a $q$-deformed epsilon tensor from $q$-oscillator states.

Showed the $q$-epsilon tensor invariance yields the quantum group $SL_q(n)$.

Connected the $q$-epsilon tensor to the volume element of the quantum hyperplane.

Abstract

Considering a multi-dimensional $q$-oscillator invariant under the (non quantum) group $U(n)$, we construct a $q$-deformed Levi-Civita epsilon tensor from the inner product states. The invariance of this $q$-epsilon tensor is shown to yield the quantum group $SL_{q}(n)$ and establishes the relationship of the $U(n)$ invariant $q$-oscillator to quantum groups and quantum group related oscillators. Furthermore the $q$-epsilon tensor provides the connection between $SL_{q}(n)$ and the volume element of the quantum hyper plane.