Isomorphisms between Quantum Group Covariant q-Oscillator Systems Defined for q and 1/q

TL;DR

This paper demonstrates an isomorphism between q-oscillator systems covariant under quantum groups SU_q(n) and SU_{q^{-1}}(n), revealing a symmetry that connects systems with q and 1/q parameters, and extends to quantum supergroups.

Contribution

It establishes a novel isomorphism between covariant q-oscillator systems for q and 1/q, and explores its implications for q-deformed Lie (super)algebras.

Findings

Isomorphism exists between SU_q(n) and SU_{q^{-1}}(n) covariant systems.

Similar isomorphism extends to quantum supergroups SU_q(n/m).

Direct generalization to q-deformed Lie (super)algebras is not possible.

Abstract

It is shown that there exists an isomorphism between q-oscillator systems covariant under $ SU_q(n) $ and $ SU_{q^{-1}}(n) $. By the isomorphism, the defining relations of $ SU_{q^{-1}}(n) $ covariant q-oscillator system are transmuted into those of $ SU_q(n) $. It is also shown that the similar isomorphism exists for the system of q-oscillators covariant under the quantum supergroup $ SU_q(n/m) $. Furthermore the cases of q-deformed Lie (super)algebras constructed from covariant q-oscillator systems are considered. The isomorphisms between q-deformed Lie (super)algebras can not obtained by the direct generalization of the one for covariant q-oscillator systems.