Deformed Traces and Covariant Quantum Algebras for Quantum Groups $GL_{qp}(2)$ and $GL_{qp}(1|1)$

TL;DR

This paper develops q-deformed traces, orbits, and covariant algebras for the quantum groups $GL_{qp}(2)$ and $GL_{qp}(1|1)$, revealing new algebraic structures and supersymmetric subalgebras.

Contribution

It introduces q-deformed traces and covariant algebras for $GL_{qp}(2)$ and $GL_{qp}(1|1)$, including invariant relations and supersymmetric structures.

Findings

Deformed traces and orbits for $GL_{qp}(2)$ and $GL_{qp}(1|1)$ are defined.

Covariant algebras include a central extension of a Witten-type $sl(2)$ algebra.

Supersymmetric quantum mechanical subalgebras are identified in the supergroup case.

Abstract

The q-deformed traces and orbits for the two parametric quantum groups $GL_{qp}(2)$ and $GL_{qp}(1|1)$ are defined. They are subsequently used in the construction of $q$-orbit invariants for these groups. General $qp$-(super)oscillator commutation relations are obtained which remain invariant under the coactions of groups $GL_{qp}(2)$ and $GL_{qp}(1|1)$. The $GL_{qp}(2)$ covariant deformed algebra is deduced in terms of the bilinears of bosonic $qp$-oscillators which turns out to be a central extension of the Witten-type deformation of $sl(2)$ algebra. In the case of the supergroup $GL_{qp}(1|1)$, the corresponding covariant algebras contain $N = 2$ supersymmetric quantum mechanical subalgebras.