An explicit formula of powers of the $2\times 2$ quantum matrices and its applications

TL;DR

This paper derives an explicit formula for powers of 2x2 quantum matrices, providing a quantum analogue to classical matrix powers and revealing new non-commutative relations among matrix entries.

Contribution

It introduces a novel explicit formula for powers of 2x2 quantum matrices, extending classical results into the quantum setting.

Findings

Explicit formula for powers of 2x2 quantum matrices

New non-commutative relations among matrix entries

Simplified proof of previous results by Vokos-Zumino-Wess

Abstract

We present an explicit formula of the powers for the $2\times 2$ quantum matrices, that is a natural quantum analogue of the powers of the usual $2\times 2$ matrices. As applications, we give some non-commutative relations of the entries of the powers for the $2\times 2$ quantum matrices, which is a simple proof of the results of Vokos-Zumino-Wess (1990).