A quantum-trace determinantal formula for matrix commutators, and applications

TL;DR

This paper introduces a determinantal formula for 2x2 matrix commutators over commutative rings, utilizing quantum traces, with applications in quadratic forms, matrix factorization, and Diophantine equations.

Contribution

It presents a novel determinantal formula for matrix commutators involving quantum traces, extending to trace and supertrace versions, with diverse mathematical applications.

Findings

Derived a determinantal formula for 2x2 matrix commutators.

Applied the formula to quadratic forms, matrix factorization, and Diophantine equations.

Demonstrated the utility of quantum traces in matrix invariants.

Abstract

In this paper, we establish a determinantal formula for 2 x 2 matrix commutators [X,Y] = XY - YX over a commutative ring, using (among other invariants) the quantum traces of X and Y. Special forms of this determinantal formula include a "trace version", and a "supertrace version". Some applications of these formulas are given to the study of value sets of binary quadratic forms, the factorization of 2 x 2 integral matrices, and the solution of certain simultaneous diophantine equations over commutative rings.