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

Abstract

In this paper, we establish a determinantal formula for matrix commutators over a commutative ring, using (among other invariants) the quantum traces of and . 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 integral matrices, and the solution of certain simultaneous diophantine equations over commutative rings.