Fully explicit FRT-based proof in arbitrary dimension

Develop a fully explicit, inductive, and algebraically detailed proof within the Faddeev–Reshetikhin–Takhtajan (FRT) framework of the identity q^{binom(n,2)} · Pf_q(A)^2 = det_q(A) for all even sizes 2n, where A is a 2n × 2n matrix with entries in a noncommutative algebra satisfying the q-skew-symmetry relations a_{ji} = −q a_{ij} and a_{ii} = 0, Pf_q(A) is the quantum Pfaffian defined via the quantum exterior algebra, and det_q(A) is the quantum determinant arising from the FRT construction.

Background

The paper constructs the quantum Pfaffian Pf_q(A) and quantum determinant det_q(A) in the setting of q-skew-symmetric matrices using the FRT construction and quantum exterior algebras. It verifies the quantum Pfaffian–determinant identity in low-dimensional cases and outlines how the exponent depends on dimension.

While the authors present evidence and outlines for the general identity, they note that a complete, fully detailed algebraic proof valid for all even dimensions 2n, carried out entirely within the FRT framework and carefully handling braided antisymmetry and noncommutativity, is still lacking. Establishing such a proof would confirm the identity at full generality and clarify the precise bookkeeping of q-powers arising from braid crossings.