Multiparameter quantum Pfaffians and identity

Extend the definition of the quantum Pfaffian to the multiparameter quantum group setting with deformation parameters q_{ij}, by defining a consistent q_{ij}-skew-symmetry for matrix entries and constructing a corresponding quantum Pfaffian Pf_{q_{ij}}(A); then establish a generalized Pfaffian–determinant identity relating Pf_{q_{ij}}(A)^2 to the appropriate multiparameter quantum determinant in O_{q_{ij}}(GL_{2n}).

Background

Most results in the paper and literature are formulated for single-parameter quantum groups U_q(g). Multiparameter deformations introduce a family of parameters q_{ij} that modify commutation relations and braided antisymmetry.

A coherent theory of multiparameter quantum Pfaffians compatible with q_{ij}-skew-symmetry and an accompanying Pfaffian–determinant identity would generalize the single-parameter case and likely require new combinatorial and categorical tools.