Quantum algebra of multiparameter Manin matrices

Multiparametric quantum semigroups ${\mathrm{M}}_{\widehat{q},\widehat{p}}(n)$ are generalization of the one-parameter general linear semigroups ${\mathrm{M}}_q(n)$, where $\widehat{q}=(q_{ij})$ and $\widehat{p}=(p_{ij})$ are $2n^2$ parameters satisfying certain conditions. In this paper, we study the algebra of multiparametric Manin matrices using the R-matrix method. The systematic approach enables us to obtain several classical identities such as Muir identities, Newton’s identities, Capelli-type identities, Cauchy-Binet’s identity both for determinant and permanent as well as a rigorous proof of the MacMahon master equation for the quantum algebra of multiparametric Manin matrices. Some of the generalized identities are also generalized to multiparameter $q$-Yangians.