A quantum shuffle approach to quantum determinants

Let $\bigwedge_\sigma V=\bigoplus_{k\geq 0}\bigwedge_\sigma^kV$ be the quantum exterior algebra associated to a finite-dimensional braided vector space $(V,\sigma)$. For an $\mathfrak{A}$, we consider convolution product on graded $\bigoplus_{k\geq 0}\mathrm{Hom}\big(\bigwedge_\sigma^kV,\bigwedge_\sigma^kV\otimes \mathfrak{A}\big)$. Using this product, define notion of minor determinant map from $V$ $V\otimes \mathfrak{A}$, which coincides with classical one in case that $\mathfrak{A}$ is FRT corresponding $U_q(\mathfrak{sl}_N)$. We establish Laplace expansion formulas and multiplicative for these determinants.