A quantum shuffle approach to quantum determinants

Abstract

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