A quantum shuffle approach to quantum determinants

Abstract

Let ${\bigwedge }_{\sigma }V={⨁}_{k\ge 0}{\bigwedge }_{\sigma }^{k}V$ be the quantum exterior algebra associated to a finite-dimensional braided vector space $(V,\sigma )$. For an algebra $\mathfrak{A}$, we consider the convolution product on the graded space ${⨁}_{k\ge 0}\mathrm{Hom}({\bigwedge }_{\sigma }^{k}V,{\bigwedge }_{\sigma }^{k}V\otimes \mathfrak{A})$. 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 quantum Laplace expansion formulas and multiplicative formulas for these determinants.

22 pages; some new references are added; comments are welcome

References in corpus (1)

• Quantum function algebras from finite-dimensional Nichols algebras