A quantum shuffle approach to quantum determinants

TL;DR

This paper introduces a quantum shuffle method to define and analyze quantum determinants within the framework of quantum exterior algebras, extending classical concepts to the quantum setting with new formulas and properties.

Contribution

It develops a new approach using quantum shuffle techniques to define quantum minors and determinants, including Laplace expansion and multiplicative formulas, in the context of quantum exterior algebras.

Findings

Defined quantum minor determinants via convolution product

Established quantum Laplace expansion formulas

Proved multiplicative properties of quantum determinants

Abstract

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 algebra $\mathfrak{A}$, we consider the convolution product on the graded space $\bigoplus_{k\geq 0}\mathrm{Hom}\big(\bigwedge_\sigma^kV,\bigwedge_\sigma^kV\otimes \mathfrak{A}\big)$. 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 quantum Laplace expansion formulas and multiplicative formulas for these determinants.