Row reducing quantum matrices, the quantum determinant, and the Dieudonné determinant
Let \(k\) be an algebraically closed field, \(q\in k^*\), \(M_q(n)\) the bialgebra of quantum \(n\times n\) matrices, \(Z\) the \(q\)-generic matrix \(Z=(z_{ij})\), \(1\leq i, j\leq n\), \(\text{det}_qZ\) the quantum determinant. Let \(D\) be the total quotient ring of \(M_q(n)\). The author considers the ``elementary'' row operation on \(Z\) which produces \(Z'\) where \(z_{1j}'= z_{1j}\), \(z_{i1}'= 0\) for \(i\geq 2\), \(z_{ij}'= z_{ij}- z_{i1}z^{-1}_{11}z_{1j}\) for \(i,j\geq 2\). The author shows that \(Z'\) is a \(q\)-generic matrix, and \(\text{det}_qZ'= \text{det}_qZ\). He points out that this result can be false for certain other elementary row operations. Let \(\pi: D\to D^{*ab}= D^*|[D^*, D^*]\) be the canonical surjection of \(D\) onto its abelianization. The author shows that \(\pi(\text{det}_qZ)\) is equal to the Dieudonné determinant \(\text{det}_D(Z)\) of \(Z\). Finally, he obtains analogous results for the multiparameter quantum determinant.
scientific article; zbMATH DE number 1051523
scientific article