Commutation relations for arbitrary quantum minors.
Let \(A\) be the quantum coordinate algebra on square matrices. It is generated by elements \(X_{ij}\), \(i,j=1,\dots,n\), with defining relations \(X_{ij}X_{lj}=qX_{lj}X_{ij}\), \(i<l\), \(X_{ij}X_{im}=qX_{im}X_{ij}\), \(j<m\), \(X_{ij}X_{lm}=X_{lm}X_{ij}\), \(i<l\), \(j>m\), and \(X_{ij}X_{lm}-X_{im}X_{ij}=(q-q^{-1})X_{im}X_{lj}\) if \(i<l\), \(j<m\). It is shown that \(A\) is coquasitriangular with the bilinear form \(r\) such that \(r(X_{ii},X_{ii})=q\), \(r(X_{ii},X_{jj})=1\), \(i\neq j\), \(r(X_{ij},X_{ji})=q-q^{-1}\), \(i>j\), and \(r(X_{ij},X_{lm})=0\) otherwise. The author considers quantum minors \([I\mid J]\) which are quantum determinants, corresponding to a choice of a set of rows \(I\) and a set of columns \(J\). The main result of the paper establishes commutation relations of the form \(q^{|I\cap M|}[I\mid J][M\mid N]=q^{|J\cap N|}[M\mid N][I\mid J]+\cdots\) where \(\cdots \) is presented explicitly as a sum of minors of smaller sizes.
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