The cylinder product and cylinder matrices (Q1969366)

Let \(A\) be a coquasitriangular (CQT) bialgebra over a field \(k\). Let \(\sigma\) denote the invertible braid form on \(A\otimes A\). \textit{T.~tom Dieck} and \textit{R.~Häring-Oldenburg} introduced CQT bialgebras with a cylinder form [Forum Math. 10, No. 5, 619-639 (1998; Zbl 0974.17017)]. In the paper under review, the author generalizes their relations to non-commutative algebras. If \(A\) is a CQT bialgebra, \(E\) an algebra, a cylinder homomorphism is a linear map \(f\) from \(A\) to \(E\) satisfying \(f(1)=1\) and \(f(ab)=\sum\sigma(b_1,a_1)f(a_2)\sigma(a_3,b_2)f(b_3)\). An example is the CQT bialgebra \(A(R)\) arising by the Faddeev-Reshetikhin-Takhtajan construction from an \(n^2\) by \(n^2\) quantum \(R\)-matrix \(R_q\). An \(n\) by \(n\) matrix \(T=(t_{ij})\) with \(t_{ij}\) in \(E\) is called a cylinder matrix for \(R_q\) if \(((I\otimes T)R)^2=(R(I\otimes T))^2\). Then there is a unique cylinder homomorphism from \(A(R)\) to \(E\) sending the canonical generator \(x_{ij}\) to \(t_{ij}\). The author illustrates these ideas for \(M_n(q)\), quantum \(n\) by \(n\) matrices. He defines a determinant and a cofactor matrix for a \(q\)-cylinder matrix for \(R_q\). He shows that a \(q\)-cylinder matrix for \(R_q\) is invertible if and only if its determinant is invertible.