Quantum algebra structures on \(n\times n\) matrices (Q1282342)

A quantum algebra is a triple consisting of an algebra \(A\) over a field \(k\), an invertible element \(R\in A\otimes A\), and an algebra antiisomorphism \(s\colon A\to A\) such that \(R^{-1}=(s\otimes 1_A)(R)\), \(R=(s\otimes s)(R)\), and \(R_{1,2}R_{1,3}R_{2,3}=R_{2,3}R_{1,3}R_{1,2}\). This concept can be thought of as a mechanism for computing solutions to the quantum Yang-Baxter equation or as a vehicle for generating knot invariants, 1-1 tangle invariants, or invariants of 3-manifolds. A general description of quantum algebra structures on the matrix algebra \(M_n(k)\) is developed. As a special case, the quantum algebra structures on \(M_2(k)\), \(k\) algebraically closed of characteristic 0, are determined by using the classification of the solutions to the quantum Yang-Baxter equation in dimension 2. The dual space of a finite dimensional quantum algebra is equipped with a natural structure of a quantum coalgebra, a concept defined by the authors in previous work. The 1-1 tangle invariants resulting from the quantum algebra structures of \(M_2(k)\), \(k\) algebraically closed of characteristic 0, are determined by computing in the dual quantum coalgebra.