On matrix quantum groups of type \(A_n\) (Q2706844)

Let \(k\) be a field of characteristic zero and let \(q\in k^\times\), not equal to a root of unity other than \(1\). A Hecke symmetry is a closed invertible linear operator \(R\colon V\otimes V\to V\otimes V\) on a finite-dimensional vector space \(V\), which satisfies the Yang-Baxter equation \((R\otimes\text{id})(\text{id}\otimes R)(R\otimes\text{id})=(\text{id}\otimes R)(R\otimes\text{id})(\text{id}\otimes R)\) and the Hecke equation \((R+1)(R-q)=0\). With a Hecke symmetry one can associate a bialgebra \(E_R\) and a Hopf algebra \(H_R\) (the Hopf envelope of \(E_R\)) which correspond, respectively, to the function algebras on the ``quantum matrix (semi)-groups'' associated to \(R\). \(R\) is called even if a certain graded algebra \(\Lambda_R\) (analogous to the exterior algebra) is finite-dimensional; in this case the degree of the highest non-vanishing homogeneous component of \(\Lambda_R\) is called the rank of \(R\).NEWLINENEWLINENEWLINEIn this paper the author studies the category of \(H_R\)-comodules for an even Hecke symmetry \(R\): this corresponds to the category of rational representations on the ``quantum groups of type \(A_n\)''. He shows that this category depends only on \(q\) and the rank of \(R\), up to braided Abelian equivalence; he also proves that every \(H_R\)-comodule is absolutely reducible.NEWLINENEWLINENEWLINEThe author shows also that the category of \(H_R\)-comodules is a ribbon category and computes the quantum dimension of simple comodules. An explicit formula for the integral on \(H_R\) is given.