On the category of modules over some semisimple bialgebras. (Q1925580)

Let \(k\) be an algebraically closed field, and let \(H\) be a finite dimensional semisimple \(k\)-bialgebra with direct sum decomposition \(H=(\bigoplus_{g\in G}ke_g)\oplus(\bigoplus_{j=1}^n\mathrm{Mat}(d_j,k))\), where \(G=G(H^*)\) is the monoid of all group-like elements of the dual bialgebra \(H^*\), \(\{e_g;\;g\in G\}\) is a system of central idempotents in \(H\), and \(n\), \(d_j\) are natural numbers. Additionally assume that \(1<d_1<d_2<\cdots<d_n\), which just means that all irreducible \(H\)-modules of the same dimension \(>1\) are isomorphic. In the paper under review, the authors study properties of the Clebsch-Gordan coefficients, that is, the multiplicities of occurrences of irreducible \(H\)-modules in semisimple decompositions of tensor products of irreducible ones. Under some restrictions on the Clebsch-Gordan coefficients, they prove that \(n\leq 2\) in the above direct sum decomposition of \(H\). For the case when \(n=2\), they compare the number of one-dimensional direct summands in this decomposition and the sizes of matrix components. Further properties of the Clebsch-Gordan coefficients are also studied.