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On the category of modules over some semisimple bialgebras. (English) Zbl 1270.16026

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.

MSC:

16T10 Bialgebras
16D60 Simple and semisimple modules, primitive rings and ideals in associative algebras
16D70 Structure and classification for modules, bimodules and ideals (except as in 16Gxx), direct sum decomposition and cancellation in associative algebras)
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