Quantum groups and quantum semigroups (Q1270992)

For a finite non-empty set \(\mathcal V\), a \(\mathcal V\)-face algebra is an algebra equipped with a coalgebra structure such that the comultiplication is an algebra morphism and the counit satisfies relations involving two complete sets of orthogonal idempotents indexed by \(\mathcal V\) and subject to certain conditions. If \(\mathcal V\) is a singleton, a \(\mathcal V\)-face algebra is just a bialgebra. Concepts as antipodes, (co)-quasitriangularity, group-like elements, double crossed products, are generalized to the case of \(\mathcal V\)-face algebras. Some special objects, called closable, are defined in the class of co-quasitriangular \(\mathcal V\)-face algebras. For such an object \(A\), a Hopf closure \(Hc(A)\) is defined, and this is a co-quasitriangular Hopf \(\mathcal V\)-face algebra. The association \(A\mapsto Hc(A)\) is functorial and \(Hc(A)\) is described for several types of objects \(A\).