Compact quantum groups of face type (Q1922293)

A generalization of compact matrix pseudo groups -- quantum groups -- on the algebraic level is given. Face algebras are compounds of a finite set and a Hopf algebra, including thereby all Hopf algebras as a special case. Coalgebras and \(*\)-structures are introduced. Compact Hopf face algebras as finite dimensional such algebras related to graphs are studied. The framework covers algebras of \(L\)-operators of lattice models of face type without spectral parameter and ``Galois quantum groups'' of AFD \(\text{II}_1\)-subfactors of index \(<4\).