Face algebras. I: A generalization of quantum group theory (Q1127664)

Let \(R\) be a commutative separable algebra over a field \(K\). An \(R\)-face algebra \(H\) is a \(K\)-algebra with a coalgebra structure, with connecting axioms involving \(R\). This is a generalization of the idea of a bialgebra. In fact, every \(K\)-bialgebra is a \(K\)-face algebra, and conversely, an \(R\)-face algebra is a \(K\)-bialgebra only if \(R\) is isomorphic to \(K\). As with bialgebras, the category of comodules over an \(R\)-face algebra \(H\) is a monoidal abelian category. In this paper, the author studies elementary properties of face algebras and their comodule categories. By considering additional structures on \(H\) (e.g., an antipode, a universal \(R\)-matrix, a ribbon structure and a *-structure), the comodule category has also some additional structure.