  The zx-calculus and related theories are based on so-called interacting
Frobenius algebras, where a pair of dagger-special commutative Frobenius
algebras jointly form a pair of Hopf algebras. In this setting we introduce a
generalisation of this structure, Hopf-Frobenius algebras, starting from a
single Hopf algebra which is not necessarily commutative or cocommutative. We
provide a few necessary and sufficient conditions for a Hopf algebra to be a
Hopf-Frobenius algebra, and show that every Hopf algebra in the category of
finite dimensional vector spaces is a Hopf-Frobenius algebra. In addition, we
show that this construction is unique up to an invertible scalar. Due to this
fact, Hopf-Frobenius algebras provide two canonical notions of duality, and
give us a "dual" Hopf algebra that is isomorphic to the usual dual Hopf algebra
in a compact closed category. We use this isomorphism to construct a Hopf
algebra isomorphic to the Drinfeld double, but has a much simpler presentation.

Quantum Algebra;Category Theory