In algebra, the Pareigis Hopf algebra is the Hopf algebra over a field k whose left comodules are essentially the same as complexes over k, in the sense that the corresponding monoidal categories are isomorphic.
It was introduced by Pareigis (1981) as a natural example of a Hopf algebra that is neither commutative nor cocommutative.
The antipode takes x to xy and y to its inverse and has order 4.
If M = ⊕Mn is a complex with differential d of degree –1, then M can be made into a comodule over H by letting the coproduct take m to Σ yn⊗mn + yn+1x⊗dmn, where mn is the component of m in Mn.
This gives an equivalence between the monoidal category of complexes over k with the monoidal category of comodules over the Pareigis Hopf algebra.