Pareigis Hopf algebra

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 as a natural example of a Hopf algebra that is neither commutative nor cocommutative.

Construction
As an algebra over k, the Pareigis algebra is generated by elements x,y, 1/y, with the relations xy + yx = x2 = 0. The coproduct takes x to x⊗1 + (1/y)⊗x and y to y⊗y, and the counit takes x to 0 and y to 1. The antipode takes x to xy and y to its inverse and has order 4.

Relation to complexes
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.