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.
As an algebra over k, the Pareigis algebra is generated by elements x,y, 1/y, with the relations xy + yx = x<sup>2</sup> = 0. The coproduct takes x to xâÂÂ1 + (1/y)âÂÂx and y to yâÂÂy, 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.
If M = âÂÂM<sub>n</sub> is a complex with differential d of degree âÂÂ1, then M can be made into a comodule over H by letting the coproduct take m to ã y<sup>n</sup>âÂÂm<sub>n</sub> + y<sup>n+1</sup>xâÂÂdm<sub>n</sub>, where m<sub>n</sub> is the component of m in M<sub>n</sub>. This gives an equivalence between the monoidal category of complexes over k with the monoidal category of comodules over the Pareigis Hopf algebra.