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 + ⊗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 = ⊕M_n 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⊗m_n + yⁿ⁺¹x⊗''dmn'', where m_n is the component of m in M_n. This gives an equivalence between the monoidal category of complexes over k with the monoidal category of comodules over the Pareigis Hopf algebra.