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 (Pareigis 1981) as a natural example of a Hopf algebra that is neither commutative nor cocommutative.

## Contents

* 1 Construction
* 2 Relation to complexes
* 3 See also
* 4 References

## 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.

## See also

* Sweedler's Hopf algebra is the quotient of the Pareigis Hopf algebra obtained by putting y2 = 1.

## References

* Pareigis, Bodo (1981), "A noncommutative noncocommutative Hopf algebra in "nature"", J. Algebra 70 (2): 356–374, doi:10.1016/0021-8693(81)90224-6, https://epub.ub.uni-muenchen.de/7116/

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