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 {{harvtxt|Pareigis|1981}} 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.
See also
- Sweedler's Hopf algebra is the quotient of the Pareigis Hopf algebra obtained by putting y2 = 1.
References
- {{citation|mr=0623814
|last=Pareigis|first= Bodo
|title=A noncommutative noncocommutative Hopf algebra in "nature"
|journal=J. Algebra|volume= 70 |year=1981|issue= 2|pages= 356–374|doi=10.1016/0021-8693(81)90224-6|url=https://epub.ub.uni-muenchen.de/7116/|doi-access=free}}