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.

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 = ⊕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⊗dm_n, 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.

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, MR 0623814