# Sweedler's Hopf algebra

In mathematics, Moss E. Sweedler (1969, p. 89–90) introduced an example of an infinite-dimensional Hopf algebra, and Sweedler's Hopf algebra H4 is a certain 4-dimensional quotient of it that is neither commutative nor cocommutative.

## Definition

The following infinite dimensional Hopf algebra was introduced by Sweedler (1969, pages 89–90). The Hopf algebra is generated as an algebra by three elements x, g, and g−1.

The coproduct Δ is given by

Δ(g) = gg, Δ(x) = 1⊗x + xg

The antipode S is given by

S(x) = –x g−1, S(g) = g−1

The counit ε is given by

ε(x)=0, ε(g) = 1

Sweedler's 4-dimensional Hopf algebra H4 is the quotient of this by the relations

x2 = 0, g2 = 1, gx = –xg

so it has a basis 1, x, g, xg (Montgomery 1993, p.8). Note that Montgomery describes a slight variant of this Hopf algebra using the opposite coproduct, i.e. the coproduct described above composed with the tensor flip on H4H4.

Sweedler's 4-dimensional Hopf algebra is a quotient of the Pareigis Hopf algebra, which is in turn a quotient of the infinite dimensional Hopf algebra.

## References

• Armour, Aaron; Chen, Hui-Xiang; Zhang, Yinhuo (2006), "Structure theorems of H4-Azumaya algebras", Journal of Algebra, 305 (1): 360–393, doi:10.1016/j.jalgebra.2005.10.020, ISSN 0021-8693, MR 2264134
• Montgomery, Susan (1993), Hopf algebras and their actions on rings, CBMS Regional Conference Series in Mathematics, 82, Published for the Conference Board of the Mathematical Sciences, Washington, DC, ISBN 978-0-8218-0738-5, MR 1243637
• Sweedler, Moss E. (1969), Hopf algebras, Mathematics Lecture Note Series, W. A. Benjamin, Inc., New York, MR 0252485
• Van Oystaeyen, Fred; Zhang, Yinhuo (2001), "The Brauer group of Sweedler's Hopf algebra H4", Proceedings of the American Mathematical Society, 129 (2): 371–380, doi:10.1090/S0002-9939-00-05628-8, ISSN 0002-9939, MR 1706961