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) = g ⊗g, Δ(x) = 1⊗x + x ⊗g

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 H₄⊗H₄. This Hopf algebra is isomorphic to the Hopf algebra described here by the Hopf algebra homomorphism $g\mapsto g$ and $x\mapsto gx$.

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, vol. 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, ISBN 9780805392548, 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, hdl:10067/378420151162165141, ISSN 0002-9939, MR 1706961