Sweedler's Hopf algebra explained

In mathematics, 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 . 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 = –xgso it has a basis 1, x, g, xg . 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.