Sweedler's Hopf algebra

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 = –xg so 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.

References