grouplike element

If primitive elements are the infinitesimal directions, grouplike elements are the honest 'group points' hiding inside a Hopf algebra. A grouplike element coproducts to its own tensor square — it splits into a perfect copy of itself on each side — which is exactly how a genuine group element behaves under the coproduct of a group algebra. Grouplikes are the discrete symmetries packaged inside the algebra.

Formally, an element g of a coalgebra is grouplike if Δ(g) = g ⊗ g and ε(g) = 1; the second condition rules out the trivial case g = 0. In a Hopf algebra every grouplike element is automatically invertible with inverse S(g), and the set G(H) of grouplike elements forms a group under multiplication. Moreover distinct grouplike elements are always linearly independent.

For the group algebra k[G], the grouplike elements are exactly the elements of G sitting inside k[G], so G(k[G]) = G recovers the original group on the nose. For the coordinate Hopf algebra O(G) of an algebraic group, grouplike elements correspond to characters G -> k^×, that is one-dimensional representations. So 'grouplike' captures group elements on one side of the duality and characters on the other.