Examples
| Depending on | Comultiplication | Counit | Antipode | Commutative | Cocommutative | Remarks | |
|---|---|---|---|---|---|---|---|
| group algebra KG | group G | Δ(g) = g ⊗ g for all g in G | ε(g) = 1 for all g in G | S(g) = g −1 for all g in G | if and only if G is abelian | yes | |
| functions f from a finite group to K, KG (with pointwise addition and multiplication) | finite group G | Δ(f)(x,y) = f(xy) | ε(f) = f(1G) | S(f)(x) = f(x−1) | yes | if and only if G is commutative | |
| Regular functions on an algebraic group | Δ(f)(x,y) = f(xy) | ε(f) = f(1G) | S(f)(x) = f(x−1) | yes | if and only if G is commutative | Conversely, every commutative Hopf algebra over a field arises from a group scheme in this way, giving an antiequivalence of categories. | |
| Tensor algebra T(V) | vector space V | Δ(x) = x ⊗ 1 + 1 ⊗ x, x in V | ε(x) = 0 | S(x) = -x for all x in T1(V) (and extended to higher tensor powers) | no | yes | symmetric algebra and exterior algebra (which are quotients of the tensor algebra) are also Hopf algebras with this definition of the comultiplication, counit and antipode |
| Universal enveloping algebra U(g) | Lie algebra g | Δ(x) = x ⊗ 1 + 1 ⊗ x for every x in g (this rule is compatible with commutators and can therefore be uniquely extended to all of U) | ε(x) = 0 for all x in g (again, extended to U) | S(x) = -x | no | yes | |
| Sweedler's Hopf algebra H=K/c2 = 1, x2 = 0 and xc = - cx. | K is a field with characteristic different from 2 | Δ (c) = c ⊗ c, Δ (x) = c ⊗ x + x ⊗ 1, Δ (1) = 1 ⊗ 1 | ε(c) = 1 and ε(x) = 0 | S(c) = c-1 = c and S(x) = -cx | no | no | The underlying vector space is generated by {1, c, x, cx} and thus has dimension 4. This is the smallest example of a Hopf algebra that is both non-commutative and non-cocommutative. |
Note that functions on a finite group can be identified with the group ring, though these are more naturally thought of as dual – the group ring consists of finite sums of elements, and thus pairs with functions on the group by evaluating the function on the summed elements.
Read more about this topic: Hopf Algebras
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