Hopf Algebras - Examples


Depending on Comultiplication Counit Antipode Commutative Cocommutative Remarks
group algebra KG group G Δ(g) = gg 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) = cc, Δ (x) = cx + 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.

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