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
Famous quotes containing the word examples:
“There are many examples of women that have excelled in learning, and even in war, but this is no reason we should bring ‘em all up to Latin and Greek or else military discipline, instead of needle-work and housewifry.”
—Bernard Mandeville (1670–1733)
“In the examples that I here bring in of what I have [read], heard, done or said, I have refrained from daring to alter even the smallest and most indifferent circumstances. My conscience falsifies not an iota; for my knowledge I cannot answer.”
—Michel de Montaigne (1533–1592)
“No rules exist, and examples are simply life-savers answering the appeals of rules making vain attempts to exist.”
—André Breton (1896–1966)