Hopf algebra
A Hopf algebra is the algebraic distillation of a group. A group has three operations: multiply, take the identity, and invert. Translate these into linear algebra on the function space of the group and you get, respectively, a multiplication, a unit, and — dual to inversion — an antipode, sitting on top of the comultiplication and counit that encode the group law. A Hopf algebra is any structure with all of these axioms, whether or not it comes from an actual group.
Formally, a Hopf algebra over k is a bialgebra (H, m, u, Δ, ε) together with a linear map S : H -> H, the antipode, satisfying m ∘ (S ⊗ id) ∘ Δ = u ∘ ε = m ∘ (id ⊗ S) ∘ Δ. In Sweedler notation this is Σ S(c(1)) c(2) = ε(c) 1 = Σ c(1) S(c(2)). The antipode is the convolution inverse of the identity map in the convolution algebra Hom(H, H).
The two foundational examples are dual to each other. The group algebra k[G] is cocommutative with Δ(g) = g ⊗ g, ε(g) = 1, S(g) = g^{-1}. Its restricted dual is the algebra of representative functions on G, which is commutative; for G algebraic this is the coordinate ring O(G). Quantum groups interpolate between these worlds as one-parameter deformations that are neither commutative nor cocommutative.
On the polynomial Hopf algebra k[x] set Δ(x) = x ⊗ 1 + 1 ⊗ x, ε(x) = 0, S(x) = -x. Then Σ S(x(1)) x(2) = S(x)·1 + S(1)·x = -x + x = 0 = ε(x)1, confirming the antipode axiom. This is the enveloping algebra of the one-dimensional abelian Lie algebra.
k[x] as the enveloping algebra of an abelian line, with x primitive.
Named after Heinz Hopf, who introduced the structure in 1941 in topology: the cohomology ring of a Lie group carries a comultiplication, and his classification theorem of such rings is a statement about Hopf algebras avant la lettre. The antipode is automatically an algebra anti-homomorphism: S(ab) = S(b)S(a).