primitive element

Among the elements of a Hopf algebra, some behave like 'infinitesimal' group elements — the directions you could move in if the group were a Lie group. These are the primitive elements. Where a grouplike element copies itself under the coproduct (a finite symmetry), a primitive element splits as itself-tensor-one plus one-tensor-itself, which is exactly the Leibniz rule for a derivative. Primitives are the linearized, additive part of the structure.

Formally, in a bialgebra H, an element x is primitive if Δ(x) = x ⊗ 1 + 1 ⊗ x. Applying the counit axiom forces ε(x) = 0. The set of primitives P(H) = { x : Δ(x) = x ⊗ 1 + 1 ⊗ x } is a linear subspace, and it is closed under the commutator bracket [x, y] = xy - yx, so P(H) is a Lie algebra. In a Hopf algebra the antipode acts on primitives by S(x) = -x.

This is the algebraic source of the Lie-algebra/group dictionary. For the universal enveloping algebra U(g) of a Lie algebra g, the primitive elements are exactly g itself (in characteristic zero, by the Milnor-Moore theorem), recovering g from its enveloping Hopf algebra. In characteristic p extra primitives can appear, such as p-th powers, which is why one passes to restricted Lie algebras there.