# Lie polynomial

Let $\mathbf{Z}\langle X \rangle$ denote the free associative algebra over $\mathbf{Z}$ in the indeterminates $X = \{ X_i \ :\ i \in I \}$. Give $\mathbf{Z}\langle X \rangle$ bi-algebra and Hopf algebra structures by means of the co-multiplication, antipode, and augmentation defined by: $$\mu(X_i) = 1 \otimes X_i + X_i \otimes 1\ ,$$ $$\epsilon(X_i) = 0 \ ,$$ $$\iota(X_i) = -X_i \ .$$
Then $\mathbf{Z}\langle X \rangle$ becomes the Leibniz–Hopf algebra. A Lie polynomial is an element $P$ of $\mathbf{Z}\langle X \rangle$ such that $\mu(P) = 1 \otimes P + P \otimes 1$, i.e., the Lie polynomials are the primitive elements of the Hopf algebra $\mathbf{Z}\langle X \rangle$ (see Primitive element in a co-algebra). These form a Lie algebra $L$ under the commutator difference product $[P,Q] = PQ - QP$. The Lie algebra $L$ is the free Lie algebra on $X$ over $\mathbf{Z}$ (Friedrich's theorem; cf. also Lie algebra, free) and $\mathbf{Z}\langle X \rangle$ is its universal enveloping algebra.
For bases of $L$ viewed as a submodule of $\mathbf{Z}\langle X \rangle$, see Hall set; Shirshov basis; Lyndon word. Still other bases, such as the Meier–Wunderli basis and the Spitzer–Foata basis, can be found in [a3].