Lie polynomial

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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].


[a1] N. Bourbaki, "Groupes de Lie" , II: Algèbres de Lie libres , Hermann (1972)
[a2] C. Reutenauer, "Free Lie algebras" , Oxford Univ. Press (1993)
[a3] X. Viennot, "Algèbres de Lie libres et monoïdes libres" , Springer (1978)
[a4] J.-P. Serre, "Lie algebras and Lie groups" , Benjamin (1965)
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Lie polynomial. Encyclopedia of Mathematics. URL:
This article was adapted from an original article by M. Hazewinkel (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article