# Lie polynomial

Let denote the free associative algebra over in the indeterminates . Give bi-algebra and Hopf algebra structures by means of the co-multiplication, antipode, and augmentation defined by:

Then becomes the Leibniz–Hopf algebra. A Lie polynomial is an element of such that , i.e., the Lie polynomials are the primitive elements of the Hopf algebra (see Primitive element in a co-algebra). These form a Lie algebra under the commutator difference product . The Lie algebra is the free Lie algebra on over (Friedrich's theorem; cf. also Lie algebra, free) and is its universal enveloping algebra.

For bases of viewed as a submodule of , 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].

#### References

[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: http://encyclopediaofmath.org/index.php?title=Lie_polynomial&oldid=17076