# Lie algebra, free

*over a ring *

A Lie algebra over in which one can distinguish a free generating set , a mapping from which into an arbitrary algebra over can be be extended to a homomorphism from into . The cardinality of completely determines and is called its rank. A free Lie algebra is a free -module (for bases of it see Basic commutator). A subalgebra of a free Lie algebra over a field is itself a free Lie algebra (Shirshov's theorem, [1]). If , then this is true only under the condition that is a free Abelian group [2]. The finitely-generated subalgebras of a free Lie algebra over a field form a sublattice of the lattice of all subalgebras [3]. W. Magnus [4] established canonical connections between free Lie algebras and both free groups and free associative algebras.

#### References

[1] | A.I. Shirshov, "Subalgebras of free Lie algebras" Mat. Sb. , 33 : 2 (1953) pp. 441–452 (In Russian) |

[2] | E. Witt, "Die Unterringe der freien Lieschen Ringe" Math. Z. , 64 (1956) pp. 195–216 |

[3] | G.P. Kukin, "Intersection of subalgebras of a free Lie algebra" Algebra and Logic , 16 (1977) pp. 387–394 Algebra i Logika , 16 (1977) pp. 577–587 |

[4] | W. Magnus, "Ueber Beziehungen zwischen höheren Kommutatoren" J. Reine Angew. Math. , 177 (1937) pp. 105–115 |

[5] | Yu.A. Bakhturin, "Identical relations in Lie algebras" , VNU , Utrecht (1987) (Translated from Russian) |

#### Comments

To construct one can start from the free associative algebra generated by , which is made into a Lie algebra by taking as Lie product . Then is the Lie subalgebra of generated by , and is the universal enveloping algebra of .

In case is a field of characteristic 0, more precise results on which elements of belong to are given by the Specht–Wever theorem and the Friedrichs theorem, respectively. The first one says that a homogeneous element of degree belongs to if and only if , where is the linear mapping defined by

for . Friedrichs' theorem says for the case of finite that belongs to if and only if , where is the homomorphism defined by for .

Free Lie algebras are the best context for the formulation of the Campbell–Baker–Hausdorff formula in its most general form.

#### References

[a1] | N. Jacobson, "Lie algebras" , Interscience (1962) ((also: Dover, reprint, 1979)) |

[a2] | N. Bourbaki, "Elements of mathematics. Lie groups and Lie algebras" , Addison-Wesley (1975) (Translated from French) |

**How to Cite This Entry:**

Lie algebra, free. Yu.A. Bakhturin (originator),

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