Free Lie algebra

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In mathematics, a free Lie algebra, over a given field K, is a Lie algebra generated by a set X, without any imposed relations.

Contents 1 Formulation 2 Description 3 See also 4 References

[edit] Formulation

More precisely, we have the following definition:

Let X be a set and i: X → L a morphism from X into a Lie algebra L. The Lie algebra L is called free on X if for any Lie algebra A with a mapping f: X → A, there is a unique Lie algebra morphism g: L → A such that f = g o i.

Given a set X, one can show that there exists a unique free Lie algebra L(X) generated by X.

In the language of category theory, the functor sending a set X to the Lie algebra generated by X is the free functor from the category of sets to the category of Lie algebras. That is, it is left adjoint to the forgetful functor.

[edit] Description

A more concrete description can be given in terms of a Hall set, which is a particular kind of subset inside the free magma on X. Elements of the free magma are binary trees, with their leaves labelled by elements of X. Hall sets are named after Philip Hall. Subsequently Wilhelm Magnus showed that they arise as the graded Lie algebra associated with the filtration on a free group given by the lower central series. This correspondence was motivated by commutator identities in group theory due to Hall and Witt.

The universal enveloping algebra of a free Lie algebra on a set X is the free associative algebra generated by X. By the Poincaré-Birkhoff-Witt theorem it is the "same size" as the symmetric algebra of the free Lie algebra (meaning that if both sides are graded by giving elements of X degree 1 then they are isomorphic as graded vector spaces). This can be used to describe the dimension of the piece of the free Lie algebra of any given degree.

Any Lie subalgebra of a free Lie algebra is itself a free Lie algebra. This was proved by Anatoly Illarionovich Shirshov (1921-1981), a student of Kurosh, in 1953, and by Ernst Witt (1956). This result is known as Shirshov's theorem or the Shirshov-Witt theorem.

[edit] See also

• Free object
• Free algebra
• Free group

[edit] References

• N. Bourbaki, "Lie Groups and Lie Algebras," Chapter II: Free Lie Algebras, Springer, 1989. ISBN 0-387-50218-1
• W. Magnus, A. Karrass, D. Solitar, "Combinatorial group theory". Reprint of the 1976 second edition, Dover, 2004. ISBN 0-486-43830-9
• Yu.A. Bakhturin (2001), “Free Lie algebra over a ring”, in Hazewinkel, Michiel, Encyclopaedia of Mathematics, Kluwer Academic Publishers, ISBN 978-1556080104 
• G. Melançon (2001), “Hall set”, in Hazewinkel, Michiel, Encyclopaedia of Mathematics, Kluwer Academic Publishers, ISBN 978-1556080104 
• G. Melançon (2001), “Hall word”, in Hazewinkel, Michiel, Encyclopaedia of Mathematics, Kluwer Academic Publishers, ISBN 978-1556080104 
• Reutenauer, Christophe (1993). Free Lie Algebras. Oxford University Press. ISBN 0198536798. .