In particular there is a basis of the free Lie algebra corresponding to Lyndon words, called the Lyndon basis. (This is also called the Chen–Fox–Lyndon basis or the Lyndon–Shirshov basis, and is essentially the same as the Shirshov basis.) There is a bijection γ from the Lyndon words in an ordered alphabet to a basis of the free Lie algebra on this alphabet defined as follows.
- If a word w has length 1 then γ(w)=w (considered as a generator of the free Lie algebra).
- If w has length at least 2, then write w=uv for Lyndon words u, v with v as long as possible. Then γ(w) =
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