In this paper, K is a field of characteristic zero. A left pre-Lie algebra is a K-vector space with product ○ such that X ○ (Y ○ Z) - (X ○ Y) ○ Z = Y ○ (X ○ Z) - (Y ○ X) ○ Z. In this paper, we construct a product denoted by {white right-pointing small triangle} on the vector space T{script}′ spanned by the rooted trees:we show first that (T{script}′, {white right-pointing small triangle}) is a pre-Lie algebra spanned by two generators. Finally, we show that the pre-lie algebra (T{script}′, {white right-pointing small triangle}) is not free, by giving two families of relations.
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