Abstract
Let 𝔤 be a free brace algebra. This structure implies that 𝔤 is also a pre-Lie algebra and a Lie algebra. It is already known that 𝔤 is a free Lie algebra. We prove here that 𝔤 is also a free pre-Lie algebra, using a description of 𝔤 with the help of planar rooted trees, a permutative product, and manipulations on the Poincaré–Hilbert series of 𝔤.
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