Pre-Lie algebras and Incidence Categories of Colored Rooted Trees

The incidence category ${\text{C}}_{\text{F}}$ of a family $\text{F}$ of colored posets closed under disjoint unions and the operation of taking convex sub-posets was introduced by the author in \cite{Sz}, where the Ringel-Hall algebra \H_{\F} of ${\text{C}}_{\text{F}}$ was also defined. We show that if the Hasse diagrams underlying $\text{F}$ are rooted trees, then the subspace ${\text{n}}_{\text{F}}$ of primitive elements of \H_{\F} carries a pre-Lie structure, defined over $\mathbb{Z}$, and with positive structure constants. We give several examples of ${\text{n}}_{\text{F}}$, including the nilpotent subalgebras of ${\mathfrak{s}\mathfrak{l}}_n$, $L{\mathfrak{g}\mathfrak{l}}_n$, and several others.