Pre-Lie algebras and Incidence Categories of Colored Rooted Trees

The incidence category $\C_{\F}$ of a family $\F$ 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 $\C_{\F}$ was also defined. We show that if the Hasse diagrams underlying $\F$ are rooted trees, then the subspace $\n_{\F}$ 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 $\n_{\F}$, including the nilpotent subalgebras of $\mathfrak{sl}_n$, $L \mathfrak{gl}_n$, and several others.