On $q$-pre-Lie algebras

In this paper, we introduce the notion of $q$-pre-Lie algebras from the perspective of representations of Lie algebras, providing a parametrized generalization that unifies pre-Lie algebras and anti-pre-Lie algebras.

For a $q$-pre-Lie algebra $(A,\circ)$, the commutator of $\circ$ is a Lie bracket and the left multiplication operator scaled by $q$ gives a representation of the associated commutator Lie algebra.

We also introduce the notions of $q$-$\mathcal{O}$-operators and $q$-Novikov algebras, and investigate their relationships with $q$-pre-Lie algebras.

Several explicit constructions of $q$-pre-Lie algebras are provided.

Moreover, we give a complete classification of graded $q$-pre-Lie algebra structures on the Witt algebra and prove the nonexistence of such structures on the Virasoro algebra when $q\neq 1$.

Finally, for finite-dimensional complex simple Lie algebras, we show that compatible root-graded $q$-pre-Lie algebras exist on $\mathfrak{sl}_2(\mathbb{C})$ precisely when $q=2$ or $q=-1$, and do not exist on any other simple Lie algebra.