$n\text{-}Lie_d$ Operad and its Koszul Dual

Abstract: We study the operad $n\text{-}Lie_d$, whose algebras are graded $n$-Lie algebras with degree $d$ $n$-arity operations, which were introduced in Nambu mechanics and later studied in the algebraic setting with Filippov. We compute the Koszul dual of $n\text{-}Lie_d$, called $n\text{-}Com_{-d+n-2}$, whose relations are derived from the Specht module $S^{(n,n-1)}$ for a partition $(n,n-1)$ of $2n-1$. The intrinsic connection between these two operads come from the eigenvalues of the sequence of graphs ${\mathcal{O}n}{n\geq 0}$, called the Odd graphs, whose spectrum is related to the lower triangular sequence ${\mathcal{E}_{r,n}}$, called the Catalan triangle.