$n\text{-}Li{e}_d$ Operad and its Koszul Dual

Abstract

We study the operad $n\text{-}Li{e}_d$, 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 $n\text{-}Li{e}_d$, called $n\text{-}Co{m}_{-d+n-2}$, 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 $\{{\mathcal{O}}_n{\}}_{n\ge 0}$, called the Odd graphs, whose spectrum is related to the lower triangular sequence $\{{\mathcal{E}}_{r,n}\}$, called the Catalan triangle.