The Koszul dual to n-Lie, n-Com algebras, and Young tableaux

Abstract

We study the operad $n\text{-}Lie_d$, whose algebras are $n$-Lie algebras, which was first introduced in Nambu mechanics to extend Hamiltonian mechanics to more than one Hamiltonian. We find the Koszul dual of $n\text{-}Lie_{-d+n-2}$ to be the operad $n\text{-}Com_d$, whose relations come from the Specht module $S^{(n,n-1)}$. We combine the operads $n$-Lie and $m$-Com to construct the operad $(m,n)-Poiss$, where the rewriting rule that relates them is through a generalized Leibniz rule. We generalize the above Koszul duality to different types of generalization of $Lie$ and $Com$ which arise from the eigenspaces of the general Kneser graphs $\cO_{n,s}$, where the operads $n\text{-}Lie$ ($n\text{-}Com)$ and $Lie_n^d$ ($Com_n^d)$ appear on different sides of the spectrum based on the parameter $s$. With the introduction of the new class of $n$-Com algebras through the Koszul duality, we take our first step in exploring these new types of algebras. Specifically, we start the work of trying to classify finite-dimensional simple $3$-Com algebras $C$ using the Peirce decomposition through semisimple idempotents $e$ through $\chi_e=m_3(e,e,-)$ to obtain important structural properties. In particular, it decomposes the $3$-Com algebra $C=\bigoplus C_e(t)$ for eigenvalues $t$ for $\chi_e$ in which $C_e(1)$ is a commutative unital associative $k$-algebra acting on the other components, which is almost associative. Furthermore, we briefly construct an analog of the Killing form for $3$-Com algebras, denoted as $\kappa$, and define non-degeneracy when the form is non-degenerate and fully degenerate when $\kappa=0$. We use this to show that every non-degenerate $3$-Com algebra is a direct product of non-degenerate simple $3$-Com algebras. However, one very interesting aspect of this is that not every finite-dimensional $3$-Com algebra is non-degenerate, as our main example is fully degenerate. Towards some classifications of simple $3$-Com algebras, if $e$ is primitive semisimple idempotent with exactly two eigenvalues, and $C$ is simple, then its eigenvalues consist of $\{1,-1\}$ which help give a classification in dimension $2$ and $3$. Finally, we construct combinatorial objects, called Young $n$-trees, which are just rooted trees with a local Young tableaux structure at each edge following what is happening in the $n$-Com operad. In particular, we use these to give an upper bound to the arities of the dimension for the operad $n\text{-}Com_d$, which gives a description for it.