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

Cody Tipton

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.

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

Abstract

We study the operad , whose algebras are graded -Lie algebras with degree -arity operations, which were introduced in Nambu mechanics and later studied in the algebraic setting with Filippov. We compute the Koszul dual of , called , whose relations are derived from the Specht module for a partition of . The intrinsic connection between these two operads come from the eigenvalues of the sequence of graphs , called the Odd graphs, whose spectrum is related to the lower triangular sequence , called the Catalan triangle.
Paper Structure (24 sections, 17 theorems, 136 equations, 3 figures)

This paper contains 24 sections, 17 theorems, 136 equations, 3 figures.

Table of Contents

  1. Preliminaries
  2. Rooted Trees
  3. Operads
  4. Free Operad
  5. $n$-Quadratic Operads
  6. Description of $n$-Quadratic Operad
  7. Description of $F(E)^{(2)}$
  8. Koszul Dual of $n$-Quadratic Operad
  9. (anti)-Commutative $n$-Quadratic Operads and their Koszul dual
  10. $n\text{-}Lie_d$ and $n\text{-}Com_d$ Operads
  11. General Definitions for $n$-Lie algebras of degree $d$
  12. Definition of $n\text{-}Lie_d$ Operad
  13. Quadratic Representations for $n\text{-}Lie_d$
  14. General Definition of $n\text{-}Com_d$ Operad
  15. Quadratic Representation for $n\text{-}Com$
  16. ...and 9 more sections

Key Result

Theorem 1

For all $n\geq 2$ and $d\in\mathbb{Z}$, the operads $n\text{-}Lie_d$ and $n\text{-}Com_{-d+n-2}$ are Koszul dual.

Figures (3)

  • Figure 1: Grafting of $Cor_3$ and $Cor_2$ at $e$, where $e$ is the $3$rd input edge of $Cor_3$.
  • Figure 2:
  • Figure 3: The Young tableau $T^{\lambda_n}$

Theorems & Definitions (56)

  • Theorem 1
  • Definition 1.1
  • Definition 1.2
  • Definition 1.3
  • Definition 1.4
  • Definition 1.5
  • Example 1
  • Definition 1.6
  • Lemma 1.1
  • Definition 1.7
  • ...and 46 more