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| Format: | Preprint |
| Published: |
2024
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| Online Access: | https://arxiv.org/abs/2401.15310 |
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| _version_ | 1866914671811887104 |
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| author | Tipton, Cody |
| author_facet | Tipton, Cody |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2401_15310 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | $n\text{-}Lie_d$ Operad and its Koszul Dual Tipton, Cody Rings and Algebras Combinatorics 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. |
| title | $n\text{-}Lie_d$ Operad and its Koszul Dual |
| topic | Rings and Algebras Combinatorics |
| url | https://arxiv.org/abs/2401.15310 |