Operads of decorated cliques II: Noncrossing cliques
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866914664416280576 |
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| author | Giraudo, Samuele |
| author_facet | Giraudo, Samuele |
| contents | A complete study of an operad $\mathrm{NC} \mathcal{M}$ of noncrossing configurations of chords introduced in previous work of the author is performed. This operad is defined on the linear span of all noncrossing $\mathcal{M}$-cliques. These are noncrossing configurations of chords with arcs labeled by a unitary magma $\mathcal{M}$. The magmatic product of $\mathcal{M}$ intervenes for the computation of the operadic composition of $\mathcal{M}$-cliques. We show that this operad is binary, quadratic, and Koszul by considering techniques coming from rewrite systems on trees. We also compute a presentation for its Koszul dual. Finally, we explain how $\mathrm{NC} \mathcal{M}$ allows one to obtain alternative constructions of already known operads like operads of formal fractions and the operad of bicolored noncrossing configurations. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2402_01500 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Operads of decorated cliques II: Noncrossing cliques Giraudo, Samuele Combinatorics Quantum Algebra 05C05, 05C76, 18D99, 05E99 A complete study of an operad $\mathrm{NC} \mathcal{M}$ of noncrossing configurations of chords introduced in previous work of the author is performed. This operad is defined on the linear span of all noncrossing $\mathcal{M}$-cliques. These are noncrossing configurations of chords with arcs labeled by a unitary magma $\mathcal{M}$. The magmatic product of $\mathcal{M}$ intervenes for the computation of the operadic composition of $\mathcal{M}$-cliques. We show that this operad is binary, quadratic, and Koszul by considering techniques coming from rewrite systems on trees. We also compute a presentation for its Koszul dual. Finally, we explain how $\mathrm{NC} \mathcal{M}$ allows one to obtain alternative constructions of already known operads like operads of formal fractions and the operad of bicolored noncrossing configurations. |
| title | Operads of decorated cliques II: Noncrossing cliques |
| topic | Combinatorics Quantum Algebra 05C05, 05C76, 18D99, 05E99 |
| url | https://arxiv.org/abs/2402.01500 |