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Operads of decorated cliques II: Noncrossing cliques

Samuele Giraudo

Abstract

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.

Operads of decorated cliques II: Noncrossing cliques

Abstract

A complete study of an operad 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 -cliques. These are noncrossing configurations of chords with arcs labeled by a unitary magma . The magmatic product of intervenes for the computation of the operadic composition of -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 allows one to obtain alternative constructions of already known operads like operads of formal fractions and the operad of bicolored noncrossing configurations.
Paper Structure (43 sections, 29 theorems, 178 equations, 1 figure)

This paper contains 43 sections, 29 theorems, 178 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Elementary definitions and tools
  3. Trees and rewrite rules
  4. Trees
  5. Syntax trees
  6. Rewrite rules
  7. Free operads and Koszul duality
  8. Free operads
  9. Evaluations and treelike expressions
  10. Presentations by generators and relations
  11. Koszul duality and Koszulity
  12. Algebras over operads
  13. Operads of noncrossing decorated cliques
  14. General properties
  15. First properties
  16. ...and 28 more sections

Key Result

Lemma 2.2.1

Let $\mathcal{O}$ be an operad admitting a quadratic presentation $(G, \mathfrak{R})$. If there exists an orientation $\mathop{\mathrm{\to}}\nolimits$ of $\mathfrak{R}$ such that $\mathop{\mathrm{\to}}\nolimits$ is a convergent rewrite rule, then $\mathcal{O}$ is Koszul.

Figures (1)

  • Figure 1: The partial composition of $\mathrm{NC}\mathcal{M}$ realized on $\mathcal{M}$-Schröder trees. Here, the two cases \ref{['subfig:composition_NC_M_Schroder_trees_2']} and \ref{['subfig:composition_NC_M_Schroder_trees_3']} for the computation of $\mathfrak{s} \circ_i \mathfrak{t}$ are shown, where $\mathfrak{s}$ and $\mathfrak{t}$ are two $\mathcal{M}$-Schröder trees. In these drawings, the triangles denote subtrees.

Theorems & Definitions (29)

  • Lemma 2.2.1
  • Proposition 3.1.1
  • Proposition 3.1.2
  • Lemma 3.1.3
  • Proposition 3.1.4
  • Proposition 3.1.5
  • Proposition 3.1.6
  • Proposition 3.1.7
  • Lemma 3.2.1
  • Proposition 3.2.2
  • ...and 19 more