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Input/output coloring and Gröbner basis for dioperads

Anton Khoroshkin

Abstract

By selecting a specific input or output of a dioperadic tree, we transform it into a rooted tree and induce a corresponding colored operadic structure. This fundamental pictorial construction demonstrates how the machinery of Gröbner bases and the theory of Hilbert series (well-established for (colored) operads) can be adapted to the dioperadic setting. We illustrate this framework by providing several examples and applications: (1) we compute the dimensions of the spaces of operations for the dioperad of Lie bialgebras; (2) we describe a Gröbner basis and a minimal resolution for the dioperad of triangular Lie bialgebras; (3) we provide computations for the dioperad of ``algebraic string operations''; (4) we present a graphical construction that establishes the existence of quadratic Gröbner bases and the Koszul property for a broad class of dioperads originating from cyclic operads.

Input/output coloring and Gröbner basis for dioperads

Abstract

By selecting a specific input or output of a dioperadic tree, we transform it into a rooted tree and induce a corresponding colored operadic structure. This fundamental pictorial construction demonstrates how the machinery of Gröbner bases and the theory of Hilbert series (well-established for (colored) operads) can be adapted to the dioperadic setting. We illustrate this framework by providing several examples and applications: (1) we compute the dimensions of the spaces of operations for the dioperad of Lie bialgebras; (2) we describe a Gröbner basis and a minimal resolution for the dioperad of triangular Lie bialgebras; (3) we provide computations for the dioperad of ``algebraic string operations''; (4) we present a graphical construction that establishes the existence of quadratic Gröbner bases and the Koszul property for a broad class of dioperads originating from cyclic operads.
Paper Structure (20 sections, 21 theorems, 93 equations)

This paper contains 20 sections, 21 theorems, 93 equations.

Table of Contents

  1. Introduction
  2. Structure of the paper
  3. Dioperads -- recollection
  4. From Dioperads to colored operads
  5. The pictorial map $\Psi$
  6. Koszul property and Hilbert series
  7. Colored shuffle operads and Gröbner bases for dioperads
  8. Shuffle (colored) operads
  9. Rewriting systems for Shuffle operads
  10. Koszulness and Anick-type resolution
  11. Coloring of inputs/outputs in a (cyclic) operad
  12. The pictorial maps $\Theta$ with examples $\Theta(\mathsf{Lie})$ and $\Theta(\mathcal{A} \mathit{ss})$
  13. Defining relations and Gröbner basis for dioperads $\Theta_{\mathfrak{c}}(\mathcal{P})$
  14. Examples
  15. The Frobenius dioperad ${{\mathcal{F}} rob}$
  16. ...and 5 more sections

Key Result

Proposition 2.1.2

The collection of all possible root choices in a dioperad defines a faithful exact functor The space of operations $\Psi({\mathcal{P}})^{{|}}(m,n-1)$ with a straight output, $m$ straight inputs, and $n-1$ dotted inputs, and the space $\Psi({\mathcal{P}})^{ \text{\bf\scriptsize $\vdots$} }(m-1,n)$ with a dotted output, $m-1$ straight inputs, and $n$ dotted inputs, are both canonically iso Moreove

Theorems & Definitions (65)

  • Definition 1.0.1
  • Definition 1.0.2
  • Definition 1.0.3
  • Proposition 2.1.2
  • proof
  • Example 2.1.3
  • Theorem 2.1.4
  • proof
  • Corollary 2.2.1
  • proof
  • ...and 55 more