Input/output coloring and Gr\"obner basis for dioperads

TL;DR

This paper introduces a functor that transforms dioperads into two-colored operads, enabling the use of operadic tools like Gr"obner bases to analyze dioperads with various applications and explicit computations.

Contribution

It presents a novel functorial construction that relates dioperads to colored operads, allowing operadic techniques to be applied to dioperads for the first time.

Findings

Computed dimensions for the dioperad of Lie bialgebras.

Described Gr"obner basis and minimal resolution for triangular Lie bialgebras.

Established Koszul property for a class of dioperads from cyclic operads.

Abstract

We introduce a functor $\Psi$ that associates to a dioperad $P$ acting on a vector space $V$ a two-colored operad $\Psi(P)$ acting on the pair $(V, V^*)$. The construction is based on a simple pictorial idea: by selecting one input or output and dualizing, if necessary, the remaining ones, any dioperadic tree can be ``rerooted'' as a colored operadic tree. This transformation allows one to apply the standard operadic machinery -- such as Gr\"obner bases and Hilbert series -- to the study of dioperads. We illustrate the method with several examples and applications. (1) We compute the dimensions of the spaces of operations for the dioperad of Lie bialgebras. (2) We describe a Gr\"obner basis and construct a minimal resolution for the dioperad of triangular Lie bialgebras. (3) We perform explicit computations for the dioperad of ``algebraic string operations''. (4) We give a pictorial construction proving the existence of quadratic Gr\"obner bases and establishing the Koszul property for a broad class of dioperads arising from cyclic operads.