[Paper Review] Completing the operadic butterfly

Jean-Louis Loday|ArXiv.org|Sep 10, 2004

Advanced Topics in Algebra5 references12 citations

TL;DR

This paper completes the operadic butterfly by introducing two new operads, $\mathcal{X}^+$ and $\mathcal{X}^-$, which are self-dual and have 8 binary operations and 16 relations. These operads unify dendriform and diassociative algebras via symmetry and symmetrization, resolving a long-standing gap in the operadic duality framework.

ABSTRACT

We complete a certain diagram (the operadic butterfly) of categories of algebras involving Com, As, and Lie by constructing a type of algebras which have 4 generating operations and 16 relations. The associated operad is self-dual for Koszul duality.

Motivation & Objective

• To complete the operadic butterfly diagram by identifying a missing category of algebras ${\mathcal{X}}$ at the top of the symmetry axis.
• To construct an operad with 8 binary operations and 16 relations that satisfies self-duality under Koszul duality.
• To ensure that dendriform algebras embed into ${\mathcal{X}}$-algebras via symmetry, and that ${\mathcal{X}}$-algebras map to diassociative algebras via symmetrization.
• To verify that the resulting diagram commutes, preserving the duality and functorial structure of the original butterfly.

Proposed method

• Define the free operad $Dend \square Dias$ on four operations: $\nwarrow, \nearrow, \swarrow, \searrow$, with 15 relations from dendriform and diassociative axioms.
• Introduce two additional relations, (16+) and (16−), to quotient the free operad and form the candidate operads $\mathcal{X}^+$ and $\mathcal{X}^-$.
• Prove that both $\mathcal{X}^\pm$ are self-dual by showing that the dual of the defining relations matches the original up to sign, satisfying $-\alpha^2 + \beta^2 = 0$.
• Establish that $\mathcal{X}^\pm$-algebras restrict to dendriform algebras when $\nwarrow = \nearrow$ and $\swarrow = \searrow$, and symmetrize to diassociative algebras via $x \dashv y = x\nwarrow y + x\swarrow y$, $x \vdash y = x\nearrow y + x\searrow y$.
• Verify that the composite functors $Dend \hookrightarrow \mathcal{X}^\pm \xrightarrow{+} Dias$ match the composition $Dend \xrightarrow{+} As \hookrightarrow Dias$, ensuring diagram commutativity.
• Confirm linear independence of the 16 relations, proving a complete and minimal presentation of the operad.

Experimental results

Research questions

• RQ1 Can the operadic butterfly be completed by a self-dual operad with 8 binary operations at the top of the symmetry axis?
• RQ2 What algebraic structure with four operations and 16 relations realizes the missing link between dendriform and diassociative algebras?
• RQ3 Are there two distinct solutions $\mathcal{X}^+$ and $\mathcal{X}^-$ that satisfy the duality and functorial constraints of the butterfly?
• RQ4 Does the resulting operad preserve the Koszul duality symmetry and allow consistent symmetrization to diassociative algebras?
• RQ5 Is the operad $\mathcal{X}^\pm$ self-dual and minimal, with a complete presentation of 16 linearly independent relations?

Key findings

• The paper constructs two new operads, $\mathcal{X}^+$ and $\mathcal{X}^-$, which complete the operadic butterfly by satisfying all required duality and functorial properties.
• Each operad has exactly 8 binary operations and 16 linearly independent relations, forming a minimal and complete presentation.
• The operads are self-dual under Koszul duality, as verified by the condition $-\alpha^2 + \beta^2 = 0$ for the defining relations.
• Dendriform algebras embed into $\mathcal{X}^\pm$-algebras via the symmetry $\nwarrow = \nearrow$, $\swarrow = \searrow$, satisfying all 16 relations.
• Any $\mathcal{X}^\pm$-algebra gives rise to a diassociative algebra through the symmetrization $x \dashv y = x\nwarrow y + x\swarrow y$, $x \vdash y = x\nearrow y + x\searrow y$.
• The composite functors $Dend \hookrightarrow \mathcal{X}^\pm \xrightarrow{+} Dias$ and $Dend \xrightarrow{+} As \hookrightarrow Dias$ are equal, confirming the commutativity of the upper square in the completed butterfly.