[Paper Review] Distributive laws between the Three Graces

Murray R. Bremner, Martin Markl|arXiv (Cornell University)|Sep 21, 2018

Advanced Topics in Algebra25 references4 citations

TL;DR

This paper classifies all homogeneous distributive laws between the three fundamental algebraic operads—associative (Ass), commutative associative (Com), and Lie (Lie)—using categorical and computational methods. It proves that only the classical, trivial, and truncated distributive laws exist, demonstrating a surprising rigidity in these algebras, with key results confirmed via Maple-based Gröbner basis computations on systems of quadratic equations.

ABSTRACT

By the Three Graces we refer, following J.-L. Loday, to the algebraic operads Ass, Com, and Lie, each generated by a single binary operation; algebras over these operads are respectively associative, commutative associative, and Lie. We classify all distributive laws (in the categorical sense of Beck) between these three operads. Some of our results depend on the computer algebra system Maple, especially its packages LinearAlgebra and Groebner.

Motivation & Objective

• To classify all homogeneous distributive laws between the three fundamental algebraic operads: associative (Ass), commutative associative (Com), and Lie (Lie).
• To investigate whether new, non-classical distributive laws exist beyond the well-known examples like Poisson and Gerstenhaber algebras.
• To explore the role of computer algebra in solving large systems of quadratic equations arising from distributive law conditions.
• To determine whether bizarre-looking distributive laws outside the Three Graces are isomorphic to standard ones, using algebraic transformations.
• To extend the analysis to associative-magmatic distributive laws, revealing that even exotic-looking laws may be isomorphic to known types.

Proposed method

• Employ Jon Beck’s categorical framework for distributive laws between operads, translating them into systems of quadratic equations in the operadic context.
• Use the computer algebra system Maple, particularly its LinearAlgebra and Groebner packages, to solve large systems of polynomial equations arising from distributive law axioms.
• Apply Gröbner basis techniques over polynomial rings to classify solutions and identify isomorphism classes of distributive laws.
• Utilize algebraic substitutions, such as Eulerian substitution (13), to transform complex-looking laws into standard forms, proving isomorphism to known laws.
• Verify coherence and consistency of distributive laws by hand in selected cases to understand structural constraints.
• Compare results across operad pairs, including Ass–Mag and Lie–Com, to identify patterns in rigidity and isomorphism classes.

Experimental results

Research questions

• RQ1 Are there any non-trivial, non-isomorphic distributive laws between the operads Ass, Com, and Lie beyond the classical examples?
• RQ2 How many isomorphism classes of homogeneous distributive laws exist between the three operads of the Three Graces?
• RQ3 Can computer algebra systems like Maple effectively classify distributive laws involving hundreds of quadratic equations?
• RQ4 Do seemingly bizarre distributive laws outside the Three Graces, such as those between associative and magmatic operations, turn out to be isomorphic to standard ones?
• RQ5 What structural properties of operads lead to a finite number of distributive laws, and can this be generalized to other operad pairs?

Key findings

• Only the trivial, truncated, and classical distributive laws exist between the operads Ass, Com, and Lie, with no new non-isomorphic laws discovered.
• The classification of distributive laws between Ass and Mag (magmatic) operations yields exactly two non-isomorphic types: the trivial and the truncated law.
• Maple-based Gröbner basis computations confirmed that the system of quadratic equations for distributive laws between the Three Graces has only finitely many solutions.
• The Eulerian substitution (13) transforms complex-looking distributive laws into standard truncated forms, proving isomorphism to known laws.
• For the pair (Lie, Com), the only nontrivial distributive law corresponds to the Poisson algebra structure, confirming its uniqueness.
• In the category of sets, no distributive laws exist between Ass and Mag, as shown by Corollary 7.3, highlighting the role of the ground field in such constructions.