[Paper Review] (Non-)Koszulity of operads for n-ary algebras, cohomology and deformations

Martin Markl, Elisabeth Remm|arXiv (Cornell University)|Jul 9, 2009

Advanced Topics in Algebra12 references13 citations

TL;DR

This paper investigates the non-Koszul nature of operads for n-ary algebras, particularly focusing on algebras with a single anti-associative operation. It constructs the minimal model of the antiassociative operad to describe the deformation cohomology, revealing deviations from standard cohomological frameworks due to non-Koszulity, and explores free partially associative algebras while posing open problems in the field.

ABSTRACT

We investigate Koszulity of operads for various n-ary algebras. We then focus to algebras with one anti-associative operation. Since the corresponding operad is not Koszul, the deformation cohomology differs from the standard one. We describe the relevant part of the deformation cohomology for this type of algebras using the minimal model for the antiassociative operad. In the remaining sections we discuss free partially associative algebras and formulate open problems.

Motivation & Objective

• To analyze the Koszulity of operads associated with n-ary algebras, especially those with anti-associative operations.
• To identify why the standard deformation cohomology fails for anti-associative algebras due to non-Koszulity of the underlying operad.
• To construct and utilize the minimal model of the antiassociative operad to describe the correct deformation cohomology.
• To investigate free partially associative algebras as a related class of structures.
• To formulate open problems in the cohomology and deformation theory of n-ary algebras beyond the antiassociative case.

Proposed method

• Analyzing the operadic structure of n-ary algebras to determine Koszulity using homological algebra techniques.
• Applying the minimal model construction to the antiassociative operad to resolve non-Koszulity and compute the relevant cohomology.
• Using the minimal model to derive the correct deformation complex, which differs from the standard one due to non-Koszulity.
• Employing operadic duality and homological perturbation theory to handle the non-Koszul setting.
• Studying free partially associative algebras as a class with intermediate structural properties between associative and anti-associative algebras.
• Formulating open problems based on structural and cohomological observations in the studied algebras.

Experimental results

Research questions

• RQ1 Is the operad governing anti-associative n-ary algebras Koszul, and what are the consequences for deformation theory?
• RQ2 How does the deformation cohomology of anti-associative algebras differ from the standard cohomology when the operad is not Koszul?
• RQ3 What is the structure of the minimal model of the antiassociative operad, and how does it encode the correct deformation complex?
• RQ4 What are the cohomological properties of free partially associative algebras, and how do they relate to the antiassociative case?
• RQ5 What open problems arise in the deformation theory of n-ary algebras when Koszulity fails?

Key findings

• The operad for anti-associative n-ary algebras is not Koszul, invalidating the standard deformation cohomology framework.
• The minimal model of the antiassociative operad provides a correct description of the deformation cohomology, which differs from the standard one due to non-Koszulity.
• The deformation cohomology complex is derived from the minimal model, revealing non-trivial higher-order cohomological structures.
• Free partially associative algebras are identified as a natural class for further study in the context of n-ary algebras.
• The paper formulates open problems concerning cohomology and deformation theory in non-Koszul n-ary algebraic structures.