A∞-algebras on directed graphs

Consider a directed graph Γ and a field k. We construct an associative algebra A(Γ) over k from Γ, whose elements are linear combinations of directed edges, where the multiplication is defined to be zero. The algebra on A(Γ) can be equipped with a graded structure by assigning integer degrees to the edges of Γ. We are particularly interested in the Hochschild cohomology CH∗(A(Γ)) of this graded algebra. Our primary focus is the A∞-algebras on A(Γ), which emerge as solutions to the Maurer-Cartan equation in CH∗(A(Γ)). Under this structure, the product of edges corresponds to their concatenation, though degree constraints on the edges significantly limit the number of such non-trivial structures. The second chapter develops the A∞-algebraic structures on directed graphs. This begins with a discussion of Hochschild cohomology on ungraded algebras, which then extends to graded ones. A∞-algebras are subsequently defined as degree 2 Hochschild cochains that satisfy the Maurer-Cartan equation. Through deformed Hochschild cohomology, an A∞-algebraic structure is then constructed for directed graphs. Later sections introduce tools and notations essential for simplifying work on these structures. The third chapter explores Directed Acyclic Graphs (DAGs), defined as graphs lacking loops. The absence of loops ensures that the Hochschild cochains associated with these graphs are finite-dimensional, which ensures that the MC equation becomes trivial beyond a certain degree. This chapter investigates such graphs, exploring degree assignments that maximize the number of products possible on them, referred to as MCAGs. Additionally, it introduces methods for visualizing and classifying the A∞-algebraic structures, documenting computational results derived using Wolfram Mathematica for these graphs with 5 and 6 vertices. The fourth chapter suggests various possible extensions to this research, revolving around discovering patterns and defining graph operations. The appendix provides the Mathematica file containing the code used to obtain computational results.

Name thesis:degree_name

Ph.D.

Level thesis:degree_level

Dissertation

Discipline thesis:degree_discipline

Mathematics

Grantor

University of Illinois Urbana-Champaign

Year dc:date

2025

Author dc:creator

• Musleh, Omar

Contributors dc:contributor

• Pascaleff, James
• Dunfield, Nathan M
• Lerman, Eugene M
• Janda, Felix

• A∞-algebras
• Hochschild cohomology
• Directed graphs
• Maurer-Cartan equation

Statement dc:rights

• Copyright 2025 Omar Musleh

Language dc:language

en, eng

Handle dc:identifier

https://hdl.handle.net/2142/129459

OAI record oai:identifier

https://www.ideals.illinois.edu/oai-pmh?verb=GetRecord&metadataPrefix=etdms&identifier=oai%3Awww.ideals.illinois.edu%3A2142%2F129459