[Paper Review] Deformations of path algebras of quivers with relations

Severin Barmeier, Zhengfang Wang|arXiv (Cornell University)|Feb 23, 2020

Algebraic structures and combinatorial models99 references4 citations

TL;DR

This paper establishes an equivalence between deformations of path algebras of quivers with relations and deformations of reduction systems, introducing an explicit L∞ algebra that controls the latter. It shows that any formal deformation of the associative multiplication can be realized via a combinatorially defined star product, providing a complete and concrete description of deformation theory for such algebras, with applications to deformation quantization and PBW-type deformations.

ABSTRACT

Let $A = \Bbbk Q / I$ be the path algebra of any finite quiver $Q$ modulo any two-sided ideal $I$ of relations and let $R$ be any reduction system satisfying the diamond condition for $I$. We introduce an intrinsic notion of deformation of reduction systems and show that there is an equivalence of deformation problems between deformations of the associative algebra $A$ and deformations of the reduction system $R$, the latter being controlled by a natural, explicit L$_\infty$ algebra. It follows in particular that any formal deformation of the associative multiplication on $A$ can, up to gauge equivalence, be given by a combinatorially defined star product, and the approach via reduction systems can be used to give a concrete and complete description of the deformation theory of $A$. For the polynomial algebra in a finite number of variables, this combinatorial star product can be described via bidifferential operators associated to graphs, which we compare to the graphs appearing in Kontsevich's universal quantization formula. Using the notion of admissible orders on the set of paths of the quiver $Q$, we give criteria for the existence of algebraizations of formal deformations, which we also interpret geometrically via algebraic varieties of reduction systems. In this context the Maurer-Cartan equation of the L$_\infty$ algebra can be viewed as a generalization of the Braverman-Gaitsgory criterion for Poincaré-Birkhoff-Witt deformations of Koszul algebras.

Motivation & Objective

• To establish a systematic deformation theory for path algebras of quivers with relations using reduction systems.
• To show that deformations of the associative algebra A = kQ/I are equivalent to deformations of a reduction system R satisfying the diamond condition.
• To construct an explicit L∞ algebra that controls deformations of reduction systems, enabling a complete description of the deformation problem.
• To provide a combinatorial realization of star products for formal deformations, particularly in the context of polynomial algebras and Kontsevich-type quantization.
• To give criteria for algebraization of formal deformations and interpret them geometrically as algebraic varieties of reduction systems.

Proposed method

• Introduce an intrinsic notion of deformation for reduction systems, generalizing classical deformation theory to combinatorial structures.
• Use the Diamond Lemma and noncommutative Gröbner basis theory to define reduction systems R from ideals I in path algebras kQ.
• Construct an L∞ algebra structure on the Hochschild cochain complex of the reduction system, which governs its deformation theory.
• Define a combinatorial star product via bidifferential operators associated to graphs, linking to Kontsevich’s formality theorem.
• Apply the Maurer–Cartan equation of the L∞ algebra to characterize deformations, generalizing the Braverman–Gaitsgory criterion for PBW algebras.
• Use admissible orders on paths to define finiteness conditions and construct algebraic varieties parametrizing actual deformations and PBW deformations.

Experimental results

Research questions

• RQ1 How can the deformation theory of path algebras of quivers with relations be reformulated in terms of reduction systems?
• RQ2 What is the role of the L∞ algebra in controlling deformations of reduction systems, and how does it relate to classical Hochschild cohomology?
• RQ3 Can every formal deformation of an associative algebra A = kQ/I be realized via a combinatorial star product constructed from reduction systems?
• RQ4 How do the graphical calculus and bidifferential operators from graphs relate to Kontsevich’s universal formality formula?
• RQ5 Under what conditions can formal deformations be algebraized, and how can this be interpreted geometrically as a variety of reduction systems?

Key findings

• There is a canonical equivalence between the deformation problem of the associative algebra A = kQ/I and the deformation problem of a reduction system R satisfying the diamond condition.
• Deformations of reduction systems are controlled by a natural, explicit L∞ algebra, which provides a complete and computable framework for deformation theory.
• Any formal deformation of the multiplication on A is, up to gauge equivalence, given by a combinatorially defined star product built from bidifferential operators associated to graphs.
• For the polynomial algebra, the star product can be described via graphs and matches the structure of Kontsevich’s universal quantization formula.
• The Maurer–Cartan equation of the L∞ algebra generalizes the Braverman–Gaitsgory criterion for Poincaré–Birkhoff–Witt deformations of Koszul algebras.
• Algebraization of formal deformations exists under finitely generated and admissible order conditions, and such deformations correspond to algebraic varieties of reduction systems.