[Paper Review] Fukaya Type Categories for Associative Algebras

Ryszard Nest, Boris Tsygan|ArXiv.org|Mar 27, 1998

Algebraic structures and combinatorial models9 references3 citations

TL;DR

This paper introduces an $A_{ u}$-category structure for associative algebras, where objects are automorphisms of the algebra, drawing conceptual parallels to Fukaya categories in symplectic topology. The construction uses curved $A_{ u}$-structures and provides a homotopical framework for studying algebraic automorphisms via higher homotopy theory, extending ideas from Floer cohomology to non-geometric settings.

ABSTRACT

We define for an associative algebra an $A_{\infty}$ category whose objects are automorphisms of this algebra. This construction has some resemblance with Fukaya'a categories related to Floer cohomology.

Motivation & Objective

• To develop a homotopical category structure for associative algebras analogous to Fukaya categories in symplectic geometry.
• To define an $A_{\infty}$-category whose objects are algebra automorphisms, generalizing categorical frameworks in quantum algebra.
• To establish a link between algebraic automorphisms and higher homotopy structures, akin to those in Floer theory.
• To provide a categorical framework for studying deformations and symmetries of associative algebras using $A_{\infty}$-technology.

Proposed method

• Constructing an $A_{\infty}$-category where the objects are automorphisms of a fixed associative algebra.
• Equipping the category with curved $A_{\infty}$-structure to handle non-unital and higher-order operations.
• Using Hochschild cochain complexes and higher operations to define the higher composition maps.
• Applying techniques from deformation theory and quantum algebra to ensure $A_{\infty}$-axioms are satisfied.
• Drawing analogies with Fukaya categories by modeling composition laws after those in Floer cohomology.
• Ensuring the category is well-defined via explicit construction of higher homotopies and associativity constraints.

Experimental results

Research questions

• RQ1 Can a Fukaya-type $A_{\infty}$-category be constructed for associative algebras using automorphisms as objects?
• RQ2 How do curved $A_{\infty}$-structures model algebraic symmetries and deformations in this context?
• RQ3 What is the relationship between this category and Floer cohomology or symplectic geometry?
• RQ4 How do the higher composition maps in this category reflect the algebraic properties of automorphisms?
• RQ5 What role do Hochschild cochains play in defining the $A_{\infty}$-structure on the category of automorphisms?

Key findings

• An $A_{\infty}$-category is explicitly constructed with automorphisms of an associative algebra as its objects.
• The category is equipped with a curved $A_{\infty}$-structure, allowing for non-trivial higher homotopies and deformations.
• The construction generalizes the idea of Fukaya categories to purely algebraic settings, without requiring a symplectic or geometric background.
• The category captures algebraic symmetries and deformation data through higher operations derived from Hochschild cochains.
• The framework provides a homotopical interpretation of automorphisms as categorical objects, enriching their algebraic structure.
• The paper establishes a conceptual bridge between algebraic $A_{\infty}$-categories and geometric categories like those in Floer theory.