[Paper Review] Differential modules with $\infty$-simplicial faces and $A_\infty$-algebras

С. В. Лапин|arXiv (Cornell University)|Sep 6, 2018

Homotopy and Cohomology in Algebraic Topology6 references4 citations

TL;DR

This paper introduces differential modules with ∞-simplicial faces using colored Koszul duality, establishing their homotopy invariance and linking them to Aₙ-algebras. The key result is that the B-construction of any Aₙ-algebra is isomorphic as a differential coalgebra to the chain realization of its tensor differential module with ∞-simplicial faces, unifying homotopy-theoretic and algebraic structures via chain-level models.

ABSTRACT

In the present paper, by using the colored version of the Koszul duality, the concept of a differential module with $\infty$-simplicial faces is introduced. The homotopy invariance of the structure of a differential module with $\infty$-simplicial faces is proved. The relationships between differential modules with $\infty$-simplicial faces and $A_\infty$-algebras are established. The notion of a chain realization of a differential module with $\infty$-simplicial faces and the concept of a tensor product of differential modules with $\infty$-simplicial faces are introduced. It is proved that for an arbitrary $A_\infty$-algebra the chain realization of the tensor differential module with $\infty$-simplicial faces, which corresponds to this $A_\infty$-algebra, and the $B$-construction of this $A_\infty$-algebra are isomorphic differential coalgebras.

Motivation & Objective

• To generalize Koszul duality to colored quadratic algebras and coalgebras.
• To define and study differential modules with ∞-simplicial faces as differential Lie modules over colored coalgebras.
• To establish a chain-level isomorphism between the B-construction and the chain realization of tensor differential modules with ∞-simplicial faces.
• To connect Aₙ-algebras to Eₙ-coalgebra structures and Steenrod operations via B-constructions.
• To provide a homotopy-invariant framework for Aₙ-algebras using ∞-simplicial face structures.

Proposed method

• Uses the colored Koszul duality theory to define the co-∞-simplicial coalgebra dual to the quadratic colored algebra of simplicial faces.
• Introduces differential modules with ∞-simplicial faces as differential Lie modules over the dual colored coalgebra (F^!, ∇), or equivalently as modules over the co-B-construction.
• Defines the chain realization of a differential module with ∞-simplicial faces as a differential coalgebra via a canonical isomorphism to the B-construction.
• Constructs the tensor product of differential modules with ∞-simplicial faces using the colored tensor product of graded modules.
• Applies the standard B-construction to Aₙ-algebras and proves its isomorphism to the chain realization of the corresponding ∞-simplicial module.
• Uses sign conventions involving degrees and Koszul signs to ensure compatibility of differentials and comultiplications.

Experimental results

Research questions

• RQ1 How can the concept of ∞-simplicial faces be generalized to colored algebras and coalgebras using Koszul duality?
• RQ2 What is the homotopy invariance structure of differential modules with ∞-simplicial faces, and how does it relate to homotopy simplicial faces?
• RQ3 Is the B-construction of an Aₙ-algebra isomorphic to the chain realization of its associated tensor differential module with ∞-simplicial faces?
• RQ4 Can the B-construction of an Aₙ-algebra be interpreted as a chain-level model of a differential module with ∞-simplicial faces?
• RQ5 What is the role of the ∞-simplicial face structure in realizing Steenrod operations on cohomology via Eₙ-coalgebra structures?

Key findings

• The chain realization of the tensor differential module with ∞-simplicial faces associated to any Aₙ-algebra is isomorphic as a differential coalgebra to the B-construction of that Aₙ-algebra.
• The B-construction of an Aₙ-algebra inherits an Eₙ-coalgebra structure via the chain realization, implying the existence of Steenrod operations on its cohomology satisfying Adem relations and the Cartan formula.
• The differential coalgebra structure of the B-construction arises naturally as the chain realization of the ∞-simplicial face module, establishing a canonical isomorphism.
• The homotopy invariance of the ∞-simplicial face structure is proven, ensuring stability under homotopy equivalences of the underlying Aₙ-algebra.
• For differential associative algebras (i.e., A₁-algebras), the B-construction coincides with the classical B-construction as defined in earlier works, validating consistency.
• The sign conventions in the comultiplication formula are consistent with standard Koszul sign rules, ensuring compatibility with the differential and graded module structures.