[Paper Review] A-infinity Algebras Derived from Associative Algebras with a Non-Derivation Differential

Kaj Börjeson|arXiv (Cornell University)|Apr 23, 2013

Advanced Topics in Algebra3 references3 citations

TL;DR

This paper introduces an $A_\infty$-structure on a graded associative algebra equipped with a degree $+1$ differential $\Delta$ that is not necessarily a derivation, measuring the failure of $\Delta$ to commute with the product. The key contribution is a non-commutative analog of BV-algebras, where $\Delta$ of associative order 2 yields a strictly associative degree $+1$ product compatible with the original product, generalizing constructions in Hochschild cohomology and bar resolutions.

ABSTRACT

Given an associative graded algebra equipped with a degree +1 differential we define an A-infinity structure that measures the failure of the differential to be a derivation. This can be seen as a non-commutative analog of generalized BV-algebras. In that spirit we introduce a notion of associative order for the differential and prove that it satisfies properties similar to the commutative case. In particular when it has associative order 2 the new product is a strictly associative product of degree +1 and there is a compatibility between the products, similar to ordinary BV-algebras. We consider several examples of structures obtained in this way. In particular we obtain an A-infinity structure on the bar complex of an A-infinity algebra that is strictly associative if the original algebra is strictly associative. We also introduce strictly associative degree +1 products for any degree +1 action on a graded algebra. Moreover, an A-infinity structure is constructed on the Hochschild cocomplex of an associative algebra with a non-degenerate inner product by using Connes' B-operator.

Motivation & Objective

• To define an $A_\infty$-structure on a graded associative algebra with a degree $+1$ differential $\Delta$ that is not required to be a derivation.
• To introduce the notion of associative order for $\Delta$, generalizing the concept of order in generalized BV-algebras.
• To establish compatibility between the original product and a new degree $+1$ product when $\Delta$ has associative order 2.
• To apply the construction to specific cases, including the bar complex of an $A_\infty$-algebra and the Hochschild cocomplex with Connes' $B$-operator.
• To show that the resulting $A_\infty$-structure on Hochschild cohomology induces the Gerstenhaber bracket as $m_2$, up to sign.

Proposed method

• Define $A_\infty$-operations $m_n$ using the differential $\Delta$ and the associative product, where $m_n$ measures the failure of $\Delta$ to be a derivation.
• Use the Koszul sign rule in the symmetric monoidal category of complexes to handle graded signs in the operations.
• Construct the $A_\infty$-structure on the bar complex of an $A_\infty$-algebra, showing it becomes strictly associative if the original algebra is strictly associative.
• Apply the construction to the Hochschild cocomplex of a finite-dimensional associative algebra with a non-degenerate invariant inner product.
• Use the inner product to dualize Connes' $B$-operator to a degree $+1$ differential on the cocomplex, then apply the general construction.
• Prove that the resulting $A_\infty$-structure on cohomology has $m_1 = \Delta$ and $m_2$ equal to the Gerstenhaber bracket up to sign.

Experimental results

Research questions

• RQ1 How can one define an $A_\infty$-structure on a graded associative algebra equipped with a degree $+1$ differential $\Delta$ that is not a derivation?
• RQ2 What is the appropriate generalization of the notion of order for $\Delta$ in the non-commutative setting, and how does it relate to the vanishing of higher $A_\infty$-operations?
• RQ3 What compatibility conditions arise between the original associative product and the new degree $+1$ product when $\Delta$ has associative order 2?
• RQ4 Can this construction be applied to the bar complex of an $A_\infty$-algebra, and what structure does it induce?
• RQ5 Does the construction yield the Gerstenhaber bracket on Hochschild cohomology when applied to the cocomplex with Connes' $B$-operator?

Key findings

• The $A_\infty$-operations $m_n$ are defined as a generalization of the Koszul hierarchy to non-commutative algebras, measuring the failure of $\Delta$ to be a derivation.
• When $\Delta$ has associative order 2, the operation $m_2$ becomes a strictly associative product of degree $+1$, and there is a compatibility between the original and new products.
• The construction yields a strictly associative degree $+1$ product on the bar complex of an $A_\infty$-algebra, which is strict if the original algebra is strictly associative.
• On the Hochschild cocomplex with a non-degenerate invariant inner product, the $A_\infty$-structure induced by Connes' $B$-operator has $m_2$ equal to the Gerstenhaber bracket up to a sign.
• The $A_\infty$-structure on the cohomology is well-defined and induces the standard Gerstenhaber bracket, confirming consistency with known structures in deformation theory.
• The construction is valid in the homologically graded setting, and the resulting $A_\infty$-structure on $C^\bullet(A,A)[-1]$ is compatible with the Hochschild coboundary.