A∞-Algebras Derived from Associative Algebras with a Non-Derivation Differential

Abstract

Given an associative graded algebra equipped with a degree +1 differential ∆ we define an A ∞ -structure that measures the failure of ∆ 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 operator ∆ 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 ∞structure on the bar complex of an A ∞ -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 ∞ -structure is constructed on the Hochschild cocomplex of an associative algebra with a non-degenerate inner product by using Connes' B-operator.

On the origin of higher braces and higher-order derivations

“…[September 30, 2013] It will follow from Theorem 4.11 that assumptions (3a)-(3c) already imply that l ∆ k = Φ ∆ k for each k ≥ 1. Let us start our discussion of the non-commutative case by recalling one construction from a recent preprint [6] of Börjeson.…”

Higher Braces Via Formal (Non)Commutative Geometry

“…The noncommutative analog of the Koszul hierarchy was found in April 2013 by Kay Börjeson [5] who also proved that the result is an A ∞ -algebra; we recall his braces {b ∇ n } n≥1 in Subsection 2.2. Amazingly, Börjeson's braces are very different from the Koszul ones.…”

“…For a graded associative, not necessarily commutative, algebra A and a degree +1 differential ∇ : A → A, he defined in [5] linear degree +1 operators b…”

Uniqueness and intrinsic properties of non-commutative Koszul brackets

“…Gauge fixing variation and Börjeson's brackets. Changing the gauge fixing condition in Theorem 5.1 one can obtain different hierarchies of higher brackets: for instance, setting Φ Id = µ 0 we get Φ n = 0 for every n ≥ 2, while setting Φ Id = µ 0 − µ 1 , i.e., K 1 = 1 and K n = 0 for every n > 1, it is easy to see that the resulting higher brackets are the graded symmetrizations of the Börjeson's brackets [8,18].…”