Journal of Generalized Lie Theory and Applications

$A_∞$-Algebras Derived from Associative Algebras with a Non-Derivation Differential

Kaj Borjeson

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Given an associative graded algebra equipped with a degree $+1$ differential Δ we define an $A_\infty$-structure that measures the failure of Δ to be a derivation. This can be seen as a non-commutative analog of generalized BValgebras. In that spirit we introduce a notion of associative order for the operator Δ 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 compatibility between the products, similar to ordinary BV-algebras. We consider several examples of structures obtained in this way. In particular we obtain an $A_\infty$-structure on the bar complex of an $A_\infty$-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 $A_\infty$-structure is constructed on the Hochschild cocomplex of an associative algebra with a non-degenerate inner product by using Connes’ B-operator.

Article information

J. Gen. Lie Theory Appl., Volume 9, Number 1 (2015), 5 pages.

First available in Project Euclid: 30 September 2015

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Zentralblatt MATH identifier

$A_∞$-Algebras Hochschild cocomplex BV-algebra coassociative coalgebra


Borjeson, Kaj. $A_∞$-Algebras Derived from Associative Algebras with a Non-Derivation Differential. J. Gen. Lie Theory Appl. 9 (2015), no. 1, 5 pages. doi:10.4172/1736-4337.1000214.

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