## Journal of Generalized Lie Theory and Applications

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

Kaj Borjeson

#### Abstract

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

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

Dates
First available in Project Euclid: 30 September 2015

https://projecteuclid.org/euclid.jglta/1443617954

Digital Object Identifier
doi:10.4172/1736-4337.1000214

Mathematical Reviews number (MathSciNet)
MR3624036

Zentralblatt MATH identifier
1357.16022

#### Citation

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. https://projecteuclid.org/euclid.jglta/1443617954