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2015 $A_∞$-Algebras Derived from Associative Algebras with a Non-Derivation Differential
Kaj Borjeson
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J. Gen. Lie Theory Appl. 9(1): 1-5 (2015). DOI: 10.4172/1736-4337.1000214


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.


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Kaj Borjeson. "$A_∞$-Algebras Derived from Associative Algebras with a Non-Derivation Differential." J. Gen. Lie Theory Appl. 9 (1) 1 - 5, 2015.


Published: 2015
First available in Project Euclid: 30 September 2015

zbMATH: 1357.16022
MathSciNet: MR3624036
Digital Object Identifier: 10.4172/1736-4337.1000214

Keywords: $A_∞$-Algebras , BV-algebra , coassociative coalgebra , Hochschild cocomplex

Rights: Copyright © 2015 Ashdin Publishing (2009-2013) / OMICS International (2014-2016)

Vol.9 • No. 1 • 2015
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