A Algebras Derived from Associative Algebras with a Non-Derivation Differential
Department of Mathematics, Stockholm University, 10691 Stockholm, Sweden
- *Corresponding Author:
- Borjeson K
Department of Mathematics
Stockholm University, 10691 Stockholm, Sweden
Tel: 08-16 4531
E-mail: [email protected]
Received date: November 09, 2014; Accepted date: February 25, 2015; Published date: March 28, 2015
Citation: Borjeson K (2015) A∞-Algebras Derived from Associative Algebras with a Non-Derivation Differential. J Generalized Lie Theory Appl 9: 214. doi:10.4172/1736-4337.1000214
Copyright: © 2015 Borjeson K. This is an open-access article distributed under the terms of the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited.
Given an associative graded algebra equipped with a degree +1 differential delta we define an A∞-structure that measures the failure of delta 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 delta 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∞-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.