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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

Tensor products of $ A_\infty$-algebras with homotopy inner products


Authors: Thomas Tradler and Ronald Umble
Journal: Trans. Amer. Math. Soc. 365 (2013), 5153-5198
MSC (2010): Primary 55S15, 52B05, 18D50, 55U99
Published electronically: May 22, 2013
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Abstract: We show that the tensor product of two cyclic $ A_\infty $-algebras is, in general, not a cyclic $ A_\infty $-algebra, but an $ A_\infty $-algebra with homotopy inner product. More precisely, we construct an explicit combinatorial diagonal on the pairahedra, which are contractible polytopes controlling the combinatorial structure of an $ A_\infty $-algebra with homotopy inner products, and use it to define a categorically closed tensor product. A cyclic $ A_\infty $-algebra can be thought of as an $ A_\infty $-algebra with homotopy inner products whose higher inner products are trivial. However, the higher inner products on the tensor product of cyclic $ A_\infty $-algebras are not necessarily trivial.


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Additional Information

Thomas Tradler
Affiliation: Department of Mathematics, College of Technology, City University of New York, 300 Jay Street, Brooklyn, New York 11201
Email: ttradler@citytech.cuny.edu

Ronald Umble
Affiliation: Department of Mathematics, Millersville University of Pennsylvania, Millersville, Pennsylvania 17551
Email: ron.umble@millersville.edu

DOI: http://dx.doi.org/10.1090/S0002-9947-2013-05803-5
PII: S 0002-9947(2013)05803-5
Keywords: $A_\infty$-algebra with homotopy inner product, colored operad, cyclic $A_\infty$-algebra, diagonal, pairahedron, tensor product, $W$-construction
Received by editor(s): August 26, 2011
Received by editor(s) in revised form: January 20, 2012
Published electronically: May 22, 2013
Additional Notes: The research of the first author was funded in part by the PSC-CUNY grant PSCREG-41-316.
The research of the second author was funded in part by a Millersville University faculty research grant.
Article copyright: © Copyright 2013 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.