Tensor products of algebras with homotopy inner products
Authors:
Thomas Tradler and Ronald Umble
Journal:
Trans. Amer. Math. Soc. 365 (2013), 51535198
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 algebras is, in general, not a cyclic algebra, but an 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 algebra with homotopy inner products, and use it to define a categorically closed tensor product. A cyclic algebra can be thought of as an algebra with homotopy inner products whose higher inner products are trivial. However, the higher inner products on the tensor product of cyclic 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/S000299472013058035
PII:
S 00029947(2013)058035
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 PSCCUNY grant PSCREG41316.
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.
