Tensor products of -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

DOI:
https://doi.org/10.1090/S0002-9947-2013-05803-5

Published electronically:
May 22, 2013

MathSciNet review:
3074370

Full-text PDF

Abstract | References | Similar Articles | Additional Information

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.

**[BV]**J.M. Boardman, R.M. Vogt,*Homotopy invariant algebraic structures on topological spaces*, Springer LNM 347, 1973 MR**0420609 (54:8623a)****[C]**C.-H. Cho,*Strong homotopy inner product of an A-infinity algebra*, Int. Math. Res. Not., no. 41, 2008**[L]**J.-L. Loday,*The diagonal of the Stasheff polytope*, preprint arXiv:0710.0572v2 MR**2762549 (2011m:18017)****[LT]**R. Longoni, T. Tradler,*Homotopy Inner Products for Cyclic Operads*, Journal of Homotopy and Related Structures, vol. 3(1), pp. 343-358, 2008 MR**2481462 (2010d:55013)****[MacL]**S. Mac Lane,*Categories for the Working Mathematician*, Springer, Graduate Texts in Mathematics 5, 1971 MR**0354798 (50:7275)****[MS]**M. Markl, S. Shnider,*Associahedra, cellular W-construction and products of algebras*, Trans. of the AMS, vol. 358, no. 6, pp. 2353-2372, 2005 MR**2204035 (2006j:18004)****[SU]**S. Saneblidze, R. Umble,*Diagonals on the Permutahedra, Multiplihedra and Associahedra*, Homology, Homotopy and Applications, vol. 6(1), pp. 363-411, 2004 MR**2118493 (2005j:55023)****[S]**J. Stasheff,*Homotopy Associativity of H-Spaces. I*, Trans. AMS, vol. 108, no. 2, pp. 275-292, 1963 MR**0158400 (28:1623)****[Ta]**D. Tamari,*The algebra of bracketings and their enumeration*, Nieuw Arch. Wisk. (3) 10, 1962, pp. 131-146. MR**0146227 (26:3749)****[TT]**J. Terilla, T. Tradler,*Deformations of associative algebras with inner products*, Homology, Homotopy, and Applications 8(2), pp. 115-131, 2006 MR**2246025 (2007d:16069)****[T1]**T. Tradler,*Infinity Inner Products on A-Infinity Algebras*, Journal of Homotopy and Related Structures, vol. 3(1), pp. 245-271, 2008 MR**2426181 (2010g:16016)****[T2]**T. Tradler,*The BV Algebra on Hochschild Cohomology Induced by Infinity Inner Products*, Annales de L'institut Fourier, vol. 58, no. 7, pp. 2351-2379, 2008 MR**2498354 (2010a:16020)****[TZ]**T. Tradler, M. Zeinalian,*Algebraic String Operations*, K-Theory, vol. 38, no. 1, 2007 MR**2353864 (2010d:57032)**

Retrieve articles in *Transactions of the American Mathematical Society*
with MSC (2010):
55S15,
52B05,
18D50,
55U99

Retrieve articles in all journals with MSC (2010): 55S15, 52B05, 18D50, 55U99

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:
https://doi.org/10.1090/S0002-9947-2013-05803-5

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.