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Extending work of Saneblidze-Umble and others, we use diagonals for the associahedron and multiplihedron to define tensor products of A-infinity algebras, modules, algebra homomorphisms, and module morphisms, as well as to define a bimodule analogue of twisted complexes (type DD structures, in the language of bordered Heegaard Floer homology) and their one- and two-sided tensor products. We then give analogous definitions for 1-parameter deformations of A-infinity algebras; this involves another collection of complexes. These constructions are relevant to bordered Heegaard Floer homology.
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The Hochschild cohomology of a tensor product of algebras is isomorphic to a graded tensor product of Hochschild cohomology algebras, as a Gerstenhaber algebra. A similar result holds when the tensor product is twisted by a bicharacter. We present new proofs of these isomorphisms, using Volkovs homotopy liftings that were introduced for handling Gerstenhaber brackets expressed on arbitrary bimodule resolutions. Our results illustrate the utility of homotopy liftings for theoretical purposes.
We find three families of twisting maps of K^m with K^n. One of them is related to truncated quiver algebras, the second one consists of deformations of the first and the third one requires m=n and yields algebras isomorphic to M_n(K). Using these families and some exceptional cases we construct all twisting maps of K^3 with K^3.
The zeta function of an integral lattice $Lambda$ is the generating function $zeta_{Lambda}(s) = sumlimits_{n=0}^{infty} a_n n^{-s}$, whose coefficients count the number of left ideals of $Lambda$ of index $n$. We derive a formula for the zeta function of $Lambda_1 otimes Lambda_2$, where $Lambda_1$ and $Lambda_2$ are $mathbb{Z}$-orders contained in finite-dimensional semisimple $mathbb{Q}$-algebras that satisfy a locally coprime condition. We apply the formula obtained above to $mathbb{Z}S otimes mathbb{Z}T$ and obtain the zeta function of the adjacency algebra of the direct product of two finite association schemes $(X,S)$ and $(Y,T)$ in several cases where the $mathbb{Z}$-orders $mathbb{Z}S$ and $mathbb{Z}T$ are locally coprime and their zeta functions are known.
We construct four families of Artin-Schelter regular algebras of global dimension four. Under some generic conditions, this is a complete list of Artin-Schelter regular algebras of global dimension four that are generated by two elements of degree 1. These algebras are also strongly noetherian, Auslander regular and Cohen-Macaulay. One of the main tools is Kellers higher-multiplication theorem on A-infinity Ext-algebras.
We show that Kraus property $S_{sigma}$ is preserved under taking weak* closed sums with masa-bimodules of finite width, and establish an intersection formula for weak* closed spans of tensor products, one of whose terms is a masa-bimodule of finite width. We initiate the study of the question of when operator synthesis is preserved under the formation of products and prove that the union of finitely many sets of the form $kappa times lambda$, where $kappa$ is a set of finite width, while $lambda$ is operator synthetic, is, under a necessary restriction on the sets $lambda$, again operator synthetic. We show that property $S_{sigma}$ is preserved under spatial Morita subordinance. En route, we prove that non-atomic ternary masa-bimodules possess property $S_{sigma}$ hereditarily.
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