A simplicial construction for noncommutative settings

Abstract

In this paper we present a general construction that can be used to define the higher order Hochschild homology for a noncommutative algebra. We also discuss other examples where this construction can be used.

Secondary Hochschild cohomology and derivations

“…Triples have now been studied quite broadly, and examples, extensions, and applications can be found in a number of places (see [1], [2], [3], [4], [5], [8], [9], [13], or [18], for example). When convenient and appropriate, we will denote a triple by T = (A, B, ε), so as to make it easier with notation.…”

Simplicial structures over the 3-sphere and generalized higher order Hochschild homology

“…We omit it here, but the construction combines a pre-simplicial left module X and a pre-cosimplicial left module Y (both over a pre-simplicial k-algebra A) to generate a precosimplicial k-module, which we denote Hom A (X , Y). Using simplicial structures, one can then define the secondary Hochschild (co)homologies, which are studied in [4], [9], [16], [17], [23], and [24].…”

“…Roughly speaking, each d * i map represents a different way to take an upper tetrahedral matrix in an n × n × n integer lattice and collapse it onto an upper tetrahedral matrix that fits into an (n−1)×(n−1)×(n−1) integer lattice. We first work through a low dimensional example: consider + consists of the elements (2, 3, 4), (1,3,4), (1,2,4), and (1, 2, 3). Then the maps d * i are as follows:…”

“…Higher order Hochschild (co)homology has been commonly used to study deformations of algebras and modules (see [2], [5], [10], [11], or [12]), and has recent applications to string topology and topological chiral homology (see [6] and [14], respectively). In the correct setting, the definition was extended to accommodate multi-module coefficients in [7], and generalized to include a not necessarily commutative algebra in both [4] and [8]. Higher order Hochschild (co)homology was even recently generalized over a pair of simplicial sets in [9].…”

“…Figure 4: Upper tetrahedral matrix for M ⊗ A 4 (n = 4) n = 4. Recall x 4+ consists of the elements (2, 3, 4),(1,3,4),(1,2,4), and (1, 2, 3). Then the maps d * i are as follows:…”