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Operations on the Hochschild Bicomplex of a Diagram of Algebras

Eli Hawkins

Abstract

A diagram of algebras is a functor valued in a category of associative algebras. I construct an operad acting on the Hochschild bicomplex of a diagram of algebras. Using this operad, I give a direct proof that the Hochschild cohomology of a diagram of algebras is a Gerstenhaber algebra. I also show that the total complex is an $L_\infty$-algebra. The same results are true for the reduced and asimplicial subcomplexes and asimplicial cohomology. This structure governs deformations of diagrams of algebras through the Maurer-Cartan equation.

Operations on the Hochschild Bicomplex of a Diagram of Algebras

Abstract

A diagram of algebras is a functor valued in a category of associative algebras. I construct an operad acting on the Hochschild bicomplex of a diagram of algebras. Using this operad, I give a direct proof that the Hochschild cohomology of a diagram of algebras is a Gerstenhaber algebra. I also show that the total complex is an -algebra. The same results are true for the reduced and asimplicial subcomplexes and asimplicial cohomology. This structure governs deformations of diagrams of algebras through the Maurer-Cartan equation.

Paper Structure

This paper contains 40 sections, 38 theorems, 246 equations, 5 figures.

Table of Contents

  1. Introduction
  2. Synopsis
  3. Some Combinatorial Operads
  4. Extensions
  5. The $\mathop{\mathrm{\textit{Brace}}}\nolimits$ operad
  6. The Gerstenhaber-Voronov operad
  7. Words
  8. The allowed words
  9. Boundary
  10. Composition
  11. Computing extensions
  12. The $\mathop{\mathrm{\textit{Quilt}}}\nolimits$ Operad
  13. Quilts
  14. Diagrams
  15. Examples
  16. ...and 25 more sections

Key Result

Lemma 2.13

If $W\in\mathop{\mathrm{\mathsf{F_2S}}}\nolimits(n)$, and $s$ is the number of last-first pairs in $W$, then $\deg W = n-s-1$. That is, $\lvert W\rvert = 2n-s-1$. If $W$ can be written as a concatenation of $m$ words with disjoint alphabets then $\deg W \leq n-m$.

Figures (5)

  • Figure 1: Construction of $Q^{(1)}_u$.
  • Figure 2: Construction of $Q^{(2)}_{\underline u}$.
  • Figure 3: Construction of $Q^{(3)}_u$.
  • Figure 4: Construction of $Q^{(4)}_u$.
  • Figure 5: Construction of $Q^{(5)}_{u,v,w}$.

Theorems & Definitions (190)

  • Remark
  • Remark
  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Definition 2.4
  • Remark
  • Remark
  • Definition 2.5
  • Definition 2.6
  • ...and 180 more