Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

A Step towards Computational Derived Algebraic Geometry: The RepHomology Package For Macaulay2

Guanyu Li

Abstract

We introduce the \verb|Macaulay2| package \verb|RepHomology| for the computations of representation homology of certain spaces. The main methods implement computing the representation homology of surfaces (with group coefficients, and analogies with algebra and Lie algebra coefficients), and the representation homology of link complements.

A Step towards Computational Derived Algebraic Geometry: The RepHomology Package For Macaulay2

Abstract

We introduce the \verb|Macaulay2| package \verb|RepHomology| for the computations of representation homology of certain spaces. The main methods implement computing the representation homology of surfaces (with group coefficients, and analogies with algebra and Lie algebra coefficients), and the representation homology of link complements.

Paper Structure

This paper contains 15 sections, 4 theorems, 37 equations.

Table of Contents

  1. Introduction
  2. Acknowledgements
  3. The complexes to compute representation homology
  4. Representation Homology of Surfaces
  5. Representations in groups
  6. Representations in algebras
  7. Representations in Lie algebras
  8. Representation Homology of Links
  9. The type Link
  10. Representation Homology of Link Complements
  11. Representation homology
  12. Notations and conventions
  13. Simplicial methods
  14. Representation homology
  15. Other types of representation homology

Key Result

Proposition 2.2

Let $\Sigma_g$ be the compact orientable surface with genus $g$ and let $G$ be an affine matrix group scheme over $k$ with generic matrix $X$, whose coordinate ring is denoted by $k[X]$. Then the DG algebra has homology groups isomorphic to $HR_*(\Sigma_g,G)$, where the matrix equation means termwise equal for two matrices.

Theorems & Definitions (11)

  • Example 2.1
  • Proposition 2.2: See also li2024commuting*Lemma 3.3
  • Proposition 2.3: See also BRY22*Theorem 6.1
  • Remark 1
  • Example 3.1
  • Example 3.2
  • Example 3.3
  • Example 4.1
  • Lemma A.1: BRY22*Lemma 3.1
  • Definition A.1
  • ...and 1 more