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Tannaka Reconstruction and the Monoid of Matrices

John C. Baez, Todd Trimble

TL;DR

This work proves that the monoid $M(n,k)$ of $n\times n$ matrices over a field $k$ of characteristic zero is the walking monoid with an $n$-dimensional representation, and that the $2$-rig of algebraic representations $ extsf{Rep}(M(n,k))$ is the free $2$-rig on an object of subdimension $n$. The authors develop a framework combining quotient $2$-rigs and $Tannaka$ reconstruction to identify $ extsf{Rep}(M(n,k))$ with the category of finite-dimensional comodules of the coordinate bialgebra $ ext{O}(M(n,k))$, showing $ extsf{Rep}(M(n,k)) \\cong extsf{Comod}( ext{O}(M(n,k)))$. A key step is the quotient of the free $2$-rig on one generator by the $2$-ideal generated by $\Lambda^{n+1}$, yielding the free $2$-rig on an object of subdimension $n$, which is then shown to be equivalent to $ extsf{Rep}(M(n,k))$ via the $Tannaka$ reconstruction. The paper also conjectures universal properties for the representation $2$-rigs of classical groups (e.g., $ ext{GL}$, $ ext{SL}$, $ ext{Sp}$, $ ext{O}$, $ ext{SO}$) and discusses extensions to other characteristics and to exceptional groups. Overall, the work provides a precise categorified universal property characterization of matrix monoid representations and lays groundwork for a broader theory of representation $2$-rigs for classical groups.

Abstract

Settling a conjecture from an earlier paper, we prove that the monoid $\mathrm{M}(n,k)$ of $n \times n$ matrices in a field $k$ of characteristic zero is the "walking monoid with an $n$-dimensional representation". More precisely, if we treat $\mathrm{M}(n,k)$ as a monoid in affine schemes, the 2-rig $\mathrm{Rep}(\mathrm{M}(n,k))$ of algebraic representations of $\mathrm{M}(n,k)$ is the free 2-rig on an object $x$ with $Λ^{n+1}(x) \cong 0$. Here a "2-rig" is a symmetric monoidal $k$-linear category that is Cauchy complete. Our proof uses Tannaka reconstruction and a general theory of quotient 2-rigs and 2-ideals. We conclude with a series of conjectures about the universal properties of representation 2-rigs of classical groups.

Tannaka Reconstruction and the Monoid of Matrices

TL;DR

This work proves that the monoid of matrices over a field of characteristic zero is the walking monoid with an -dimensional representation, and that the -rig of algebraic representations is the free -rig on an object of subdimension . The authors develop a framework combining quotient -rigs and reconstruction to identify with the category of finite-dimensional comodules of the coordinate bialgebra , showing . A key step is the quotient of the free -rig on one generator by the -ideal generated by , yielding the free -rig on an object of subdimension , which is then shown to be equivalent to via the reconstruction. The paper also conjectures universal properties for the representation -rigs of classical groups (e.g., , , , , ) and discusses extensions to other characteristics and to exceptional groups. Overall, the work provides a precise categorified universal property characterization of matrix monoid representations and lays groundwork for a broader theory of representation -rigs for classical groups.

Abstract

Settling a conjecture from an earlier paper, we prove that the monoid of matrices in a field of characteristic zero is the "walking monoid with an -dimensional representation". More precisely, if we treat as a monoid in affine schemes, the 2-rig of algebraic representations of is the free 2-rig on an object with . Here a "2-rig" is a symmetric monoidal -linear category that is Cauchy complete. Our proof uses Tannaka reconstruction and a general theory of quotient 2-rigs and 2-ideals. We conclude with a series of conjectures about the universal properties of representation 2-rigs of classical groups.

Paper Structure

This paper contains 7 sections, 21 theorems, 71 equations.

Table of Contents

  1. Introduction
  2. Representations of affine monoids
  3. Tannaka reconstruction
  4. Quotient 2-rigs
  5. Quotients of the free 2-rig on one generator
  6. Proof of the main theorem
  7. Conclusions

Key Result

Theorem 1

If $k$ is a field of characteristic zero, the 2-rig $\mathsf{Rep}(\mathrm{M}(n,k))$ is the free 2-rig on an object of subdimension $n$. That is, given any 2-rig $\mathsf{R}$ containing an object $r$ of subdimension $n$, there is a map of 2-rigs $F \colon \mathsf{Rep}(\mathrm{M}(n,k)) \to \mathsf{R}$

Theorems & Definitions (48)

  • Theorem 1
  • Definition 2
  • Definition 3
  • Definition 4
  • Definition 5
  • Definition 6
  • Lemma 7
  • proof
  • Definition 8
  • Lemma 9
  • ...and 38 more