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Classification of simple commutative algebras in the Delannoy category

Pavel Etingof, Andrew Snowden

Abstract

The Delannoy category is an interesting pre-Tannakian category associated to the oligomorphic group $\mathbb{G}$ of automorphisms of the totally ordered set $(\mathbf{R}, <)$. By construction, it admits some obvious simple commutative algebras, corresponding to certain transitive $\mathbb{G}$-sets. We show that these account for all of the simple commutative algebras in the Delannoy category. Previous results of this kind have been limited to interpolation categories; since the Delannoy category cannot be obtained by interpolation, new methods are required.

Classification of simple commutative algebras in the Delannoy category

Abstract

The Delannoy category is an interesting pre-Tannakian category associated to the oligomorphic group of automorphisms of the totally ordered set . By construction, it admits some obvious simple commutative algebras, corresponding to certain transitive -sets. We show that these account for all of the simple commutative algebras in the Delannoy category. Previous results of this kind have been limited to interpolation categories; since the Delannoy category cannot be obtained by interpolation, new methods are required.

Paper Structure

This paper contains 33 sections, 38 theorems, 44 equations.

Table of Contents

  1. Introduction
  2. Statement of results
  3. Sketch of proof
  4. Tensor category terminology
  5. Notation
  6. Algebras in tensor categories
  7. Algebras
  8. Frobenius algebras
  9. Étale algebras
  10. Subétale algebras
  11. The Delannoy category
  12. Oligomorphic tensor categories
  13. The Delannoy group
  14. The Delannoy category
  15. Schwartz spaces
  16. ...and 18 more sections

Key Result

Theorem A

Any simple commutative algebra in $\mathop{\mathrm{\text{ \uline{\space} \contour{white}{\rm Rep}}}}\nolimits(\mathbb{G})$ is isomorphic to some $\mathcal{C}(\mathbf{R}^{(n)})$.

Theorems & Definitions (87)

  • Theorem A
  • Theorem B
  • Theorem C
  • Remark 1.1
  • Remark 1.2
  • Proposition 2.1
  • proof
  • Corollary 2.2
  • proof
  • Remark 2.3
  • ...and 77 more