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Tensor $2$-Product for $\mathfrak{sl}_{2}$: Extensions to the Negative Half

Matthew McMillan

Abstract

In a recent paper, the author defined an operation of tensor product for a large class of $2$-representations of $\mathcal{U}^{+}$, the positive half of the $2$-category associated to $\mathfrak{sl}_{2}$. In this paper, we prove that the operation extends to give an operation of tensor product for $2$-representations of the full $2$-category $\mathcal{U}$: when the inputs are $2$-representations of the full $\mathcal{U}$, the $2$-product is also a $2$-representation of the full $\mathcal{U}$. As in the previous paper, the $2$-product is given for a simple $2$-representation $\mathcal{L}(1)$ and an abelian $2$-representation $\mathcal{V}$ taken from the $2$-category of algebras. This is the first construction of an operation of tensor product for higher representations of a full Lie algebra in the abelian setting.

Tensor $2$-Product for $\mathfrak{sl}_{2}$: Extensions to the Negative Half

Abstract

In a recent paper, the author defined an operation of tensor product for a large class of -representations of , the positive half of the -category associated to . In this paper, we prove that the operation extends to give an operation of tensor product for -representations of the full -category : when the inputs are -representations of the full , the -product is also a -representation of the full . As in the previous paper, the -product is given for a simple -representation and an abelian -representation taken from the -category of algebras. This is the first construction of an operation of tensor product for higher representations of a full Lie algebra in the abelian setting.
Paper Structure (23 sections, 17 theorems, 62 equations)

This paper contains 23 sections, 17 theorems, 62 equations.

Table of Contents

  1. Introduction
  2. Background and motivation
  3. Result
  4. Outline summary
  5. Acknowledgments
  6. Background: extending $\mathcal{U}^{+}$ actions to $\mathcal{U}$ actions
  7. $2$-Representations of $\mathcal{U}$
  8. Commutator morphisms
  9. Conventions
  10. Adding a dual
  11. Integrability
  12. Background: $2$-product for $\mathcal{U}^{+}$
  13. More bimodules
  14. Adjunction
  15. Isomorphisms $\tilde{\rho}_{\lambda}$
  16. ...and 8 more sections

Key Result

Theorem 1

Suppose $(A,E,x,\tau)$ gives the data of a $2$-representation $\mathcal{V}$ of $\mathcal{U}^{+}$ such that $\mathcal{V}$ has a weight decomposition. Define the left-dual $(A,A)$-bimodule $F=\mathrm{Hom}_{A}(_{A}E,A)$. Suppose $E$ has the following properties: These properties imply that $(A,E,F,x,\tau,\eta,\varepsilon)$ determines an integrable $2$-representation of $\mathcal{U}$. Now let $C$ be

Theorems & Definitions (39)

  • Theorem : Main result
  • Proposition 2.1
  • Lemma 2.2
  • Definition 2.3: Def. 3.1 of mcmillanTensor2product2representations2022
  • Definition 2.4: Def. 3.2 of mcmillanTensor2product2representations2022
  • Lemma 2.5: Lemma 3.3 of mcmillanTensor2product2representations2022
  • Definition 2.6: Def. 3.5 of mcmillanTensor2product2representations2022
  • Proposition 2.7: Prop. 3.6 of mcmillanTensor2product2representations2022
  • Definition 2.8: Def. 3.16 of mcmillanTensor2product2representations2022
  • Proposition 2.9: Props. 3.18, 3.20, and 3.22 together with 3.27 and 3.28 of mcmillanTensor2product2representations2022
  • ...and 29 more