A tensor 2-product of 2-representations of $\mathfrak{sl}_{2}^{+}$

Matthew McMillan

A tensor 2-product of 2-representations of $\mathfrak{sl}_{2}^{+}$

Representation Theory 2023-06-23 v3 Quantum Algebra

Authors: Matthew McMillan

Abstract

We construct an explicit abelian model for the operation of tensor $2$ -product of $2$ -representations of $\mathfrak{sl}_{2}^{+}$ , specifically the product of a simple $2$ -representation $\mathcal{L}(1)$ with a given abelian $2$ -representation $\mathcal{V}$ taken from the $2$ -category of algebras. We study the case $\mathcal{V}=\mathcal{L}(1)$ in detail, and we show that the $2$ -product in this case recovers the expected structure. Our construction partially verifies a conjecture of Rouquier from 2008.

Keywords

tensor 2-product 2-representations sl2+ abelian model rouquier conjecture 2-category of algebras

Cite

@article{arxiv.2209.06782,
  title  = {A tensor 2-product of 2-representations of $\mathfrak{sl}_{2}^{+}$},
  author = {Matthew McMillan},
  journal= {arXiv preprint arXiv:2209.06782},
  year   = {2023}
}

Comments

50 pages, 5 diagrams. v3 is compatible with arXiv:2303.17115v2. Addition of Remark 3.15. Numbering of propositions etc. has changed

A tensor 2-product of 2-representations of $\mathfrak{sl}_{2}^{+}$

Abstract

Keywords

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