Tensor $2$-Product for $\mathfrak{sl}_{2}$: Extensions to the Negative Half
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arXiv
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| Format: | Preprint |
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2023
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| _version_ | 1866913186215624704 |
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| author | McMillan, Matthew |
| author_facet | McMillan, Matthew |
| contents | 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2303_17115 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Tensor $2$-Product for $\mathfrak{sl}_{2}$: Extensions to the Negative Half McMillan, Matthew Representation Theory Quantum Algebra 17B10, 17B55, 18N25, 57K16, 57K18 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. |
| title | Tensor $2$-Product for $\mathfrak{sl}_{2}$: Extensions to the Negative Half |
| topic | Representation Theory Quantum Algebra 17B10, 17B55, 18N25, 57K16, 57K18 |
| url | https://arxiv.org/abs/2303.17115 |