The Temperley-Lieb Tower and the Weyl Algebra
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arXiv
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| Main Authors: | , |
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| Format: | Preprint |
| Published: |
2024
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| _version_ | 1866908315649310720 |
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| author | Harper, Matthew Samuelson, Peter |
| author_facet | Harper, Matthew Samuelson, Peter |
| contents | We define a monoidal category $\operatorname{\mathbf{W}}$ and a closely related 2-category $\operatorname{\mathbf{2Weyl}}$ using diagrammatic methods. We show that $\operatorname{\mathbf{2Weyl}}$ acts on the category $\mathbf{TL} :=\bigoplus_n \operatorname{TL}_n\mathrm{-mod}$ of modules over Temperley-Lieb algebras, with its generating 1-morphisms acting by induction and restriction. The Grothendieck groups of $\operatorname{\mathbf{W}}$ and a third category we define $\operatorname{\mathbf W}^\infty$ are closely related to the Weyl algebra. We formulate a sense in which $K_0(\operatorname{\mathbf W}^\infty)$ acts asymptotically on $K_0(\mathbf{TL})$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2401_02545 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | The Temperley-Lieb Tower and the Weyl Algebra Harper, Matthew Samuelson, Peter Quantum Algebra Representation Theory 16D90 We define a monoidal category $\operatorname{\mathbf{W}}$ and a closely related 2-category $\operatorname{\mathbf{2Weyl}}$ using diagrammatic methods. We show that $\operatorname{\mathbf{2Weyl}}$ acts on the category $\mathbf{TL} :=\bigoplus_n \operatorname{TL}_n\mathrm{-mod}$ of modules over Temperley-Lieb algebras, with its generating 1-morphisms acting by induction and restriction. The Grothendieck groups of $\operatorname{\mathbf{W}}$ and a third category we define $\operatorname{\mathbf W}^\infty$ are closely related to the Weyl algebra. We formulate a sense in which $K_0(\operatorname{\mathbf W}^\infty)$ acts asymptotically on $K_0(\mathbf{TL})$. |
| title | The Temperley-Lieb Tower and the Weyl Algebra |
| topic | Quantum Algebra Representation Theory 16D90 |
| url | https://arxiv.org/abs/2401.02545 |