Naturality of ${\rm SL}_3$ quantum trace maps for surfaces
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| Format: | Preprint |
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2021
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| _version_ | 1866917682021924864 |
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| author | Kim, Hyun Kyu |
| author_facet | Kim, Hyun Kyu |
| contents | Fock-Goncharov's moduli spaces $\mathscr{X}_{{\rm PGL}_3,\frak{S}}$ of framed ${\rm PGL}_3$-local systems on punctured surfaces $\frak{S}$ provide prominent examples of cluster $\mathscr{X}$-varieties and higher Teichmüller spaces. In a previous paper of the author (arXiv:2011.14765), building on the works of others, the so-called ${\rm SL}_3$ quantum trace map is constructed for each triangulable punctured surface $\frak{S}$ and an ideal triangulation $Δ$ of $\frak{S}$, as a homomorphism from the stated ${\rm SL}_3$-skein algebra of the surface to a quantum torus algebra that deforms the ring of Laurent polynomials in the cube-roots of the cluster coordinate variables for the cluster $\mathscr{X}$-chart for $\mathscr{X}_{{\rm PGL}_3,\frak{S}}$ associated to $Δ$. We develop quantum mutation maps between special subalgebras of the cube-root quantum torus algebras for different triangulations and show that the ${\rm SL}_3$ quantum trace maps are natural, in the sense that they are compatible under these quantum mutation maps. As an application, the quantum ${\rm SL}_3$-${\rm PGL}_3$ duality map constructed in the previous paper is shown to be independent of the choice of an ideal triangulation. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2104_06286 |
| institution | arXiv |
| publishDate | 2021 |
| record_format | arxiv |
| spellingShingle | Naturality of ${\rm SL}_3$ quantum trace maps for surfaces Kim, Hyun Kyu Quantum Algebra Mathematical Physics Geometric Topology 13F60, 57K31, 53D55, 57K20 Fock-Goncharov's moduli spaces $\mathscr{X}_{{\rm PGL}_3,\frak{S}}$ of framed ${\rm PGL}_3$-local systems on punctured surfaces $\frak{S}$ provide prominent examples of cluster $\mathscr{X}$-varieties and higher Teichmüller spaces. In a previous paper of the author (arXiv:2011.14765), building on the works of others, the so-called ${\rm SL}_3$ quantum trace map is constructed for each triangulable punctured surface $\frak{S}$ and an ideal triangulation $Δ$ of $\frak{S}$, as a homomorphism from the stated ${\rm SL}_3$-skein algebra of the surface to a quantum torus algebra that deforms the ring of Laurent polynomials in the cube-roots of the cluster coordinate variables for the cluster $\mathscr{X}$-chart for $\mathscr{X}_{{\rm PGL}_3,\frak{S}}$ associated to $Δ$. We develop quantum mutation maps between special subalgebras of the cube-root quantum torus algebras for different triangulations and show that the ${\rm SL}_3$ quantum trace maps are natural, in the sense that they are compatible under these quantum mutation maps. As an application, the quantum ${\rm SL}_3$-${\rm PGL}_3$ duality map constructed in the previous paper is shown to be independent of the choice of an ideal triangulation. |
| title | Naturality of ${\rm SL}_3$ quantum trace maps for surfaces |
| topic | Quantum Algebra Mathematical Physics Geometric Topology 13F60, 57K31, 53D55, 57K20 |
| url | https://arxiv.org/abs/2104.06286 |