HOMFLY parabolic restriction, defect skein theory and the Turaev coproduct
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arXiv
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| Format: | Preprint |
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2026
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| _version_ | 1866918274173763584 |
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| author | García, Juan Ramón Gómez |
| author_facet | García, Juan Ramón Gómez |
| contents | We define a HOMFLY version of the category $\text{Rep}_q\text{P}$ of quantum representations of a parabolic subgroup $\text{P}\subseteq\text{GL}_{m+n}$ of block triangular matrices. Alongside this category, we construct functors that interpolate the usual restriction functors between $\text{GL}_{m+n}$, $\text{P}$ and the subgroup $\text{L}\subseteq\text{GL}_{m+n}$ of block-diagonal matrices, yielding a universal version of the formalism of parabolic restriction. Based on this formalism, we construct central algebras and centred bimodules which serve as algebraic ingredients for defining a skein theory on $3$-manifolds with surface and line defects. We recover the Turaev coproduct on the HOMFLY skein algebra as a particular instance of this theory. In particular, this coproduct is compatible with the cutting and gluing of surfaces. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2601_03196 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | HOMFLY parabolic restriction, defect skein theory and the Turaev coproduct García, Juan Ramón Gómez Quantum Algebra Algebraic Topology Category Theory We define a HOMFLY version of the category $\text{Rep}_q\text{P}$ of quantum representations of a parabolic subgroup $\text{P}\subseteq\text{GL}_{m+n}$ of block triangular matrices. Alongside this category, we construct functors that interpolate the usual restriction functors between $\text{GL}_{m+n}$, $\text{P}$ and the subgroup $\text{L}\subseteq\text{GL}_{m+n}$ of block-diagonal matrices, yielding a universal version of the formalism of parabolic restriction. Based on this formalism, we construct central algebras and centred bimodules which serve as algebraic ingredients for defining a skein theory on $3$-manifolds with surface and line defects. We recover the Turaev coproduct on the HOMFLY skein algebra as a particular instance of this theory. In particular, this coproduct is compatible with the cutting and gluing of surfaces. |
| title | HOMFLY parabolic restriction, defect skein theory and the Turaev coproduct |
| topic | Quantum Algebra Algebraic Topology Category Theory |
| url | https://arxiv.org/abs/2601.03196 |