arXiv subjects

Juan Ramón Gómez García

Publications and source records attributed to Juan Ramón Gómez García.

2 recordsLinked to original sources

arXiv2026

Weyl defects in skein theory and quantum cluster charts

We prove a modified invertibility property for the parabolic defects introduced in arXiv:2102.12283, arXiv:2505.14836. Namely, we show that the Borel defect cancels its dual, up to insertion of an invertible Weyl defect and up to restriction to an open-subcategory. The main result holds for an arbitrary reductive group $G$ -- for illustration we give extended computations for $G=\mathrm{SL}_3$. Along the way, we introduce a defect skein theory with defects in both codimension one and two, which is compatible with gluing of defect 3-manifolds and defect surfaces. Our results provide a potential defect skein theoretic construction of certain standard charts which appears frequently in quantum cluster varieties associated to surfaces.

math.QA↗

arXiv2026

HOMFLY parabolic restriction, defect skein theory and the Turaev coproduct

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.

math.QA↗