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arXiv · 1605.05634

Logarithmic Link Invariants of $\overline{U}_q^H(\mathfrak{sl}_2)$ and Asymptotic Dimensions of Singlet Vertex Algebras

Thomas Creutzig·Antun Milas·Matt Rupert
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Abstract

We study relationships between the restricted unrolled quantum group $\overline{U}_q^H(\mathfrak{sl}_2)$ at $2r$-th root of unity $q=e^{πi/r}, r \geq 2$, and the singlet vertex operator algebra $\mathcal M(r)$. We use deformable families of modules to efficiently compute $(1, 1)$-tangle invariants colored with projective modules of $\overline{U}_q^H(\mathfrak{sl}_2)$. These relate to the colored Alexander tangle invariants studied in [ADO, M1]. It follows that the regularized asymptotic dimensions of characters of $\mathcal M(r)$ coincide with the corresponding modified traces of open Hopf link invariants. We also discuss various categorical properties of $\mathcal M(r)$-mod in connection to braided tensor categories.

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Thomas Creutzig, Antun Milas, Matt Rupert. 2016-05-18. Logarithmic Link Invariants of $\overline{U}_q^H(\mathfrak{sl}_2)$ and Asymptotic Dimensions of Singlet Vertex Algebras. https://arxiv.org/abs/1605.05634

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