Logarithmic Link Invariants of $\\overline{U}_q^H(\\mathfrak{sl}_2)$ and\n Asymptotic Dimensions of Singlet Vertex Algebras Article Swipe
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· 2016
· Open Access
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· DOI: https://doi.org/10.48550/arxiv.1605.05634
· OA: W4293765838
We study relationships between the restricted unrolled quantum group\n$\\overline{U}_q^H(\\mathfrak{sl}_2)$ at $2r$-th root of unity $q=e^{\\pi i/r}, r\n\\geq 2$, and the singlet vertex operator algebra $\\mathcal M(r)$. We use\ndeformable families of modules to efficiently compute $(1, 1)$-tangle\ninvariants colored with projective modules of\n$\\overline{U}_q^H(\\mathfrak{sl}_2)$. These relate to the colored Alexander\ntangle invariants studied in [ADO, M1]. It follows that the regularized\nasymptotic dimensions of characters of $\\mathcal M(r)$ coincide with the\ncorresponding modified traces of open Hopf link invariants. We also discuss\nvarious categorical properties of $\\mathcal M(r)$-mod in connection to braided\ntensor categories.\n
