Pure and Applied Mathematics Quarterly

Volume 12 (2016)

Number 4

Comparing skein and quantum group representations and their application to asymptotic faithfulness

Pages: 473 – 492

DOI: http://dx.doi.org/10.4310/PAMQ.2016.v12.n4.a2

Authors

Wade Bloomquist (Department of Mathematics, University of California at Santa Barbara)

Zhenghan Wang (Microsoft Station Q and Department of Mathematics, University of California at Santa Barbara)

Abstract

We make two related observations in this paper. First, the representations of mapping class groups from the Ising TQFT and its quantum group counterpart $SU(2)_2$ are neither equivalent as representations nor Galois conjugate to each other. Hence mapping class group representations obtained from quantum skein theory are fundamentally distinct from those obtained from quantum group Reshetikhin–Turaev or geometric quantization constructions. Then we generalize the asymptotic faithfulness of the skein quantum $SU(2)_2$ representations of mapping class groups of orientable closed surfaces to skein quantum $SU(3)$. We conjecture asymptotic faithfulness holds for skein quantum $G$ representations when $G$ is a simply-connected simple Lie group. The difficulty for such a generalization lies in the lack of an explicit description of the fusion spaces with multiplicities to define an appropriate complexity of state vectors.

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The second author is partially supported by NSF grants DMS-1410144 and DMS-1411212.

Received 29 November 2017