JournalsqtVol. 3, No. 2pp. 139–180

Categorification of the Jones–Wenzl projectors

  • Benjamin Cooper

    University of Virginia, Charlottesville, USA
  • Vyacheslav Krushkal

    Krushkal
Categorification of the Jones–Wenzl projectors cover
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Abstract

The Jones–Wenzl projectors pnp_n play a central role in quantum topology, underlying the construction of SU(2) topological quantum field theories and quantum spin networks. We construct chain complexes PnP_n, whose graded Euler characteristic is the “classical” projector pnp_n in the Temperley–Lieb algebra. We show that the Pn{P}_n are idempotents and uniquely defined up to homotopy. Our results fit within the general framework of Khovanov’s categorification of the Jones polynomial. Consequences of our construction include families of knot invariants corresponding to higher representations of Uqsl(2)\mathrm{U}_q\mathfrak{sl}(2) and a categorification of quantum spin networks. We introduce 6j-symbols in this context.

Cite this article

Benjamin Cooper, Vyacheslav Krushkal, Categorification of the Jones–Wenzl projectors. Quantum Topol. 3 (2012), no. 2, pp. 139–180

DOI 10.4171/QT/27