Categorification of the Jones-Wenzl Projectors

By Benjamin C. Cooper, Vyacheslav Krushkal. 2010

86 citations recorded by the source corpus.

The Jones-Wenzl projectors 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 whose graded Euler characteristic is the "classical" projector in the Temperley-Lieb algebra. We show that they are homotopy 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 quantum su(2) and a categorification of quantum spin networks. We introduce 6j-symbols in this context.

Subfields: Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/66d88f46296555815b3a04c997b43098657152a0

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In Geometry and Topology

In Mathematical Physics

Citations in the corpus

Categorifying fractional Euler characteristics, Jones–Wenzl projectors and 3j-symbols (2012, methodology): “Based on two theorems characterizing the categorifications of the Jones–Wenzl projector, Theorem 71, and [17], Section 3, we expect that these categorifications agree (up to Koszul duality).”
BPS spectra and 3-manifold invariants (2017, methodology): “should be obtained by using the categorified Jones-Wenzl projectors [91, 92] instead of the naive cabling relations (B.”
Khovanov homology is a skew Howe 2-representation of categorified quantum sl(m) (2012, methodology): “The approach of Cooper-Krushkal uses foam based methods [22], while another approach of Frenkel, Stroppel, and Sussan uses Lie theoretic methods [25] based on categoryO for gln.”
Homological algebra of knots and BPS states (2011, methodology): “Several ways to categorify the Jones–Wenzl projectors have been proposed in the literature, eg the topological categorification ([8] and [9]) and the Lie-theoretic categorification [16] which agree (up to Koszul duality).”