zbMATH — the first resource for mathematics

Symmetric Webs, Jones-Wenzl recursions, and \(q\)-Howe duality. (English) Zbl 1404.17025
Summary: We define and study the category of symmetric \(\mathfrak{sl}_2\)-webs. This is a combinatorial description of the category of all finite-dimensional quantum \(\mathfrak{sl}_2\)-modules. Explicitly, we show that (the additive closure of) the category of symmetric \(\mathfrak{sl}_{2}\)-webs is (braided monoidally) equivalent to the latter. Our main tool is a quantum version of symmetric Howe duality. As a corollary of our construction, we provide new insight into Jones-Wenzl projectors and colored Jones polynomials.

17B37 Quantum groups (quantized enveloping algebras) and related deformations
17B10 Representations of Lie algebras and Lie superalgebras, algebraic theory (weights)
18B99 Special categories
57M27 Invariants of knots and \(3\)-manifolds (MSC2010)
Full Text: DOI