Abstract
AbstractWe define a limiting ${\mathfrak {sl}_N}$ Khovanov–Rozansky homology for semi-infinite positive multicolored braids. For a large class of such braids, we show that this limiting homology categorifies a highest-weight projector in the tensor product of fundamental representations determined by the coloring of the braid. This effectively completes the extension of Cautis’ similar result for infinite twist braids, begun in our earlier papers with Islambouli and Abel. We also present several similar results for other families of semi-infinite and bi-infinite multicolored braids.
Highlights
The Jones–Wenzl projector Pn [16] is a special idempotent element of the Temperley– Lieb algebra representing a highest-weight projector in the representation theory of Uq(sl2), used in particular to define WRT-invariants for 3-manifolds
A sequence of papers by various authors [4, 6, 13] showed that, in all of these cases, such highest-weight projectors could be categorified via infinite chain complexes associated to the stable limiting Khovanov– Rozansky complex of infinite full twists
Together with Islambouli in [7] and Abel in [1], the author has shown that, for allinfinite uni-colored braids Bthat are both positive and complete, the limiting Khovanov–Rozansky complex CN (B ) is chain homotopy equivalent to CN (F∞), the limiting complex of the infinite full twist
Summary
The Jones–Wenzl projector Pn [16] is a special idempotent element of the Temperley– Lieb algebra representing a highest-weight projector in the representation theory of Uq(sl2), used in particular to define WRT-invariants for 3-manifolds (see, for example, [8]). Theorem 1.1 ([7] Theorem 1.1, [1] Theorem 1.1) All positive complete semi-infinite unicolored braids categorify highest weight projectors in the tensor product of fundamental representations determined by the coloring of the braid. Theorem 1.2 All positive color-complete semi-infinite braids categorify highest weight projectors in the tensor product of fundamental representations determined by the coloring of the braid. Theorem 1.3 All positive color-complete bi-infinite braids Bhave limiting complexes of the form P ⊗ C ⊗ P′ where P and P′ indicate categorified projectors for two (possibly different) sequences of colors depending on the coloring of B , while C is a complex categorifying certain maps between the representations determined by the two color sequences.
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