Knot Invariants and Higher Representation Theory

By Ben Webster. 2013

164 citations recorded by the source corpus.

We construct knot invariants categorifying the quantum knot variants for all representations of quantum groups. We show that these invariants coincide with previous invariants defined by Khovanov for sl_2 and sl_3 and by Mazorchuk-Stroppel and Sussan for sl_n. Our technique is to study 2-representations of 2-quantum groups (in the sense of Rouquier and Khovanov-Lauda) categorifying tensor products of irreducible representations. These are the representation categories of certain finite dimensional algebras with an explicit diagrammatic presentation, generalizing the cyclotomic quotient of the KLR algebra. When the Lie algebra under consideration is $\mathfrak{sl}_n$, we show that these categories agree with certain subcategories of parabolic category O for gl_k. We also investigate the finer structure of these categories: they are standardly stratified and satisfy a double centralizer property with respect to their self-dual modules. The standard modules of the stratification play an important role as test objects for functors, as Vermas do in more classical representation theory. The existence of these representations has consequences for the structure of previously studied categorifications. It allows us to prove the non-degeneracy of Khovanov and Lauda's 2-category (that its Hom spaces have the expected dimension) in all symmetrizable types, and that the cyclotomic quiver Hecke algebras are symmetric Frobenius. In work of Reshetikhin and Turaev, the braiding and (co)evaluation maps between representations of quantum groups are used to define polynomial knot invariants. We show that the categorifications of tensor products are related by functors categorifying these maps, which allow the construction of bigraded knot homologies whose graded Euler characteristics are the original polynomial knot invariants.

Subfields: Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/b8460193f6a92db9227bfe2d44983b7d3899dc43

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Citations in the corpus

Tensor product categorifications and the super Kazhdan-Lusztig conjecture (2013, methodology): “ely this is already established in the literature. Brieﬂy, in view of Theorem 2.11 and [LW, Theorem 3.12], we may assume that Cis the category of modules over a tensor product algebra in the sense of [W1]. This algebra is naturally graded and its category of graded modules gives us the desired grade”
ROUQUIER’S CONJECTURE AND DIAGRAMMATIC ALGEBRA (2013, background): “ line being left of a red is not enough to conclude the diagram is 0; it must be far enough left to avoid all entanglements with ghosts. We can associate the elements of Z/eZ to roots of bgl e. As in [Weba], we’ll let Tϑ ν for νa weight of bgl e be the subalgebra where the sum of the weights λ i min”