A combinatorial approach to functorial quantum slk knot invariants

By Volodymyr Mazorchuk, C. Stroppel. 2007

75 citations recorded by the source corpus.

This paper contains a categorification of the ${\frak sl}(k)$ link invariant using parabolic singular blocks of category ${\cal{O}}$. Our approach is intended to be as elementary as possible, providing essentially combinatorial arguments for the main results of Sussan. The justification that our combinatorial arguments and steps are correct uses non-combinatorial geometric and representation theoretic results (e.g., the Kazhdan-Lusztig and Soergel's theorems). We take these results as granted and use them like axioms (called {\it Facts\/} in the text). We first construct an exact functor valued invariant of webs or ``special'' trivalent graphs labelled with $1, 2, k-1, k$ satisfying the MOY relations. Afterwards we extend it to the ${\frak sl}(k)$-invariant of links by passing to the derived categories. The approach of Khovanov using foams appears naturally in this context. More generally, we expect that our approach provides a representation theoretic interpretation of the ${\frak sl}(k)$-homology, based on foams and the Kapustin-Li formula, from Mackaay, Stosic, and Vaz. Conjecturally this implies that the Khovanov-Rozansky link homology is obtained from our invariant by restriction.

Subfields: Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/af8fc1db887e631ada907e331d8668dee37cabc9

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Citations in the corpus

Khovanov homology is a skew Howe 2-representation of categorified quantum sl(m) (2012, methodology): “When n = 3, the latter of these link homologies can then be identified [56] with Khovanov’s more elementary construction [32] of sl3 link homology defined using singular cobordisms called foams.”
Categorifying fractional Euler characteristics, Jones–Wenzl projectors and 3j-symbols (2012, background): “[47] for an interpretation in terms of tableaux combinatorics).”
The $\mathfrak{sl}_n$ foam 2-category: a combinatorial formulation of Khovanov-Rozansky homology via categorical skew Howe duality (2014, methodology): “Independently, Cautis and Kamnitzer [5, 6] construct sln link homology using derived categories of coherent sheaves on orbits in the affine Grassmannian, Mazorchuk and Stroppel [44] and Sussan [50] give constructions based on category O, and Webster [52] defines link homology associated to any simpl”
Categorifications from planar diagrammatics (2010, background): “Some of the earliest examples of categorifications of Hecke algebra and sl(k) representations [19, 5, 46, 39] came from highest weight categories of representations of sl(n), with Ei and Fi being translation functors (direct summands of tensor products with finite-dimensional modules) or Zuckerman f”
Nazarov–Wenzl algebras, coideal subalgebras and categorified skew Howe duality (2013, unknown): “glm ↷ ⋀(n,m,r) ↶ glr ×glr ∪ ∩ gl m↷ ? γ γ1 V ⊗ γ2 V ⊗⋯⊗ γ2r V ↶ gl2r In fact, the top and bottom parts can be quantized, [CKM12], and were already categoriﬁed using parabolic-singular category O’s in [MS09] and ﬁt into the framework of [LQR12]. In our categoriﬁcation of the middle part the commuting”
Highest weight categories arising from Khovanov's diagram algebra III: category O (2008, methodology): “m the Kazhdan-Lusztig conjecture; see also [BK4, §3.1]. The graded version of this result gives a categoriﬁcation of a k-fold tensor product of quantum exterior powers, which has been used in [S] and [MS] to deﬁnefunctorial knot and tangle invariants which decategorify to the slk-version of the HOMF”
A diagrammatic categorification of the q-Schur algebra (2010, methodology): “lations which arose for general colors. Webster and Williamson subsequently showed our conjecture to be true, using a generalization of their geometric approach [35]. Note that Mazorchuk and Stroppel [26] could have deﬁned the c olored Rouquier complex in their setup, but did not. Having a geometric”