Category O and slk link invariants

By Joshua Sussan. 2007

63 citations recorded by the source corpus.

We construct a functor valued invariant of oriented tangles on certain singular blocks of category O. Parabolic subcategories of these blocks categorify tensor products of various fundamental sl(k) representations. Projective functors restricted to these categories give rise to a functorial action of the Lie algebra. On the derived category, Zuckerman functors categorify sl(k)- homomorphisms. Cones of natural transformations between the identity functor and Zuckerman functors are assigned to crossings and this assignment satisfies the appropriate relations. On the Grothendieck group, the functors assigned to the crossings satisfy the sl(k)- specialization of the two variable HOMFLYPT polynomial. For the special case of links, we get a homological invariant.

Subfields: Algebra and Number Theory, Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/9af4cbc81395b7d0cb15318c4e28766898e72321

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Citations in the corpus

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”
A diagrammatic approach to categorification of quantum groups II (2008, unknown): “This model was given a categorical interpretation by Grojnowski and Lusztig [15] and later reinterpreted, for N = 2, via translation and Zuckerman functors in [6], with various generalizations constructed in [13], [44], [48], and in a very recent striking work [49].”
A diagrammatic approach to categorification of quantum groups II (2008, methodology): “This model was given a categorical interpretation by Grojnowski and Lusztig [15] and later reinterpreted, for N = 2, via translation and Zuckerman functors in [6], with various generalizations constructed in [13], [ 44 ], [48], and in a very recent striking work [49].”
Knot invariants and higher representation theory I: diagrammatic and geometric categorification of tensor products (2010, methodology): “tion of U and S agree, up to an easily understood shift, with those used in other papers on graded category O (most importantly for us, this is used in the work of MazorchukStroppel [MS08] and Sussan [Sus07] on link homologies, which we address in this paper’s sequel [Webc]). 4.3. The aﬃnecase. We n”
Khovanov homology is a skew Howe 2-representation of categorified quantum sl(m) (2012, methodology): “More generally, for link homology theories associated with fundamental sln representations, Webster describes an isomorphism relating his construction to Sussan’s category O based link homology theory [69] , which is related via Koszul duality to a theory defined by Mazorchuk and Stroppel [56].”
Quantum (sln, ∧V n ) link invariant and matrix factorizations (2009, methodology): “Note that Sussan [12] and Mazorchuk and Stroppel [8] gave a categorification of MOY diagrams colored from the set {1, .”
An introduction to diagrammatic algebra and categorified quantum sl(2) (2011, background): “Many of these are based on geometric constructions [8, 26, 69], while others have a more algebraic flavour [3, 63, 24, 64, 25, 24].”
A combinatorial approach to functorial quantum slk knot invariants (2007, methodology): “Only crossings force us to pass to derived categories (whereas the approach of [30] has to use derived categories and higher derived functors from the very beginning).”
Highest weight categories arising from Khovanov's diagram algebra III: category O (2008, methodology): “uced from 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 [Su] and [MS] to deﬁnefunctorial knot and tangle invariants which decategorify to the slk-version of”
A categorification of quantum sl(n) (2008, background): “entation theory. This 2-category or its mild modiﬁcations is expected to act on parabolic-singular blocks of highest weight categories for slN in the context of categoriﬁcation of sln representations [7, 10, 31], on derived categories of coherent sheaves on Kronheimer-Nakajima [20] and Nakajima [26]”