COHERENT SHEAVES AND CATEGORICAL sl2 ACTIONS

By Sabin Cautis, Joel Kamnitzer, Anthony M. Licata. 2009

47 citations recorded by the source corpus.

We introduce the concept of a geometric categorical sl2 action and relate it to that of a strong categorical sl2 action. The latter is a special kind of 2-representation in the sense of Rouquier. The main result is that a geometric categorical sl2 action induces a strong categorical sl2 action. This allows one to apply the theory of strong sl2 actions to various geometric situations. Our main example is the construction of a geometric categorical sl2 action on the derived category of coherent sheaves on cotangent bundles of Grassmannians.

Subfields: Discrete Mathematics and Combinatorics, Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/ca694f02e76ec2235235f37e3fce4c521da4ec00

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In Discrete Mathematics and Combinatorics

In Geometry and Topology

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Citations in the corpus

Braiding via geometric Lie algebra actions (2010, background): “erent method). However, we do not prove that a geometric categorical g action gives a strong categorical gaction, though we expect this to be the case (the proof should follow along the same lines as [CKL2], where we established this result for g= sl2). We give a quick example showing how resolution”
Khovanov homology is a skew Howe 2-representation of categorified quantum sl(m) (2012, background): “The resulting derived equivalences are highly non-trivial and have led to the resolution of several important conjectures [20, 16, 14].”
The derived category of a GIT quotient (2012, unknown): “ed analysis, one can verify Ansatz 4.11 for more examples of balanced wall crossings, and we will describe this in a future paper. Remark 4.13. This example is similar to the generalized Mukai ops of [9]. The dierence is that we are not restricting to the hyperkahler moment ber fba= 0g. We will see ”
The $\mathfrak{sl}_n$ foam 2-category: a combinatorial formulation of Khovanov-Rozansky homology via categorical skew Howe duality (2014, background): “Given a 2-representation F : ǓQ(slm) → K, the equivalences given by these complexes are highly nontrivial and have been applied to a variety of contexts ranging from the representation theory of the symmetric group [14] to coherent sheaves on cotangent bundles [9, 10].”
Categorical geometric skew Howe duality (2009, background): “us result for Es instead of Fs follows by adjunction. Remark6.3. Notice that we only needed to prove Proposition 6.2 in the case r1 = r2 = 1 (the general case then follows from the formal argument in [CKL1]). However, we proved the general statement since it was no more diﬃcult and somewhat instruct”
Quiver Hecke Algebras and 2-Lie Algebras (2011, methodology): “An earlier geometrical approach has been given by Cautis, Kamnitzer and Licata for sl2, based on coherent sheaves on cotangent bundles of flag varieties and compactifications of those [6], later generalized to arbitrary Γ [7].”