Braiding via geometric Lie algebra actions

By Sabin Cautis, Joel Kamnitzer. Compositio Mathematica, 2010

53 citations recorded by the source corpus.

Abstract We introduce the idea of a geometric categorical Lie algebra action on derived categories of coherent sheaves. The main result is that such an action induces an action of the braid group associated to the Lie algebra. The same proof shows that strong categorical actions in the sense of Khovanov–Lauda and Rouquier also lead to braid group actions. As an example, we construct an action of Artin’s braid group on derived categories of coherent sheaves on cotangent bundles to partial flag varieties.

Subfields: Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/e610f5c10a42def39988717c9ce0a5d27f38e246

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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): “Our main tool is the higher representation theory of the Khovanov-Lauda categorified quantum group [25, 26, 27] (see also work of Rouquier [48] for an independent construction), in particular, the categorical skew Howe duality of Cautis-Kamnitzer-Licata [8].”
Khovanov homology is a skew Howe 2-representation of categorified quantum sl(m) (2012, methodology): “Since the images of the Rickard complexes braid in any (integrable) 2-representation [13], the above maps are uniquely defined up to coherent isomorphism.”
Implicit structure in 2-representations of quantum groups (2011, background): “ iEj 1λ,EiEj λ) ∼= ˆ 0 if k < 0, k if k = 0. (6.2) The same results hold if we replace all the Es by Fs. Proof. If the Dynkin diagram of gis simply laced then this result follows from Lemma 4.5 of [6]. But since we are dealing with arbitrary symmetrizable Kac-Moody algebras we reproduce the proof”
Categorifications from planar diagrammatics (2010, background): “ related these rings to Ariki’s categoriﬁcations of U+ and highest weight representations. A direct relation to derived categories of coherent sheaves on Nakajima quiver 20 varieties [41] is expected [10]. Let Γ be an unoriented graph without loop and multiple edges, with set of vertices I. The quan”
New derived symmetries of some hyperkähler varieties (2011, background): “e in a diﬀerent direction, getting representations of sl2 and of other Lie algebras in later papers. Many other authors are also involved, including Rouquier and Anno; for a more complete history see [10]. Donovan [12] gave an example in which the cotwist is more interesting than just a shift or a l”
Heisenberg categorification and Hilbert schemes (2010, background): “These computations are analogous to those from [4] and [3] in the setting of categorical sln-actions.”