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  1. ; ; (, Journal für die reine und angewandte Mathematik (Crelles Journal))
    Abstract Rickard complexes in the context of categorified quantum groups can be used to construct braid group actions. We define and study certain natural deformations of these complexes which we call curved Rickard complexes. One application is to obtain deformations of link homologies which generalize those of Batson–Seed [3] [J. Batson and C. Seed, A link-splitting spectral sequence in Khovanov homology,Duke Math. J. 164 2015, 5, 801–841] and Gorsky–Hogancamp [E. Gorsky and M. Hogancamp, Hilbert schemes and y -ification of Khovanov–Rozansky homology,preprint 2017] to arbitrary representations/partitions. Another is to relate the deformed homology defined algebro-geometrically in [S. Cautis and J. Kamnitzer,Knot homology via derived categories of coherent sheaves IV, colored links,Quantum Topol. 8 2017, 2, 381–411] to categorified quantum groups (this was the original motivation for this paper). 
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  2. ; (, Quantum Topology)
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  3. (, Algebras and Representation Theory)
    In this note we give explicit isomorphisms of 2-categories between various versions of the categorified quantum group associated to a simply-laced Kac-Moody algebra. These isomorphisms are convenient when working with the categorified quantum group. They make it possible to translate results from the gln variant of the 2-category to the sln variant and transfer results between various conventions in the literature. We also extend isomorphisms of finite type KLR algebras for different choices of parameters to the level of 2-categories. 
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  4. ; ; ; (, Journal of Combinatorial Algebra)
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