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Generalized braid group actions Anno, Rina


Consider a diagrammatic category whose objects are partitions of n and whose morphisms are braids with multiplicities where strands are allowed to merge and come apart, so topologically such a braid is a trivalent graph with boundary. In addition, we add framing on edges with multiplicities greater than 1. The usual (type A) braid group is then the group of automorphisms of (1,1,...,1). We prove that any DG enhanceable triangulated category D with a braid group action (of which there are numerous examples in algebraic geometry) can be completed to a representation of this diagrammatic category. We do this by constructing a monad over D that is best described as the nil Hecke algebra generated by the generators of the braid group action, and considering suitable categories of modules over its "block subalgebras". If D=D(X), those modules would be complexes of sheaves on X with additional data. Similar structures have been known before, but they satisfy stronger conditions (i.e. the twist of framing on a multiple strand being a shift, which in our construction is not the case). This is joint work in progress with Timothy Logvinenko.

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