Homological Mirror symmetry and Quantum Groups

There is a new family of homological mirror pairs for which homological mirror symmetry can be understood as explicitly as in the simplest known examples. The resulting categories categorify braid-group representations coming from quantum groups. One application is the categorification of quantum link invariants. Another is a solution to an outstanding problem in higher representation theory: describing a categorification of the Hopf algebra structure of a quantum group.

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