We establish the equivalence between the refined topological vertex of Iqbal-Kozcaz-Vafa (or Awata-Kanno) and a certain representation theory of the quantum algebra introduced by Miki. Our construction involves trivalent intertwining operators associated with triples of the bosonic Fock modules. Resembling the topological vertex, a triple of vectors is attached to each intertwining operator, which satisfy the Calabi-Yau and smoothness conditions. (Joint work with H. Awata and B. Feigin. arXiv:1112.6074)

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