Quantum shuffles and quantum supergroups of basic type

  • Sean Clark

    Northeastern University, Boston, USA
  • David Hill

    Washington State University, Vancouver, USA
  • Weiqiang Wang

    University of Virginia, Charlottesville, USA


We initiate the study of several distinguished bases for the positive half of a quantum supergroup associated to a general super Cartan datum of basic type inside a quantum shuffle superalgebra. The combinatorics of words for an arbitrary total ordering on is developed in connection with the root system associated to . The monomial, Lyndon, and PBW bases of are constructed, and moreover, a direct proof of the orthogonality of the PBW basis is provided within the framework of quantum shuffles. Consequently, the canonical basis is constructed for associated to the standard super Cartan datum of type , , or or an arbitrary non-super Cartan datum. In the non-super case, this refines Leclerc's work and provides a new self-contained construction of canonical bases. The canonical bases of , of its polynomial modules, as well as of Kac modules in the case of quantum are explicitly worked out.

Cite this article

Sean Clark, David Hill, Weiqiang Wang, Quantum shuffles and quantum supergroups of basic type. Quantum Topol. 7 (2016), no. 3, pp. 553–638

DOI 10.4171/QT/81