Quantum shuffles and quantum supergroups of basic type

Abstract

We initiate the study of several distinguished bases for the positive half of a quantum supergroup U_q associated to a general super Cartan datum (\mathrm{I}, (\cdot,\cdot)) of basic type inside a quantum shuffle superalgebra. The combinatorics of words for an arbitrary total ordering on \mathrm{I} is developed in connection with the root system associated to \mathrm{I} . The monomial, Lyndon, and PBW bases of U_q 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 U_q associated to the standard super Cartan datum of type \mathfrak{gl}(n|1) , \mathfrak{osp}(1|2n) , or \mathfrak{osp}(2|2n) 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 U_q , of its polynomial modules, as well as of Kac modules in the case of quantum \mathfrak{gl}(2|1) are explicitly worked out.