Quantum shuffles and quantum supergroups of basic type (Q330015)

The paper under review makes a first step towards construction of canonical bases for quantum supergroups of arbitrary basic type using the approach of quantum shuffles. In particular, the paper provides a family of monomial bases and orthogonal PBW bases for the positive half \(U_q\) of a quantum supergroup of basic type, one for each total ordering on the index set \(I\) which labels simple roots. The next step gives a construction of an integral form for \(\mathfrak{gl}(m| n)\), \(\mathfrak{osp}(1| 2n)\) and \(\mathfrak{osp}(2| 2n)\) which gives a possibility to construct a canonical basis i \(U_q\) in these three cases (as well as for an arbitrary non-super Cartan datum). The construction follows a careful study of the combinatorics of various types of words. As a bonus, a direct proof of orthogonality for the PBW basis is given. The example of \(\mathfrak{gl}(2| 1)\) is worked out in detail.