  We construct a basis for a modified quantum group of finite type, extending
the PBW bases of positive and negative halves of a quantum group. Generalizing
Lusztig's classic results on PBW bases, we show that this basis is orthogonal
with respect to its natural bilinear form (and hence called a PBW basis), and
moreover, the matrix for the PBW-expansion of the canonical basis is unital
triangular. All these follow by a new construction of the modified quantum
group of arbitrary type, which is built on limits of sequences of elements in
tensor products of lowest and highest weight modules. Explicitly formulas are
worked out in the rank one case.
