An algebraic characterization of the affine canonical basis

Abstract

The canonical basis for finite type quantized universal enveloping algebras was introduced in [L3]. The principal technique is the explicit construction (via the braid group action) of a lattice L over Z[q -1 ]. This allows the algebraic characterization of the canonical basis as a certain bar-invariant basis of L. Here we present a similar algebraic characterization of the affine canonical basis. Our construction is complicated by the need to introduce basis elements to span the "imaginary" subalgebra which is fixed by the affine braid group. Once the basis is found we construct a PBW-type basis whose Z[q -1 ]span reduces to a "crystal" basis at q = , with the imaginary component given by the Schur functions.