An Algebraic Characterization of the Affine Canonical Basis

Abstract

The canonical basis for quantized universal enveloping algebras associated to the finite--dimensional simple Lie algebras, was introduced by Lusztig. The principal technique is the explicit construction (via the braid group action) of a lattice over $\bz[q^{-1}]$. This allows the algebraic characterization of the canonical basis as a certain bar-invariant basis of $\cl$. 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 $\bz[q^{-1}]$-span reduces to a ``crystal'' basis at $q=\infty,$ with the imaginary component given by the Schur functions.