An Algebraic Characterization of the Affine Canonical Basis
Abstract
The canonical basis for quantized universal enveloping algebras associated to the finitedimensional 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 barinvariant 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 PBWtype basis whose $\bz[q^{1}]$span reduces to a ``crystal'' basis at $q=\infty,$ with the imaginary component given by the Schur functions.
 Publication:

arXiv Mathematics eprints
 Pub Date:
 August 1998
 DOI:
 10.48550/arXiv.math/9808060
 arXiv:
 arXiv:math/9808060
 Bibcode:
 1998math......8060B
 Keywords:

 Quantum Algebra