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.
- Publication:
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arXiv Mathematics e-prints
- Pub Date:
- August 1998
- DOI:
- 10.48550/arXiv.math/9808060
- arXiv:
- arXiv:math/9808060
- Bibcode:
- 1998math......8060B
- Keywords:
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- Quantum Algebra