A crystal base for a representation of a quantum group-vector space-lattice in that vector space. Crystal bases appeared in the work of and also in the work of of the canonical basis defined by .

Definition

can be regarded as a Hopf algebra over the field of all rational functions of an indeterminate q.

For simple root, define

) can be uniquely decomposed into the sums

.

by

be the integral domain (i.e.).

A crystal base, such that

for all i are represented by mutual transposes, and map basis vectors to basis vectors or 0.

A crystal base can be represented by a directed graph, and a directed edge, labelled by i).

For any integrable module with a crystal base, the weight spectrum for the crystal base is the same as the weight spectrum for the module, and therefore the weight spectrum for the crystal base is the same as the weight spectrum for the corresponding module of the appropriate Kac–Moody algebra. The multiplicities of the weights in the crystal base are also the same as their multiplicities in the corresponding module of the appropriate Kac–Moody algebra.

It is a theorem of Kashiwara that every integrable highest weight module has a crystal base. Similarly, every integrable lowest weight module has a crystal base.

Tensor products of crystal bases

. For crystal bases, the coproduct, given by

, define

are given by

The decomposition of the product two integrable highest weight modules into irreducible submodules is determined by the decomposition of the graph of the crystal base into its connected components (i.e. the highest weights of the submodules are determined, and the multiplicity of each highest weight is determined).

References

External links