Crystal base

A crystal base for a representation of a quantum group on a

Crystal bases appeared in the work of Kashiwara (1990) and also in the work of Lusztig (1990).

of the canonical basis defined by Lusztig (1990).

As a consequence of its defining relations, the quantum group

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

, define In an integrable module

) can be uniquely decomposed into the sums where

be the integral domain of all rational functions in

(i.e. a rational function

, such that To put this into a more informal setting, the actions of

on the module are introduced so that the actions of

The module is then restricted to the free

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

A crystal base can be represented by a directed graph with labelled edges.

Each vertex of the graph represents an element of the

, and a directed edge, labelled by i, and directed from vertex

is the basis element represented by

is the basis element represented by

The graph completely determines the actions of

If an integrable module has a crystal base, then the module is irreducible if and only if the graph representing the crystal base is connected (a graph is called "connected" if the set of vertices cannot be partitioned into the union of nontrivial disjoint subsets

such that there are no edges joining any vertex in

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.

be an integrable module with crystal base

be an integrable module with crystal base

For crystal bases, the coproduct

, define The actions of

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).