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).