Crystal base

In algebra, a crystal base or canonical base is a base of a representation, such that generators of a quantum group or semisimple Lie algebra have a particularly simple action on it. Crystal bases were introduced by and (under the name of canonical bases).
Definition.
As a consequence of its defining relations, the quantum group formula_1 can be regarded as a Hopf algebra over the field of all rational functions of an indeterminate "q" over formula_2, denoted formula_3.
For simple root formula_4 and non-negative integer formula_5, define 
In an integrable module formula_7, and for weight formula_8, a vector formula_9 (i.e. a vector formula_10 in formula_7 with weight formula_8) can be uniquely decomposed into the sums
where formula_14, formula_15, formula_16 only if formula_17, and formula_18 only if formula_19. 
Linear mappings formula_20 can be defined on formula_21 by
Let formula_24 be the integral domain of all rational functions in formula_3 which are regular at formula_26 ("i.e." a rational function formula_27 is an element of formula_24 if and only if there exist polynomials formula_29 and formula_30 in the polynomial ring formula_31 such that formula_32, and formula_33). A crystal base for formula_7 is an ordered pair formula_35, such that
To put this into a more informal setting, the actions of formula_53 and formula_54 are generally singular at formula_26 on an integrable module formula_7. The linear mappings formula_57 and formula_58 on the module are introduced so that the actions of formula_59 and formula_60 are regular at formula_26 on the module. There exists a formula_3-basis of weight vectors formula_63 for formula_7, with respect to which the actions of formula_57 and formula_58 are regular at formula_26 for all "i". The module is then restricted to the free formula_24-module generated by the basis, and the basis vectors, the formula_24-submodule and the actions of formula_57 and formula_58 are evaluated at formula_26. Furthermore, the basis can be chosen such that at formula_26, for all formula_74, formula_57 and formula_58 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 formula_77-basis formula_40 of formula_42, and a directed edge, labelled by "i", and directed from vertex formula_80 to vertex formula_81, represents that formula_82 (and, equivalently, that formula_83), where formula_84 is the basis element represented by formula_80, and formula_86 is the basis element represented by formula_81. The graph completely determines the actions of formula_57 and formula_58 at formula_26. 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 formula_91 and formula_92 such that there are no edges joining any vertex in formula_91 to any vertex in formula_92).
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.
Let formula_7 be an integrable module with crystal base formula_35 and formula_97 be an integrable module with crystal base formula_98. For crystal bases, the coproduct formula_99, given by 
is adopted. The integrable module formula_101 has crystal base formula_102, where formula_103. For a basis vector formula_104, define 
The actions of formula_57 and formula_58 on formula_109 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).