Crystal base
A crystal base for a representation of a quantum group on a -vector space
is not a base of that vector space but rather a -base of  where  is a -lattice in that vector space. Crystal bases appeared in the work of  and also in the work of . They can be viewed as specializations as  of the canonical basis defined by . Definition

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 , denoted . For simple root  and non-negative integer , define 

 

In an integrable module , and for weight , a vector  (i.e. a vector  in  with weight ) can be uniquely decomposed into the sums

where , ,  only if , and  only if . Linear mappings  can be defined on  by

Let  be the integral domain of all rational functions in  which are regular at  (i.e. a rational function  is an element of  if and only if there exist polynomials  and  in the polynomial ring  such that , and ). A crystal base for  is an ordered pair , such that

 is a free -submodule of  such that 
 is a -basis of the vector space  over 
 and , where  and 
 and 
 and 

To put this into a more informal setting, the actions of  and  are generally singular at  on an integrable module . The linear mappings  and  on the module are introduced so that the actions of  and  are regular at  on the module. There exists a -basis of weight vectors  for , with respect to which the actions of  and  are regular at  for all i. The module is then restricted to the free -module generated by the basis, and the basis vectors, the -submodule and the actions of  and  are evaluated at . Furthermore, the basis can be chosen such that at , for all ,  and  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 -basis  of , and a directed edge, labelled by i, and directed from vertex  to vertex , represents that  (and, equivalently, that ), where  is the basis element represented by , and  is the basis element represented by . The graph completely determines the actions of  and  at . 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  and  such that there are no edges joining any vertex in  to 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. Tensor products of crystal bases

Let  be an integrable module with crystal base  and  be an integrable module with crystal base . For crystal bases, the coproduct , given by 

is adopted. The integrable module  has crystal base , where . For a basis vector , define 

  

The actions of  and  on  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

Lie algebras
Representation theory
Quantum groups