Crystal base

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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 Kashiwara (1990) and also in the work of Lusztig (1990). They can be viewed as specializations as of the canonical basis defined by Lusztig (1990).

Definition

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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

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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

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  • Jantzen, Jens Carsten (1996), Lectures on quantum groups, Graduate Studies in Mathematics, vol. 6, Providence, R.I.: American Mathematical Society, ISBN 978-0-8218-0478-0, MR 1359532
  • Kashiwara, Masaki (1990), "Crystalizing the q-analogue of universal enveloping algebras", Communications in Mathematical Physics, 133 (2): 249–260, doi:10.1007/bf02097367, ISSN 0010-3616, MR 1090425, S2CID 121695684
  • Lusztig, G. (1990), "Canonical bases arising from quantized enveloping algebras", Journal of the American Mathematical Society, 3 (2): 447–498, doi:10.2307/1990961, ISSN 0894-0347, JSTOR 1990961, MR 1035415
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