CRYSTALLIZING THE Q-ANALOG OF UNIVERSAL ENVELOPING-ALGEBRAS

For an irreducible representation of the q-analogue of a universal enveloping algebra, one can find a canonical base at q=0, named crystal base (conjectured in a general case and proven for An, Bn, Cn and Dn). The crystal base has a structure of a colored oriented graph, named crystal graph. The crystal base of the tensor product (respectively the direct sum) is the tensor product (respectively the union) of the crystal base. The crystal graph of the tensor product is also explicitly described. This gives a combinatorial description of the decomposition of the tensor product into irreducible components. © 1990 Springer-Verlag.