Minimal affinization of representations of quantum groups: The irregular case
If \({\mathfrak g}\) denotes a finite-dimensional complex simple Lie algebra, and \(V\) a finite-dimensional irreducible representation of \(U_q({\mathfrak g})\), then by definition an affinization \(\widehat{V}\) is an irreducible representation of the quantum affine algebra \(U_q(\widehat{\mathfrak g})\) containing \(V\) with multiplicity one and the property that the highest weights of all other irreducible \(U_q({\mathfrak g})\)-components of \(\widehat{V}\) are strictly smaller than the highest weight \(\lambda\) of \(V\). In this paper the authors deal with the special case that \(U_q({\mathfrak g})\) is of type \(D_4\) and \(\lambda\) is orthogonal to the simple root corresponding to the triple node in the Dynkin diagram of \({\mathfrak g}\). In this ``irregular'' case it is proved, that the number of affinizations, which are minimal with respect to a natural ordering in the set of affinizations, increases with \(\lambda\) and has no upper bound independent of \(\lambda\). This in contrast to the ``regular'' cases in which generally there is a unique minimal affinization up to \(U_q({\mathfrak g})\) isomorphisms. The proof relies on certain non minimal affinizations in the \(A_n\) case, and by the way a conjecture of the authors stated in an earlier paper [ibid. 30, 131-145 (1994; Zbl 0795.17008)] is disproved by a counterexample for \({\mathfrak g}\) of type \(A_2\).
scientific article; zbMATH DE number 874334
scientific article