ISSN:
1573-0530
Keywords:
17B37
;
81R50
;
quantum affine algebras
;
representations
Source:
Springer Online Journal Archives 1860-2000
Topics:
Mathematics
,
Physics
Notes:
Abstract Let $$\mathfrak{g}$$ be a finite-dimensional complex simple Lie algebra and Uq( $$\mathfrak{g}$$ ) the associated quantum group (q is a nonzero complex number which we assume is transcendental). IfV is a finitedimensional irreducible representation of Uq( $$\mathfrak{g}$$ ), an affinization ofV is an irreducible representationVV of the quantum affine algebra Uq( $$\hat {\mathfrak{g}}$$ ) which containsV with multiplicity one and is such that all other irreducible Uq( $$\mathfrak{g}$$ )-components ofV have highest weight strictly smaller than the highest weight λ ofV. There is a natural partial order on the set of Uq( $$\mathfrak{g}$$ ) classes of affinizations, and we look for the minimal one(s). In earlier papers, we showed that (i) if $$\mathfrak{g}$$ is of typeA, B, C, F orG, the minimal affinization is unique up to Uq( $$\mathfrak{g}$$ )-isomorphism; (ii) if $$\mathfrak{g}$$ is of typeD orE and λ is not orthogonal to the triple node of the Dynkin diagram of $$\mathfrak{g}$$ , there are either one or three minimal affinizations (depending on λ). In this paper, we show, in contrast to the regular case, that if Uq( $$\mathfrak{g}$$ ) is of typeD 4 and λ is orthogonal to the triple node, the number of minimal affinizations has no upper bound independent of λ. As a by-product of our methods, we disprove a conjecture according to which, if $$\mathfrak{g}$$ is of typeA n,every affinization is isomorphic to a tensor product of representations of Uq( $$\hat {\mathfrak{g}}$$ ) which are irreducible under Uq( $$\mathfrak{g}$$ ) (in an earlier paper, we proved this conjecture whenn=1).
Type of Medium:
Electronic Resource
URL:
http://dx.doi.org/10.1007/BF00943278
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