We obtain character formulas of minimal affinizations of representations of quantum groups when the underlying simple Lie algebra is orthogonal and the support of the highest weight is contained in the first three nodes of the Dynkin diagram. We also give a framework for extending our techniques to a more general situation. In particular, for the orthogonal algebras and a highest weight supported in at most one spin node, we realize the restricted classical limit of the corresponding minimal affinizations as a quotient of a module given by generators and relations and, furthermore, show that it projects onto the submodule generated by the top weight space of the tensor product of appropriate restricted Kirillov-Reshetikhin modules. We also prove a conjecture of Chari and Pressley regarding the equivalence of certain minimal affinizations in type D 4 .
ADRIANO MOURAThe ℓ-character of a finite-dimensional U q (g)-module V is the associated element char ℓ (V ) of the integral group ring Z[P q ] which records the dimensions of the ℓ-weight spaces of V . Given λ ∈ P + q , let us denote by V q (λ) the irreducible U q (g)-module with highest ℓ-weight λ. Finding formulas for the ℓ-character of V q (λ) is still an an open problem in general. In [22], E. Frenkel and E. Mukhin defined an algorithm, now widely known as the Frenkel-Mukhin algorithm which for agiven λ ∈ P + q returns an element of Z[P q ] that was conjectured to be char ℓ (V q (λ)). The conjecture was proved for certain situations in [22], but it has recently been shown that this is not always the case [34]. However, even in the situations for which the conjecture holds, the task of translating the information given by the algorithm into general closed formulas remains a challenge. For further details on the theory of ℓ-characters, beside the aforementioned literature, we refer the reader to the very recent survey [8] and the references therein. We remark that in [32, 33] the authors give path-tableaux descriptions of Jacobi-Trudi determinants which, conjecturally, coincide with the ℓ-characters if g is of classical type. This conjecture has been partially proved if g is of type B in [24](see also [8]).Another approach for studying minimal affinizations is by considering their classical limit. Even though most of the ℓ-character information is lost in this process, it provides an effective tool to study their U q (g)-structure, i.e., their characters. The U q (g)-structure of the minimal affinizations belonging to the family of Kirillov-Reshetikhin modules was obtained in [5] partially using this approach. The proof consisted in showing that the conjectural character was both a lower and an upper bound for the character of the given Kirillov-Reshetikhin module. While the latter was proved by working with the classical limit, the proof of the former was done in the quantum context. Later on, it was shown in [11, 12] that both "upper and lower bound" parts of the proofs of the results of [5] could be performed by working with the current algebra g[t] = g ⊗ C[t...
We study finite-dimensional representations of hyper loop algebras, that is, the hyperalgebras over an algebraically closed field of positive characteristic associated to the loop algebra over a complex finite-dimensional simple Lie algebra. The main results are the classification of the irreducible modules, a version of Steinberg's tensor product theorem, and the construction of positive characteristic analogues of the Weyl modules as defined by Chari and Pressley in the characteristic zero setting. Furthermore, we start the study of reduction modulo p and prove that every irreducible module of a hyper loop algebra can be constructed as a quotient of a module obtained by a certain reduction modulo p process applied to a suitable characteristic zero module. We conjecture that the Weyl modules are also obtained by reduction modulo p. The conjecture implies a tensor product decomposition for the Weyl modules which we use to describe the blocks of the underlying abelian category.
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