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On minimal affinizations of representations of quantum groups. (English) Zbl 1141.17011
The notion of a minimal affinization of a simple finite dimensional module of a quantum group was introduced by V. Chari [Publ. Res. Inst. Math. Sci. 31, 873–911 (1995; Zbl 0855.17010)]. An affinization is said to be special (resp. antispecial) if its \(q\)-character has a unique dominant (resp. antidominant) monomial. These types of representations were distinguished in [H. Nakajima, Ann. Math. (2) 160, 1057–1097 (2005; Zbl 1140.17015)]. In the present paper the author proves that, for the quantum groups of types \(A\), \(B\) and \(G\), all minimal affinizations are special and antispecial (Theorem 3.8). In the case of quantum groups of types \(C\), \(D\) and \(F_4\), classes of special and antispecial affinizations are identified (Theorem 3.9). It is also proven that all affinizations of quantum groups of types \(A\) and \(B\) are thin – a notion introduced by the author in [D. Hernandez, Smallness problem for quantum affine algebras and quiver varieties, Ann. Sci. Éc. Norm. Supér. (4) 41, 271–306 (2008), (Theorem 3.10)].

MSC:
17B37 Quantum groups (quantized enveloping algebras) and related deformations
17B10 Representations of Lie algebras and Lie superalgebras, algebraic theory (weights)
46L87 Noncommutative differential geometry
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