Mathematics – Quantum Algebra
Scientific paper
2004-04-08
Mathematische Zeitschrift 250 (2005), no. 2, 443--473
Mathematics
Quantum Algebra
23 pages; accepted for publication in Mathematische Zeitschrift
Scientific paper
Nakajima introduced the morphism of q,t-characters for finite dimensional representation of simply-laced quantum affine algebras : it is a t-deformation of the Frenkel-Reshetikhin's morphism of q-characters (sum of monomials in infinite variables). In math.QA/0212257 we generalized the construction of q,t-characters for non simply-laced quantum affine algebras. First in this paper we prove a conjecture of math.QA/0212257 : the monomials of q and q,t-characters of standard representations are the same in non simply-laced cases (the simply-laced cases were treated by Nakajima) and the coefficients are non negative. In particular these q,t-characters can be considered as t-deformations of q-characters. In the proof we show that for quantum affine algebras of type A, B, C and quantum toroidal algebras of type A^{(1)} the l-weight spaces of fundamental representations are of dimension 1. Eventually we show and use a generalization of results of Frenkel-Reshetikhin, -Mukhin and Nakajima : for general quantum affinizations we prove that the l-weights of a l-highest weight simple module are lower than the highest l-weight in the sense of monomials.
Hernandez David
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