Characters and blocks for finite-dimensional representations of quantum affine algebras.

*(English)*Zbl 1074.17004The authors study the category \(\mathcal{C}_{q}\) of finite-dimensional representations of a quantum loop algebra \(\mathcal{U}\). The aim is to research and to put into a common representation-theoretic framework, two kinds of characters which have been associated to an object of \(\mathcal{C}_{q}\). One is the notion of \(q\)-characters defined in [E. Frenkel and N. Reshetikhin, Contemp. Math. 248, 163–205 (1999; Zbl 0973.17015)], which is analogous in this context, to the usual notion of a character of a finite-dimensional representation of a simple Lie algebra. The other is the notion of the elliptic character defined in [P. I. Etingof and A. A. Moura, Represent. Theory 7, 346–373 (2003; Zbl 1066.17005)], which plays the role of the central character for representations of semisimple Lie algebras.

In this paper, the methods, which avoid the use of the \(\mathcal{R}-\)matrix, allow one to determine the blocks for all \(q\) not a root of unity. One of the conjectures of [E. Frenkel and N. Reshetikhin, loc. cit.], proved in [E. Frenkel and E. Mukhin, Commun. Math. Phys. 216, No.1, 23–57 (2001; Zbl 1051.17013)], is that the character of simple objects of \(\mathcal{C_{q}}\) has a certain cone-like form. The authors prove this result for the quantum affine algebras associated to a classical Lie algebra in a representation-theoretic way rather than in a combinatorial fashion. They are actually able to prove a stronger version of the result, which allows us to give a formula for the \(q\)-characters of the fundamental representations in terms of the braid group action defined in [V. Chari, Int. Math. Res. Not. 2002, No. 7, 357–382 (2002; Zbl 0990.17009)].

In this paper, the methods, which avoid the use of the \(\mathcal{R}-\)matrix, allow one to determine the blocks for all \(q\) not a root of unity. One of the conjectures of [E. Frenkel and N. Reshetikhin, loc. cit.], proved in [E. Frenkel and E. Mukhin, Commun. Math. Phys. 216, No.1, 23–57 (2001; Zbl 1051.17013)], is that the character of simple objects of \(\mathcal{C_{q}}\) has a certain cone-like form. The authors prove this result for the quantum affine algebras associated to a classical Lie algebra in a representation-theoretic way rather than in a combinatorial fashion. They are actually able to prove a stronger version of the result, which allows us to give a formula for the \(q\)-characters of the fundamental representations in terms of the braid group action defined in [V. Chari, Int. Math. Res. Not. 2002, No. 7, 357–382 (2002; Zbl 0990.17009)].

Reviewer: Li Fang (Hangzhou)