Elliptic Central Characters And Blocks Of Finite Dimensional Representations Of Quantum Affine Algebras

The category of finite dimensional (type 1) representations of a quantum affine algebra U q(ĝ) is not semisimple. However, as any abelian category with finite-length objects, it admits a unique decomposition in a direct sum of indecomposable subcategories (blocks). We define the elliptic central character of a finite dimensional (type 1) representation of U q(ĝ) and show that the block decomposition of this category is parametrized by these elliptic central characters. © Copyright 2005, American Mathematical Society.7346373Akasaka, T., Kashiwara, M., Finite dimensional representations of quantum affine algebras (1997) Publ. Res. Inst. Math. Sci., 33 (5), pp. 839-867. , MR 99d:17017Beck, J., Braid group action and quantum affine algebras (1993) Comm. Math. Phys., 165, pp. 555-568. , MR 95i:17011Chari, V., Braid group actions and tensor products (2002) Int. Math. Res. Not., pp. 357-382. , MR 2003a:17014Chari, V., Pressley, A., Fundamental representations of Yangians and singularities of R-matrices (1991) J. Reine Angew. Math. (Crelle), 417, pp. 87-128. , MR 92h:17010Chari, V., Pressley, A., Quantum affine algebras and affine Hecke algebras (1996) Pacific J. Math., 174, pp. 295-326. , MR 97i:17011Chari, V., Pressley, A., (1994) A Guide to Quantum Groups, , Cambridge University Presscorrected reprint of the 1994 original. MR 95j:17010MR 96h:17014Chari, V., Pressley, A., Quantum affine algebras and their representations (1995) CMS Conf. Proc (Banff, AB, 1994), 16, pp. 59-78. , Representations of groups. hep-th/9411145. MR 96j:17009Chari, V., Pressley, A., Yangians, integrable systems and Dorey's rule (1996) Comm. Math. Phys., 181, pp. 265-302. , hep-th/9505085. MR 92h:17010Chari, V., Pressley, A., Weyl modules for classical and quantum affine algebras (2001) Represent. Theory, 5, pp. 191-223. , MR 2002g:17027Drinfeld, V., Hopf algebras and the quantum Yang-Baxter equations (1985) Soviet Math. Dokl., 32, pp. 254-258Drinfeld, V., A new realization of Yangians and quantized affine algebras (1988) Soviet Math. Dokl., 36, pp. 212-216. , MR 88j:17020Etingof, P., Frenkel, I., Kirillov Jr., A., Lectures on representation theory and Knizhnik-Zamolodchikov equations (1998) Mathematical Surveys and Monographs, 58. , AMS. MR 2001b:32028Etingof, P., Moura, A., On the Quantum Kazhdan-Luzstig Functor, , preprint. QA/0203003Frenkel, E., Mukhin, E., Combinatorics of -characters of finite-dimensional representations of quantum affine algebras (2001) Comm. Math. Phys., 216, pp. 23-57. , MR 2002c:17023Frenkel, E., Reshetikhin, N., Deformations of -algebras associated to simple Lie algebras (1998) Comm. Math. Phys., 197, pp. 1-32. , MR 99k:17028Frenkel, E., Reshetikhin, N., The q-characters of of representation of quantum affine algebras and deformations of W-algebras (2000) Contemporary Math., 248, pp. 163-205. , MR 2002f:17022Frenkel, I.B., Reshetikhin, N.Yu., Quantum affine algebras and holonomic difference equations (1992) Comm. Math. Phys., 146 (1), pp. 1-60. , MR 94c:17024Jimbo, M.A., A q-difference analogue of U(g) and the Yang-Baxter equation (1985) Lett. Math. Phys., 10, pp. 63-69. , MR 86k:17008Jimbo, M.A., A q-analogue of U(gl(n + 1)), Hecke algebra and the Yang-Baxter equation (1986) Lett. Math. Phys., 11, pp. 257-1252. , MR 87k:17011Kashiwara, M., On level zero representations of quantized affine algebras (2002) Duke Math. J., 112, pp. 117-195. , MR 2002m:17013Kazhdan, D., Soibelman, Y., Representations of quantum affine algebras (1995) Selecta Math. (N.S.), 1 (3), pp. 537-595. , MR 96m:17031Khoroshkin, S., Tolstoy, V., Extremal projector and universal R-matrix for quantized contragredient Lie (super) algebras (1992) Quantum Groups and Related Topics, pp. 23-32. , MR 94b:17006Moura, A., Elliptic Dynamical R-matrices from the Monodromy of the Q-Knizhnik-Zamolodchikov Equations for the Standard Representation of Uq(5̃l n+1), , preprint, rt/0112145Nakajima, H., Extremal Weight Modules of Quantum Affine Algebras, , preprint. QA/020418