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Finite-dimensional representations of quantum affine algebras. (English) Zbl 0915.17011
Let \(\widehat{\mathfrak g}\) be an untwisted affine Lie algebra and \(U_q(\widehat{\mathfrak g})\) the associated quantum group over the field \(\mathbb{C} ({\mathfrak q})\) of rational functions of an indeterminate \(q\). Let \(\lambda_1, \dots, \lambda_n\) be the fundamental weights of the underlying finite-dimensional complex simple Lie algebra \({\mathfrak g}\), and let \(V(\lambda_i)_a\) be the evaluation representation with parameter \(a\in\mathbb{C} (q)^\times\). The authors propose a conjecture on the condition for irreducibility of a tensor product \(V(\lambda_{i_1})_{a_1} \otimes \cdots \otimes V(\lambda_{i_N})_{a_N}\).
This has the following consequence: this tensor product is irreducible iff the \(R\)-matrix \(V(\lambda_{i_r})_x\otimes V(\lambda_{i_s})_y\to V(\lambda_{i_s})_y \otimes V(\lambda_{i_r})_x\) has no pole at \((x,y)= (a_r,a_s)\) for all \(r<s\). Moreover, when this \(R\)-matrix has no pole at \((x,y)= (a_r,a_s)\) for all \(r>s\), the submodule of \(v(\lambda_{i_1})_{a_1} \otimes\cdots \otimes V(\lambda_{i_N})_{a_N}\) generated by the tensor product of highest weight vectors is irreducible, and every finite-dimensional irreducible integrable module is obtained in this way.
The main conjecture is proved when \({\mathfrak g}\) is of type \(A_n\) and \(C_n\), using crystal basis techniques.

17B37 Quantum groups (quantized enveloping algebras) and related deformations
17B67 Kac-Moody (super)algebras; extended affine Lie algebras; toroidal Lie algebras
Full Text: DOI arXiv
[1] Beck, J., Braid group action and quantum affme algebras, Camm. Math. Phys., 165 (1994) 555-568. · Zbl 0807.17013
[2] Chari, V. and Pressley, A., A Guide to Quantum Groups, Cambridge University Press, 1994. · Zbl 0839.17009
[3] _ . Quantum affine algebras and their representations, Canad. Math. Soc. Conf. Proc., 16 (1995). 59-78. · Zbl 0855.17009
[4] _ __, Twisted quantum affine algebras, q-alg/9611002.
[5] Date, E. and Okado, M., Calculation of excitation spectra of the spin model related with the vector represention of the quantized affine algebra of type An, Interim. J Modern Phys. A, 9 (1994), 399-417. · Zbl 0905.17004 · doi:10.1142/S0217751X94000194
[6] Drinfeld, V, G., Quantum groups, Proc. ICM 86 (Berkeley) , AMS, 1, 798-820.
[7] [ 8 ] Frenkel, I. and Reshetikhin, N., Quantum affine algebras and holonomic difference equations, Camm. Math. Phys., 146 (1992), 1-60. · Zbl 0760.17006 · doi:10.1007/BF02099206
[8] Gandenberger, G. M. and MacKay, N. J., Exact S-matrices for dj?+i affine Toda solitons and their bounded states. Nuclear Phys. £457 (1995) , 240-272. · Zbl 0996.37505 · doi:10.1016/0550-3213(95)00462-9
[9] Jimbo, M., A ^-difference analogue of U(Q) and the Yang Baxter equation, Lett. Math. Phys., 10 (1985) , 63-69. · Zbl 0587.17004 · doi:10.1007/BF00704588
[10] Jimbo, M. and Miwa, T., Algebraic Analysis of Solvable Lattice Models, CBMS Regional Conf. series in Math., 85, AMS, 1995.
[11] Kashiwara, M., On crystal bases of the ^-analogue of universal enveloping algebras, Duke Math. J., 6\( (1991), 465-516.\) · Zbl 0739.17005 · doi:10.1215/S0012-7094-91-06321-0
[12] _ _ __. Global crystal bases of quantum groups, Duke Math. J., 69 (1993) . 455-485.
[13] _ , Crystal bases of modified quantized enveloping algebra, Duke Math. J., 73 (1994) , 383-413. · Zbl 0794.17009 · doi:10.1215/S0012-7094-94-07317-1
[14] Kang, S.-J., Kashiwara, M., Misra, K., Miwa, T., Nakashima, T.,and Nakayashiki, A., Affine crystal and Vertex models, Internal. J. Modern Phys. A 7, Suppl. 1A (1992) , 449-484. · Zbl 0925.17005 · doi:10.1142/S0217751X92003896
[15] ___ , Perfect crystals of quantum affine Lie algebra, Durk Math.J., 68 (1992) 499-607.
[16] Kashiwara, M. and Nakashima, T., Crystal graphs for representations of the ^-analogue of classical Lie algebras./. Algebra, 165 (1994). 295-345. · Zbl 0808.17005 · doi:10.1006/jabr.1994.1114
[17] Tanisaki, T., Killing forms, Harish- Chandra isomorphisms, and universal R -matrices for quantum algebras, Infinite Analysis, Proceedings of the RIMS Project 1991, Part B, Adv. Ser. Math. Phys. 16, World Scientific, (1992) 941-962. · Zbl 0870.17007
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