×

Monomial realization of crystal graphs for \(U_q(A_n^{(1)})\). (English) Zbl 1155.17304

Summary: We give a new realization of crystal graphs for irreducible highest weight modules over \(U_q(A_n^{(1)})\) in terms of the monomials introduced by H. Nakajima. We also discuss the natural connection between the monomial realization and other known realizations, path realization and Young wall realization.

MSC:

17B37 Quantum groups (quantized enveloping algebras) and related deformations
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Drinfel?d, V.G.: Hopf algebras and the quantum Yang-Baxter equation. Soviet Math. Dokl. 32, 254-258 (1985)
[2] Hong, J., Kang, S.-J.: Introduction to Quantum Groups and Crystal Bases. Graduate Studies in Mathematics 42, Amer. Math. Soc., 2002 · Zbl 1134.17007
[3] Jimbo, M.: A q-difference analogue of and the Yang-Baxter equation. Lett. Math. Phys. 10, 63-69 (1985) · Zbl 0587.17004 · doi:10.1007/BF00704588
[4] Jimbo, M., Misra, K.C., Miwa, T., Okado, M.: Combinatorics of Representation of at q=0. Comm. Math. Phys. 136, 543-566 (1991) · Zbl 0749.17015 · doi:10.1007/BF02099073
[5] Kang, S.-J.: Crystal bases for quantum affine algebras and combinatorics of Young walls. Proc. London Math. Soc., 86, 29-69 (2003) · Zbl 1030.17013 · doi:10.1112/S0024611502013734
[6] Kang, S.-J., Lee, H.: Higher level affine crystal and Young walls. Preprint, arXiv:Math. QA/0310430 · Zbl 1114.17005
[7] Kashiwara, M.: Crystalizing the q-analogue of universal enveloping algebras. Comm. Math. Phys. 133, 249-260 (1990) · Zbl 0724.17009 · doi:10.1007/BF02097367
[8] Kashiwara, M.: On crystal bases of the q-analogue of universal enveloping algebras. Duke Math. J. 63, 465-516 (1991) · Zbl 0739.17005 · doi:10.1215/S0012-7094-91-06321-0
[9] Kashiwara, M.: Realizations of crystals. Contemp. Math. 325, Amer. Math. Soc., 133-139 (2003) · Zbl 1066.17006
[10] Kang, S.-J., Kashiwara, M., Misra, K.C.: Crystal bases of verma modules for the quantum affine Lie algebras. Composito Math. 92, 299-325 (1994) · Zbl 0808.17007
[11] Kang, S.-J., Kashiwara, M., Misra, K.C., Miwa, T., Nakashima, T., Nakayashiki, A.: Affine crystals and vertex models. Int. J. Mod. Phys. A. Suppl. 1A, 449-484 (1992) · Zbl 0925.17005 · doi:10.1142/S0217751X92003896
[12] Kang, S.-J., Kashiwara, M., Misra, K.C., Miwa, T., Nakashima, T., Nakayashiki, A.: Perfect crystals of quantum affine Lie algebras. Duke Math. J. 68, 499-607 (1992) · Zbl 0774.17017 · doi:10.1215/S0012-7094-92-06821-9
[13] Kang, S.-J., Kim, J.-A., Shin, D.-U.: Monomial realization of crystal bases for special linear Lie algebras. Preprint math. RT/0303232, to appear in J. Algebra. · Zbl 1055.17006
[14] Kang, S.-J., Kim, J.-A., Shin, D.-U.: Crystal bases for quantum classical algebras and Nakajima?s monomials. Publ. Res. Inst. Math. Sci. 40, 758-791 (2004) · Zbl 1072.17007 · doi:10.2977/prims/1145475492
[15] Misra, K., Miwa, T. Crystal base for the basic representation of . Comm. Math. Phys. 134, 79-88 (1990) · Zbl 0724.17010
[16] Nakajima, H.: Quiver varieties and finite dimensional representation s of quantumn affine algebras. J. Amer. Math. Soc.,14, 145-238 (2001) · Zbl 0981.17016
[17] Nakajima, H.: Quiver varieties and tensor products. Invent. Math. 146, 399-449 (2001) · Zbl 1023.17008 · doi:10.1007/PL00005810
[18] Nakajima, H.: t-analogs of q-characters of quantum affine algebras of type An, Dn. Contemp. Math. 325 (2003), Amer. Math. Soc., 141-160 · Zbl 1098.17013
[19] Nakajima, H.: t-analogue of the q-characters of finite dimensional representations of quantum affine algebras. in ?Physics and Combinatorics?, Proceedings of the Nagoya 2000 International Workshop, World Scientific, 2001, 195-218
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.