Young wall construction of level-1 highest weight crystals over \(U_q (D_4^{(3)})\) and \(U_q (G_2^{(1)})\). (English) Zbl 07874672
Summary: With the help of path realization and affine energy function, we give a Young wall construction of level-1 highest weight crystals \(B(\lambda)\) over \(U_q (G_2^{(1)})\) and \(U_q (D_4^{(3)})\). Our construction is based on four different shapes of colored blocks, O-block, I-block, L-block and LL-block, obtained by cutting the unit cube in three different ways.
MSC:
| 17B37 | Quantum groups (quantized enveloping algebras) and related deformations |
| 17B67 | Kac-Moody (super)algebras; extended affine Lie algebras; toroidal Lie algebras |
| 16G20 | Representations of quivers and partially ordered sets |
References:
| [1] | Hong, J.; Kang, S.-J.; Lee, H., Young wall realization of crystal graphs for \(U_q( C_n^{( 1 )})\), Commun. Math. Phys., 244, 111-131, 2004 · Zbl 1108.17003 |
| [2] | Jayne, R. L.; Misra, K. C., On Demazure crystals for \(U_q( G_2^{( 1 )})\), Proc. Am. Math. Soc., 139, 2343-2356, 2011 · Zbl 1278.17018 |
| [3] | Kang, S.-J., Crystal bases for quantum affine algebras and combinatorics of Young walls, Proc. Lond. Math. Soc., 86, 29-69, 2003 · Zbl 1030.17013 |
| [4] | Kang, S-J.; Kashiwara, M.; Misra, K. C.; Miwa, T.; Nakashima, T.; Nakayashiki, A., Affine crystals and vertex models, Int. J. Mod. Phys., 7, suppl. 1A, 449-484, 1992 · Zbl 0925.17005 |
| [5] | 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 |
| [6] | Kang, S.-J.; Lee, H., Higher level affine crystals and Young walls, Algebr. Represent. Theory, 9, 593-632, 2006 · Zbl 1114.17005 |
| [7] | Kang, S.-J.; Lee, H., Crystal bases for quantum affine algebras and Young walls, J. Algebra, 322, 1979-1999, 2009 · Zbl 1196.17014 |
| [8] | Kashiwara, M., Crystalizing the q-analogue of universal enveloping algebras, Commun. Math. Phys., 133, 249-260, 1990 · Zbl 0724.17009 |
| [9] | Kashiwara, M., On crystal bases of the q-analogue of universal enveloping algebras, Duke Math. J., 63, 465-516, 1991 · Zbl 0739.17005 |
| [10] | Kashiwara, M.; Misra, K. C.; Okado, M.; Yamada, D., Perfect crystals for \(U_q( D_4^{( 3 )})\), J. Algebra, 317, 392-423, 2007 · Zbl 1140.17012 |
| [11] | Kashiwara, M.; Nakashima, T., Crystal graphs for representations of the q-analogue of classical Lie algebras, J. Algebra, 165, 295-345, 1994 · Zbl 0808.17005 |
| [12] | Kim, J. A.; Shin, D. U., Strip bundle realization of the crystals over \(U_q( G_2^{( 1 )})\), J. Math. Phys., 60, Article 111703 pp., 2019 · Zbl 1429.81037 |
| [13] | Kim, J. A.; Shin, D. U., Strip bundle realization of the crystals over \(U_q( D_4^{( 3 )})\), J. Math. Phys., 63, Article 121701 pp., 2022 · Zbl 1509.17012 |
| [14] | Misra, K. C., On Demazure crystals for \(U_q( D_4^{( 3 )})\), Contemp. Math., 392, 83-93, 2005 · Zbl 1170.17300 |
| [15] | Misra, K. C.; Mohamad, M.; Okado, M., Zero action on perfect crystals for \(U_q( G_2^{( 1 )})\), SIGMA, 6, Article 022 pp., 2010 · Zbl 1241.17017 |
| [16] | Nakashima, T.; Zelevinsky, A., Polyhedral realizations of crystal bases for quantized Kac-Moody algebras, Adv. Math., 131, 253-278, 1997 · Zbl 0897.17014 |
| [17] | Nakashima, T., Polyhedral realizations of crystal bases for integrable highest weight modules, J. Algebra, 219, 571-597, 1999 · Zbl 0967.17011 |
| [18] | Yamane, S., Perfect crystals of \(U_q( G_2^{( 1 )})\), J. Algebra, 210, 440-486, 1998 · Zbl 0929.17013 |
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