Abstract
We generalize Benkart-Frenkel-Kang-Lee’s adjoint crystals and describe their crystal structure for type A (1) n , C (1) n and D (2) n+1 .
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Benkart, G., Frenkel, I., Kang, S.-J., Lee, H.: Level 1 perfect crystals and path realizations of basic representations at q=0. Int. Math. Res. Not., 10312 (2006)
Chari, V.: On the fermionic formula and the Kirillov-Reshetikhin conjecture. Int. Math. Res. Not. 12, 629–654 (2001)
Hernandez, D.: Kirillov-Reshetikhin conjecture: the general case. arXiv:0704.2838
Hong, J., Kang, S.-J.: Introduction to Quantum Groups and Crystal Bases. Graduate Studies in Mathematics, vol. 42. American Mathematical Society, Providence (2002)
Humphreys, J.E.: Introduction to Lie Algebras and Representation Theory. Graduate Texts in Mathematics, vol. 9. Springer, New York (1978); second printing, revised
Kac, V.G.: Infinite-Dimensional Lie Algebras, 3rd edn. Cambridge University Press, Cambridge (1990)
Kang, S.-J., Kashiwara, M., Misra, K.C.: Crystal bases of Verma modules for quantum affine Lie algebras. Compos. Math. 92(3), 299–325 (1994)
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(3), 499–607 (1992)
Kashiwara, M.: On crystal bases of the Q-analogue of universal enveloping algebras. Duke Math. J. 63(2), 465–516 (1991)
Kashiwara, M., Misra, K.C., Okado, M., Yamada, D.: Perfect crystals for U q (D (3)4 ). J. Algebra 317(1), 392–423 (2007)
Kashiwara, M., Nakashima, T.: Crystal graphs for representations of the q-analogue of classical Lie algebras. J. Algebra 165(2), 295–345 (1994)
Okado, M.: X=M conjecture, Combinatorial aspect of integrable systems. MSJ Mem., Math. Soc. Jpn., Tokyo 17, 43–73 (2007)
Okado, M., Schilling, A.: Existence of Kirillov-Reshetikhin crystals for nonexceptional types. Represent. Theory 12, 186–207 (2008)
Schilling, A., Sternberg, P.: Finite-dimensional crystals B 2,s for quantum affine algebras of type D (1) n . J. Algebr. Comb. 23(4), 317–354 (2006)
Shimozono, M.: Affine type A crystal structure on tensor products of rectangles, Demazure characters, and nilpotent varieties. J. Algebr. Comb. 15(2), 151–187 (2002)
Yamane, S.: Perfect crystals of U q (G (1)2 ). J. Algebra 210(2), 440–486 (1998)
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Kodera, R. A generalization of adjoint crystals for the quantized affine algebras of type A (1) n , C (1) n and D (2) n+1 . J Algebr Comb 30, 491–514 (2009). https://doi.org/10.1007/s10801-009-0174-3
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DOI: https://doi.org/10.1007/s10801-009-0174-3