Abstract
In this paper, we give a new realization of the crystal basis B(∞) using modified Nakajima monomials for the quantum finite algebras. Moreover, as an application, we obtain the image of the Kashiwara embedding Ψ ι from this realization of B(∞).
Similar content being viewed by others
References
Hoshino, A.: Polyhedral realizations of crystal bases for quantum qlgebras of finite types. J. Math. Phys. 46, 113514–113514.31 (2005)
Kang, S.-J.: Crystal bases for quantum affine algebras and combinatorics of Young walls. Proc. London Math. Soc. 86, 29–69 (2003)
Kang, S.-J., Kashiwara, M., Misra, K.C., Miwa, T., Nakashima, T., Nakayashiki, A.: Affine crystals and vertex models. Internat. J. Modern Phys. A 7(Suppl. 1A), 449–484 (1992)
Kang, S.-J., Kim, J.-A., Shin, D.-U.: Monomial realization of crystal bases for special linear Lie algebras. J. Algebra 274, 629–642 (2004)
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)
Kang, S.-J., Kim, J.-A., Shin, D.-U.: Modified Nakajima monomials and the crystal B(∞). J. Algebra 308, 524–535 (2007)
Kashiwara, M.: Crystalizing the q-analogue of universal enveloping algebras. Comm. Math. Phys. 133, 249–260 (1990)
Kashiwara, M.: On crystal bases of the q-analogue of universal enveloping algebras. Duke Math. J. 63, 465–516 (1991)
Kashiwara, M.: The crystal base and Littlemann’s refined Demazure character formula. Duke Math. J. 71, 839–858 (1993)
Kashiwara, M.: Realizations of crystals. Contemp. Math. 325, 133–139 (2003)
Kashiwara, M., Nakashima, T.: Crystal graphs for representations of the q-analogue of classical Lie algebras. J. Algebra 165, 295–345 (1994)
Kim, J.-A.: Monomial realization of crystal graphs for \(U_{q} {\left( {A^{{{\left( 1 \right)}}}_{n} } \right)}\). Math. Ann. 332, 17–35 (2005)
Littlemann, P.: Paths and root operators in representation theory. Ann. of Math. 142, 499–525 (1995)
Kim, J.-A., Shin, D.-U.: Generalized Young walls and crystal bases for \(U_{q} {\left( {A^{{{\left( 1 \right)}}}_{n} } \right)}\), submitted
Nakajima, H.: Quiver varieties and finite dimensional representations of quantumn affine algebras. J. Amer. Math. Soc. 14, 145–238 (2001)
Nakajima, H.: Quiver varieties and tensor products. Invent. Math 146, 399–449 (2001)
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, pp. 195–218. World Scientific, Singapore (2001)
Nakajima, H.: Quiver varieties and t-analogs of q-characters of quantum affine algebras. Ann. of Math. 160, 1057–1097 (2004)
Nakajima, H.: t-analogs of q-characters of quantum affine algebras of type A n , D n . Contemp. Math. 325, 141–160 (2003)
Nakashima, T., Zelevinsky, A.: Polyhedral realizations of crystal bases for quantized Kac–Moody algebras. Adv. Math. 131, 253–278 (1997)
Shin, D.-U.: Crystal bases and monomials for U q (G 2)-modules. Comm. Algebra 34, 129–142 (2006)
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Kim, JA., Shin, DU. Monomial Realization of Crystal Bases B(∞) for the Quantum Finite Algebras. Algebr Represent Theor 11, 93–105 (2008). https://doi.org/10.1007/s10468-007-9056-3
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10468-007-9056-3