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Quiver varieties and path realizations arising from adjoint crystals of type $ A_n^{(1)}$


Authors: Seok-Jin Kang and Euiyong Park
Journal: Trans. Amer. Math. Soc. 363 (2011), 5341-5366
MSC (2010): Primary 05E10, 17B67, 81R10
DOI: https://doi.org/10.1090/S0002-9947-2011-05246-3
Published electronically: May 9, 2011
MathSciNet review: 2813418
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Abstract: Let $ B(\Lambda_0)$ be the level 1 highest weight crystal of the quantum affine algebra $ U_q(A_n^{(1)})$. We construct an explicit crystal isomorphism between the geometric realization $ \mathbb{B}(\Lambda_0)$ of $ B(\Lambda_0)$ via quiver varieties and the path realization $ {\mathcal P}^{\textrm{ad}}(\Lambda_0)$ of $ B(\Lambda_0)$ arising from the adjoint crystal $ B^{\textrm{ad}}$.


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Additional Information

Seok-Jin Kang
Affiliation: Department of Mathematical Sciences and Research Institute of Mathematics, Seoul National University, 599 Gwanak-ro, Gwanak-gu, Seoul 151-747, Korea
Email: sjkang@math.snu.ac.kr

Euiyong Park
Affiliation: Department of Mathematical Sciences, Seoul National University, 599 Gwanak-ro, Gwanak-gu, Seoul 151-747, Korea
Address at time of publication: School of Mathematics, Korea Institute for Advanced Study, 85 Hoegiro, Dongdaemun-gu, Seoul 130-722, Korea
Email: pwy@snu.ac.kr, eypark@kias.re.kr

DOI: https://doi.org/10.1090/S0002-9947-2011-05246-3
Keywords: Adjoint crystal, crystal graph, Kac-Moody algebra, path realization, quiver variety
Received by editor(s): September 30, 2009
Received by editor(s) in revised form: November 12, 2009, and November 13, 2009
Published electronically: May 9, 2011
Additional Notes: The research of both authors was supported by KRF Grant # 2007-341-C00001.
The second author’s research was supported by BK21 Mathematical Sciences Division.
Article copyright: © Copyright 2011 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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