Quiver Varieties and Adjoint Crystals of Level ℓ for Type A_n⁽¹⁾

Abstract

Let B(Λ) be a level ℓ highest weight crystal of the quantum affine algebra . We construct an explicit crystal isomorphism between the geometric realization 𝔅(Λ) of the crystal B(Λ) using quiver varieties and the path realization 𝒫^ad(Λ) arising from the adjoint crystal B ^{ad, ℓ} of level ℓ.

Key Words:

• Adjoint crystals
• Crystal graphs
• Quiver varieties

2010 Mathematics Subject Classification:

• 17B10
• 17B67
• 81R10

ACKNOWLEDGMENTS

The author would like to express his deepest gratitude to Professor Seok-Jin Kang for his guidance, valuable comments, and enthusiastic encouragement. This work supported by KRF Grant 2007-341-C00001.

Notes

Communicated by K. Misra.