Connectedness of Lakshmibai-Seshadri path crystals for hyperbolic Kac-Moody algebras of rank 2

Abstract

Let $\mathfrak{g}$ be a hyperbolic Kac-Moody algebra of rank 2. We give a necessary and sufficient condition for the crystal graph of the Lakshmibai-Seshadri path crystal $\mathbb{B}(\lambda )$ to be connected for an arbitrary integral weight λ.

Keywords:

• Crystal
• hyperbolic Kac–Moody algebras
• Lakshmibai–Seshadri paths

2010 MATHEMATICS SUBJECT CLASSIFICATION:

• 17B37
• 17B67
• 81R50

Acknowledgments

The author would like to thank Daisuke Sagaki, who is his supervisor, for his kind support and advice. Also, he is grateful to the referee for valuable comments and suggestions.