$A_n^{(1)}$-Geometric Crystal corresponding to Dynkin index $i=2$ and its ultra-discretization

Kailash C. Misra; Toshiki Nakashima

$A_n^{(1)}$-Geometric Crystal corresponding to Dynkin index $i=2$ and its ultra-discretization

Quantum Algebra 2012-09-21 v1

Authors: Kailash C. Misra , Toshiki Nakashima

Abstract

Let $g$ be an affine Lie algebra with index set $I = \{0, 1, 2,..., n\}$ and $g^L$ be its Langlands dual. It is conjectured that for each $i \in I \setminus \{0\}$ the affine Lie algebra $g$ has a positive geometric crystal whose ultra-discretization is isomorphic to the limit of certain coherent family of perfect crystals for $g^L$ . We prove this conjecture for $i=2$ and $g = A_n^{(1)}$ .

Keywords

affine Lie algebra geometric crystal ultra-discretization Langlands dual perfect crystals Dynkin index $A_n^{(1)}$

Cite

@article{arxiv.1209.4565,
  title  = {$A_n^{(1)}$-Geometric Crystal corresponding to Dynkin index $i=2$ and its ultra-discretization},
  author = {Kailash C. Misra and Toshiki Nakashima},
  journal= {arXiv preprint arXiv:1209.4565},
  year   = {2012}
}

Comments

18 pages. arXiv admin note: substantial text overlap with arXiv:1003.1242, arXiv:0712.3894, arXiv:math/0612858, arXiv:0911.3556

$A_n^{(1)}$-Geometric Crystal corresponding to Dynkin index $i=2$ and its ultra-discretization

Abstract

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