Geometric Crystal and Tropical R for D^(1)_n
Quantum Algebra
2018-10-24 v1 Mathematical Physics
math.MP
Exactly Solvable and Integrable Systems
Abstract
We construct a geometric crystal for the affine Lie algebra D^{(1)}_n in the sense of Berenstein and Kazhdan. Based on a matrix realization including a spectral parameter, we prove uniqueness and explicit form of the tropical R, the birational map that intertwines products of the geometric crystals. The tropical R commutes with geometric Kashiwara operators and satisfies the Yang-Baxter equation. It is subtraction-free and yields a piecewise linear formula of the combinatorial R for crystals upon ultradiscretization.
Keywords
Cite
@article{arxiv.math/0208239,
title = {Geometric Crystal and Tropical R for D^(1)_n},
author = {Atsuo Kuniba and Masato Okado and Taichiro Takagi and Yasuhiko Yamada},
journal= {arXiv preprint arXiv:math/0208239},
year = {2018}
}
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41 pages