Scattering Rule in Soliton Cellular Automaton associated with Crystal Base of $U_q(D_4^{(3)})$

Details Scattering Rule in Soliton Cellular Automaton associated with Crystal Base of $U_q(D_4^{(3)})$ Scattering Rule in Soliton Cellular Automaton associated with Crystal Base of $U_q(D_4^{(3)})$

2006-09-03

arxiv.org/abs/math/0609073v2

Mathematics

Quantum Algebra

25 pages

Scientific paper

10.1063/1.2721347

In terms of the crystal base of a quantum affine algebra $U_q(\mathfrak{g})$, we study a soliton cellular automaton (SCA) associated with the exceptional affine Lie algebra $\mathfrak{g}=D_4^{(3)}$. The solitons therein are labeled by the crystals of quantum affine algebra $U_q(A_1^{(1)})$. The scatteing rule is identified with the combinatorial $R$ matrix for $U_q(A_1^{(1)})$-crystals. Remarkably, the phase shifts in our SCA are given by {\em 3-times} of those in the well-known box-ball system.

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