Simple algorithm for factorized dynamics of the $\frak{g}_n$-automaton

Abstract

We present an elementary algorithm for the dynamics of recently introduced soliton cellular automata associated with quantum affine algebra Uq(gn) at q = 0. Forn , the rule reproduces the ball-moving algorithm in Takahashi-Satsuma's box-ball system. For non-exceptional gn other than A(1) n , it is described as a motion of particles and anti-particles which undergo pair-annihilation and creation through a neutral bound state. The algorithm is formulated without using representation theory nor crystal basis theory.

A Quantization of Box-Ball Systems

“…It is a system of particles and antiparticles on one dimensional lattice whose dynamics is governed by the L operator constructed in section 3.3. In the limit q → 0, the dynamics become deterministic and the system reduces to the D (1) n automaton [HKT3,HKT1]. Since our results are parallel with those in subsections 2.4 -2.7, we shall only give a brief sketch and omit the details.…”

“…An interesting feature in these automata is the factorization of time evolution into a product of propagation operators of particles and antiparticles with fixed color [HKT3,KTT]. This is a consequence of the factorization of the combinatorial R shown in [HKT2].…”

Periodic Cellular Automata and Bethe Ansatz

“…· The system was identified [4,3] with a solvable lattice model [1] at q = 0, which led to a direct formulation by the crystal base theory [5] and generalizations to the soliton cellular automata with quantum group symmetry [7,6]. Here we develop the approach to the periodic case in [13] further by combining the commuting transfer matrix method [1] and the Bethe ansatz [2] at q = 0.…”

GENERALIZED ENERGIES AND INTEGRABLE $D^{(1)}_{n}$ CELLULAR UTOMATON

“…In this paper we introduce generalized energies E gi : P → Z ≥0 corresponding to g i 's, and study them from the viewpoint of the integrable cellular automaton of type D (1) n [6,7]. The latter is an integrable U q (D (1) n ) vertex model at q = 0.…”