### Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 8 (2012), 062, 33 pages      arXiv:1109.1689      https://doi.org/10.3842/SIGMA.2012.062
Contribution to the Special Issue “Geometrical Methods in Mathematical Physics”

### Affine and Finite Lie Algebras and Integrable Toda Field Equations on Discrete Space-Time

Rustem Garifullin a, Ismagil Habibullin a and Marina Yangubaeva b
a) Ufa Institute of Mathematics, Russian Academy of Science, 112 Chernyshevskii Str., Ufa, 450077, Russia
b) Faculty of Physics and Mathematics, Birsk State Social Pedagogical Academy, 10 Internationalnaya Str., Birsk, 452452, Russia

Received April 24, 2012, in final form September 14, 2012; Published online September 18, 2012

Abstract
Difference-difference systems are suggested corresponding to the Cartan matrices of any simple or affine Lie algebra. In the cases of the algebras $A_N$, $B_N$, $C_N$, $G_2$, $D_3$, $A_1^{(1)}$, $A_2^{(2)}$, $D^{(2)}_N$ these systems are proved to be integrable. For the systems corresponding to the algebras $A_2$, $A_1^{(1)}$, $A_2^{(2)}$ generalized symmetries are found. For the systems $A_2$, $B_2$, $C_2$, $G_2$, $D_3$ complete sets of independent integrals are found. The Lax representation for the difference-difference systems corresponding to $A_N$, $B_N$, $C_N$, $A^{(1)}_1$, $D^{(2)}_N$ are presented.

Key words: affine Lie algebra; difference-difference systems; $S$-integrability; Darboux integrability; Toda field theory; integral; symmetry; Lax pair.

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