Rustem Garifullin, Ismagil Habibullin, Marina Yangubaeva, “Affine and finite Lie algebras and integrable Toda field equations on discrete space-time”, SIGMA, 2012, Volume 8,062, 33 pp.

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Affine and finite Lie algebras and integrable Toda field equations on discrete space-time

Rustem Garifullin^a, Ismagil Habibullin^a, 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

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.

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

MSC: 35Q53; 37K40

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

Language: English

DOI: 10.3842/SIGMA.2012.062