Abstract
A A1 Toda system is extended via Lax pair formulations in order to probe noncommutative variables extensions. Systems, some solvable, are built using matrix generalizations.
Highlights
Noncommutative theories have been studied and probed from different viewpoints
As a first step towards a noncommutative “master system,” it has been mentioned that a noncommutative version of self-dual supersymmetric YangMills systems could provide via reductions noncommutative generalizations of integrable systems [35]
One could consider generalizations of the equations of motion based on quadratic r-matrices, such as those leading to Sklyanin brackets, where, for instance, the latter can be obtained for skew-symmetric R-operators obeying the modified Yang-Baxter equations
Summary
Noncommutative theories have been studied and probed from different viewpoints (see reviews [18, 34, 48]). An integrable isospectral deformation of an arbitrary N × N real elements matrix, which is related to a generalization of the nonperiodic Toda lattice, has been obtained [26], and integrable generalizations of the Toda chains have been formulated with Z-gradations of classical Lie algebras using a Lax formulation [54], non-Abelian versions of Toda models have been written (see [19, 20, 43] and references therein) as well as supersymmetric versions [36] and Toda-like systems [11] In this short communication, generalizations of the simple (A1) Toda system [37, 43], which is a system with 2 particles, is considered and probed at different levels. A discussion of the results and further generalizations conclude this short communication, with its main objective to present certain (to our knowledge new) examples of extensions with noncommutativity and deformations
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