Takiff algebras, Toda systems, and jet transformations

Abstract

We study a family of completely integrable systems attached to Takiff algebras gN, extending Toda systems for split simple Lie algebras g. With respect to Darboux coordinates on coadjoint orbits O, the potentials of the hamiltonians are products of polynomial and exponential functions. Explicit general solutions for their equations of motion are obtained using differential operators called jet transformations. The classical integrable systems are then lifted to families of commuting operators in an enveloping algebra, quantizing the Poisson algebra of functions on O. These results are illustrated with concrete examples, including a 3-body problem based on sl(2) and an extension of soliton solutions for A∞ to associated Takiff algebras.