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arXiv · 2207.06348

Toda systems for Takiff algebras

Michael Lau
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Abstract

We study completely integrable systems attached to Takiff algebras $\mathfrak{g}_N$, extending open Toda systems of split simple Lie algebras $\mathfrak{g}$. With respect to Darboux coordinates on coadjoint orbits $\mathcal{O}$, the potentials of the hamiltonians are products of polynomial and exponential functions. General solutions for equations of motion for $\mathfrak{g}_N$ are obtained using differential operators called jet transformations. These results are applied to a $3$-body problem based on $\mathfrak{sl}(2)$, and to an extension of soliton solutions for $A_\infty$ to associated Takiff algebras. The new classical integrable systems are then lifted to families of commuting operators in an enveloping algebra, solving a Vinberg problem and quantizing the Poisson algebra of functions on $\mathcal{O}$.

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Michael Lau. 2022-07-13. Toda systems for Takiff algebras. https://arxiv.org/abs/2207.06348

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