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# Nonlinear Sciences > Exactly Solvable and Integrable Systems

# Title: Integrable triples in semisimple Lie algebras

(Submitted on 23 Dec 2020 (v1), last revised 8 Jan 2021 (this version, v2))

Abstract: We classify all integrable triples in simple Lie algebras, up to equivalence. The importance of this problem stems from the fact that for each such equivalence class one can construct the corresponding integrable hierarchy of bi-Hamiltonian PDE. The simplest integrable triple $(f,0,e)$ in $\mathfrak{sl}_2$ corresponds to the KdV hierarchy, and the triple $(f,0,e_\theta)$, where $f$ is the sum of negative simple root vectors and $e_\theta$ is the highest root vector of a simple Lie algebra, corresponds to the Drinfeld-Sokolov hierarchy.

## Submission history

From: Mamuka Jibladze [view email]**[v1]**Wed, 23 Dec 2020 19:03:12 GMT (46kb,D)

**[v2]**Fri, 8 Jan 2021 06:27:56 GMT (46kb,D)

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