Takiff algebras, Toda systems, and jet transformations (Q6565662)

The author studies an extension of Toda systems to larger phase spaces attached to Takiff algebras \(\mathfrak{g}_n=\mathfrak{g}\otimes K[u]/(u^{M+1})\) for split simple Lie algebras \(\mathfrak{g}\) over the real or complex field \(K\). These Lie algebras are neither semisimple nor reductive when \(N>0\), but nonetheless have nondegenerate symmetric invariant bilinear forms.\N\NThen the author follows the usual procedure of construction of the Toda system. The Lie bracket of \(\mathfrak{g}_N\) is deformed using a classical \(r\)-matrix satisfying the modified Yang-Baxter equation, and then he considers a distinguished coadjoint orbits of the corresponding Lie groups. The orbits are symplectic manifolds parametrized by Lax matrices. In contrast with the split simple Lie algebra case, these matrices no longer have enough eigenvalues to generate an integrable system. To resolve this difficulty, the author enlarges the supply of functions by replacing ordinary traces of matrices with traces along superdiagonals in a faithful representation.