A bi-Hamiltonian nature of the Gaudin algebras

Abstract: Let $\mathfrak q$ be a Lie algebra over a field $\mathbb K$ and $p,\tilde p\in\mathbb K[t]$ two different normalised polynomials of degree at least 2. As vector spaces both quotient Lie algebras $\mathfrak q[t]/(p)$ and $\mathfrak q[t]/(\tilde p)$ can be identified with $W=\mathfrak q{\cdot}1\oplus\mathfrak q\bar t\oplus\ldots\oplus\mathfrak q\bar t^{n-1}$. If $\mathrm{deg}\,(p-\tilde p)$ is at most 1, then the Lie brackets $[\,\,,\,]p$, $[\,\,,\,]{\tilde p}$ induced on $W$ by $p$ and $\tilde p$, respectively, are compatible. By a general method, known as the Lenard-Magri scheme, we construct a subalgebra $Z=Z(p,\tilde p)\subset {\mathcal S}(W)^{\mathfrak q{\cdot}1}$ such that ${Z,Z}p={Z,Z}{\tilde p}=0$. If ${\mathrm{tr.deg\,}}{\mathcal S}(\mathfrak q)^{\mathfrak q}=\mathrm{ind}\,\mathfrak q$ and $\mathfrak q$ has the codim-$2$ property, then ${\mathrm{tr.deg\,}} Z$ takes the maximal possible value, which is $((n-1)\dim\mathfrak q)/2+((n+1)\mathrm{ind}\,\mathfrak q)/2$. If $\mathfrak q=\mathfrak g$ is semisimple, then $Z$ contains the Hamiltonians of a suitably chosen Gaudin model. Therefore, in a non-reductive case, we obtain a completely integrable generalisation of Gaudin models.