2021/05/03 by Oksana Yakimova, Yakimova, Oksana · 1 citation
Mathematics · #Advanced Topics in Algebra #Algebraic Geometry and Number Theory #Algebraic structures and combinatorial models #FOS: Mathematics #FOS: Physical sciences #Mathematical Physics (math-ph) #Representation Theory (math.RT)
paper · pdf · doi:10.48550/arxiv.2105.01020
openalex publication_date 2021/05/03 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Let \mathfrak q be a Lie algebra over a field \mathbb K and p, p∈\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]/( p) can be identified with W=\mathfrak q⋅1⊕\mathfrak q t⊕…⊕\mathfrak q tn-1. If deg (p- p) is at most 1, then the Lie brackets [ , ]p, [ , ] p induced on W by p and p, respectively, are compatible. By a general method, known as the Lenard-Magri scheme, we construct a subalgebra Z=Z(p, p)⊂ \mathcal S(W)^\mathfrak q⋅1 such that \Z,Z\p=\Z,Z\ p=0. If tr.deg \mathcal S(\mathfrak q)\mathfrak q=ind \mathfrak q and \mathfrak q has the codim-2 property, then tr.deg Z takes the maximal possible value, which is ((n-1)dim\mathfrak q)/2+((n+1)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.