{"paper":{"title":"Lie--Poisson pencils related to semisimple Lie algebras: towards classification","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["nlin.SI"],"primary_cat":"math.DG","authors_text":"Andriy Panasyuk","submitted_at":"2012-08-08T11:09:39Z","abstract_excerpt":"Let $\\mathfrak{g}$ be a vector space and $[,],[,]'$ be a pair of Lie brackets on $\\mathfrak{g}$. By definition they are compatible if $[,]+[,]'$ is again a Lie bracket. Such pairs play important role in bihamiltonian and $r$-matrix formalisms in the theory of integrable systems.\n  We propose an approach to a long standing problem of classification of such pairs in the case when one of them, say $[,]$, is semisimple. It is known that any such pair is determined by a linear operator on $(\\mathfrak{g},[,])$, which is defined up to adding a derivation. We propose a special fixing of this operator "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1208.1642","kind":"arxiv","version":1},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}