{"paper":{"title":"On the variety of 1-dimensional representations of finite $W$-algebras in low rank","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.RA"],"primary_cat":"math.RT","authors_text":"Jonathan Brown, Simon M. Goodwin","submitted_at":"2017-08-29T07:07:15Z","abstract_excerpt":"Let $\\mathfrak g$ be a simple Lie algebra over $\\mathbb C$ and let $e \\in \\mathfrak g$ be nilpotent. We consider the finite $W$-algebra $U(\\mathfrak g,e)$ associated to $e$ and the problem of determining the variety $\\mathcal E(\\mathfrak g,e)$ of 1-dimensional representations of $U(\\mathfrak g,e)$. For $\\mathfrak g$ of low rank, we report on computer calculations that have been used to determine the structure of $\\mathcal E(\\mathfrak g,e)$, and the action of the component group $\\Gamma_e$ of the centralizer of $e$ on $\\mathcal E(\\mathfrak g,e)$. As a consequence, we provide two examples where "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1708.08609","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"}