{"paper":{"title":"On matrix realizations of the Lie superalgebra D(2, 1 ; \\alpha)","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.RT","authors_text":"Elena Poletaeva","submitted_at":"2010-08-14T11:45:20Z","abstract_excerpt":"We obtain a realization of the Lie superalgebra $D(2, 1 ; \\alpha)$ in differential operators on the supercircle $S^{1|2}$ and in $4\\times 4$ matrices over a Weyl algebra. A contraction of $D(2, 1 ; \\alpha)$ is isomorphic to the universal central extension $\\hat{\\p\\s\\l}(2|2)$ of $\\p\\s\\l(2|2)$. We realize it in $4\\times 4$ matrices over the associative algebra of pseudodifferential operators on $S^1$. Correspondingly, there exists a three-parameter family of irreducible representations of $\\hat{\\p\\s\\l}(2|2)$ in a $(2|2)$--dimensional complex superspace."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1008.2433","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"}