{"paper":{"title":"Graded identities of the first Weyl algebra and its generalizations","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.RA","authors_text":"J. Schwarz, P. Koshlukov, V. Futorny","submitted_at":"2026-06-09T22:40:28Z","abstract_excerpt":"We study the graded polynomial identities of the first Weyl algebra $W_1$ over an infinite field. The algebra $W_1$ satisfies no ordinary polynomial identities in characteristic 0. It admits a natural grading by the infinite cyclic group $\\mathbb{Z}$. We construct a basis of the $\\mathbb{Z}$-graded identities of $W_1$, which consists of a single identity. It expresses the fact that the degree 0 component in the grading is commutative. It is also well known that if the characteristic of the base field is $p>2$, then $W_1$ satisfies the same identities as the full matrix algebra of order $p$. In"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2606.11497","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":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2606.11497/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"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"}