{"paper":{"title":"Quantum $\\frak {gl}_\\infty$, infinite $q$-Schur algebras and their representations","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.RT"],"primary_cat":"math.QA","authors_text":"Jie Du, Qiang Fu","submitted_at":"2007-08-19T05:24:47Z","abstract_excerpt":"In this paper, we investigate the structure and representations of the quantum group ${\\mathbf{U}(\\infty)}=\\mathbf U_\\upsilon(\\frak{gl}_\\infty)$. We will present a realization for $\\mathbf{U}(\\infty)$, following Beilinson--Lusztig--MacPherson (BLM) \\cite{BLM}, and show that the natural algebra homomorphism $\\zeta_r$ from $\\mathbf{U}(\\infty)$ to the infinite $q$-Schur algebra ${\\boldsymbol{\\mathcal S}}(\\infty,r)$ is not surjective for any $r\\geq 1$. We will give a BLM type realization for the image $\\mathbf{U}(\\infty,r):=\\zeta_r(\\mathbf{U}(\\infty))$ and discuss its presentation in terms of gene"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"0708.2525","kind":"arxiv","version":2},"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"}