{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,13]],"date-time":"2026-04-13T14:54:31Z","timestamp":1776092071706,"version":"3.50.1"},"reference-count":33,"publisher":"American Mathematical Society (AMS)","issue":"3","license":[{"start":{"date-parts":[[2015,11,18]],"date-time":"2015-11-18T00:00:00Z","timestamp":1447804800000},"content-version":"am","delay-in-days":365,"URL":"https:\/\/www.ams.org\/publications\/copyright-and-permissions"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Trans. Amer. Math. Soc."],"abstract":"

\n An Ore extension over a polynomial algebra\n \n \n \n \n \n F<\/mml:mi>\n <\/mml:mrow>\n [<\/mml:mo>\n x<\/mml:mi>\n ]<\/mml:mo>\n <\/mml:mrow>\n \\mathbb {F}[x]<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n is either a quantum plane, a quantum Weyl algebra, or an infinite-dimensional unital associative algebra\n \n \n \n \n \n A<\/mml:mi>\n <\/mml:mrow>\n h<\/mml:mi>\n <\/mml:msub>\n \\mathsf {A}_h<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n generated by elements\n \n \n \n \n x<\/mml:mi>\n ,<\/mml:mo>\n y<\/mml:mi>\n <\/mml:mrow>\n x,y<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n , which satisfy\n \n \n \n \n y<\/mml:mi>\n x<\/mml:mi>\n \n \u2212\n \n <\/mml:mo>\n x<\/mml:mi>\n y<\/mml:mi>\n =<\/mml:mo>\n h<\/mml:mi>\n <\/mml:mrow>\n yx-xy = h<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n , where\n \n \n \n \n h<\/mml:mi>\n \n \u2208\n \n <\/mml:mo>\n \n F<\/mml:mi>\n <\/mml:mrow>\n [<\/mml:mo>\n x<\/mml:mi>\n ]<\/mml:mo>\n <\/mml:mrow>\n h\\in \\mathbb {F}[x]<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n . We investigate the family of algebras\n \n \n \n \n \n A<\/mml:mi>\n <\/mml:mrow>\n h<\/mml:mi>\n <\/mml:msub>\n \\mathsf {A}_h<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n as\n \n \n \n h<\/mml:mi>\n h<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n ranges over all the polynomials in\n \n \n \n \n \n F<\/mml:mi>\n <\/mml:mrow>\n [<\/mml:mo>\n x<\/mml:mi>\n ]<\/mml:mo>\n <\/mml:mrow>\n \\mathbb {F}[x]<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n . When\n \n \n \n \n h<\/mml:mi>\n \n \u2260\n \n <\/mml:mo>\n 0<\/mml:mn>\n <\/mml:mrow>\n h \\neq 0<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n , the algebras\n \n \n \n \n \n A<\/mml:mi>\n <\/mml:mrow>\n h<\/mml:mi>\n <\/mml:msub>\n \\mathsf {A}_h<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n are subalgebras of the Weyl algebra\n \n \n \n \n \n A<\/mml:mi>\n <\/mml:mrow>\n 1<\/mml:mn>\n <\/mml:msub>\n \\mathsf {A}_1<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n and can be viewed as differential operators with polynomial coefficients. We give an exact description of the automorphisms of\n \n \n \n \n \n A<\/mml:mi>\n <\/mml:mrow>\n h<\/mml:mi>\n <\/mml:msub>\n \\mathsf {A}_h<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n over arbitrary fields\n \n \n \n \n F<\/mml:mi>\n <\/mml:mrow>\n \\mathbb {F}<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n and describe the invariants in\n \n \n \n \n \n A<\/mml:mi>\n <\/mml:mrow>\n h<\/mml:mi>\n <\/mml:msub>\n \\mathsf {A}_h<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n under the automorphisms. We determine the center, normal elements, and height one prime ideals of\n \n \n \n \n \n A<\/mml:mi>\n <\/mml:mrow>\n h<\/mml:mi>\n <\/mml:msub>\n \\mathsf {A}_h<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n , localizations and Ore sets for\n \n \n \n \n \n A<\/mml:mi>\n <\/mml:mrow>\n h<\/mml:mi>\n <\/mml:msub>\n \\mathsf {A}_h<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n , and the Lie ideal\n \n \n \n \n [<\/mml:mo>\n \n \n A<\/mml:mi>\n <\/mml:mrow>\n h<\/mml:mi>\n <\/mml:msub>\n ,<\/mml:mo>\n \n \n A<\/mml:mi>\n <\/mml:mrow>\n h<\/mml:mi>\n <\/mml:msub>\n ]<\/mml:mo>\n <\/mml:mrow>\n [\\mathsf {A}_h,\\mathsf {A}_h]<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n . We also show that\n \n \n \n \n \n A<\/mml:mi>\n <\/mml:mrow>\n h<\/mml:mi>\n <\/mml:msub>\n \\mathsf {A}_h<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n cannot be realized as a generalized Weyl algebra over\n \n \n \n \n \n F<\/mml:mi>\n <\/mml:mrow>\n [<\/mml:mo>\n x<\/mml:mi>\n ]<\/mml:mo>\n <\/mml:mrow>\n \\mathbb {F}[x]<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n , except when\n \n \n \n \n h<\/mml:mi>\n \n \u2208\n \n <\/mml:mo>\n \n F<\/mml:mi>\n <\/mml:mrow>\n <\/mml:mrow>\n h \\in \\mathbb {F}<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n . In two sequels to this work, we completely describe the irreducible modules and derivations of\n \n \n \n \n \n A<\/mml:mi>\n <\/mml:mrow>\n h<\/mml:mi>\n <\/mml:msub>\n \\mathsf {A}_h<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n over any field.\n <\/p>","DOI":"10.1090\/s0002-9947-2014-06144-8","type":"journal-article","created":{"date-parts":[[2014,11,18]],"date-time":"2014-11-18T16:15:48Z","timestamp":1416327348000},"page":"1993-2021","source":"Crossref","is-referenced-by-count":15,"special_numbering":"958","title":["A parametric family of subalgebras of the Weyl algebra I. 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