{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,3,18]],"date-time":"2026-03-18T01:49:19Z","timestamp":1773798559950,"version":"3.50.1"},"publisher-location":"Providence, Rhode Island","reference-count":25,"publisher":"American Mathematical Society","isbn-type":[{"value":"9780821890370","type":"print"},{"value":"9781470411053","type":"print"},{"value":"9781470411077","type":"print"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":[],"published-print":{"date-parts":[[2013]]},"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 . 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. In previous work, we studied the structure 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 and determined its automorphism group\n \n \n \n \n \n A<\/mml:mi>\n u<\/mml:mi>\n \n t<\/mml:mi>\n \n F<\/mml:mi>\n <\/mml:mrow>\n <\/mml:msub>\n <\/mml:mrow>\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 {Aut_\\mathbb {F}}(\\mathsf {A}_h)<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n and the subalgebra of invariants under\n \n \n \n \n \n A<\/mml:mi>\n u<\/mml:mi>\n \n t<\/mml:mi>\n \n F<\/mml:mi>\n <\/mml:mrow>\n <\/mml:msub>\n <\/mml:mrow>\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 {Aut_\\mathbb {F}}(\\mathsf {A}_h)<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n . Here we determine the irreducible\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 -modules. In a sequel to this paper, we completely describe the 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\/conm\/602\/12027","type":"other","created":{"date-parts":[[2013,11,27]],"date-time":"2013-11-27T12:29:08Z","timestamp":1385555348000},"page":"73-98","source":"Crossref","is-referenced-by-count":11,"title":["A Parametric Family of Subalgebras of the Weyl Algebra II. Irreducible Modules"],"prefix":"10.1090","author":[{"given":"Georgia","family":"Benkart","sequence":"first","affiliation":[]},{"given":"Samuel","family":"Lopes","sequence":"additional","affiliation":[]},{"given":"Matthew","family":"Ondrus","sequence":"additional","affiliation":[]}],"member":"14","reference":[{"issue":"5","key":"1","doi-asserted-by":"publisher","first-page":"1655","DOI":"10.1080\/00927879708825943","article-title":"Invariants du corps de Weyl sous l\u2019action de groupes finis","volume":"25","author":"Alev, J.","year":"1997","journal-title":"Comm. Algebra","ISSN":"https:\/\/id.crossref.org\/issn\/0092-7872","issn-type":"print"},{"key":"2","doi-asserted-by":"publisher","first-page":"350","DOI":"10.1063\/1.1666651","article-title":"On algebraically irreducible representations of the Lie algebra \ud835\udc60\ud835\udc59(2)","volume":"15","author":"Arnal, D.","year":"1974","journal-title":"J. Mathematical Phys.","ISSN":"https:\/\/id.crossref.org\/issn\/0022-2488","issn-type":"print"},{"issue":"2","key":"3","doi-asserted-by":"publisher","first-page":"231","DOI":"10.1007\/BF01231444","article-title":"Noncommutative graded domains with quadratic growth","volume":"122","author":"Artin, M.","year":"1995","journal-title":"Invent. Math.","ISSN":"https:\/\/id.crossref.org\/issn\/0020-9910","issn-type":"print"},{"issue":"2","key":"4","first-page":"175","article-title":"Note on derivations of graded rings and classification of differential polynomial rings","volume":"40","author":"Awami, M.","year":"1988","journal-title":"Bull. Soc. Math. Belg. S\\'{e}r. A","ISSN":"https:\/\/id.crossref.org\/issn\/0037-9476","issn-type":"print"},{"issue":"1","key":"5","first-page":"75","article-title":"Generalized Weyl algebras and their representations","volume":"4","author":"Bavula, V. 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