Finiteness properties of subalgebras of the Weyl algebra. (English) Zbl 0573.16008
The author proves two theorems. The first states that if L is a left ideal of the nth Weyl algebra \(A_ n\) over a field k of characteristic 0 then \(k+L\) is a finitely generated subalgebra. This is a special case of an unpublished result of L. W. Small who proved it for any simple left noetherian k-algebra by using ideas from his paper with S. Montgomery [Bull. Lond. Math. Soc. 13, 33-38 (1981; Zbl 0453.16021)]. The second theorem states that if c is a regular element of positive degree in a graded k-algebra A such that \(k+Ac\) is left noetherian then its centralizer \(C_ A(c)\) is a finitely generated left k[c]-module.
Reviewer: P.F.Smith
MSC:
16P40 | Noetherian rings and modules (associative rings and algebras) |
16W50 | Graded rings and modules (associative rings and algebras) |
16W20 | Automorphisms and endomorphisms |
16P10 | Finite rings and finite-dimensional associative algebras |
16D30 | Infinite-dimensional simple rings (except as in 16Kxx) |
Keywords:
left ideal; Weyl algebra; finitely generated subalgebra; simple left noetherian k-algebra; regular element; graded k-algebra; centralizerCitations:
Zbl 0453.16021References:
[1] | Dixmier, J. 1977.Enveloping Algebras, 147–148. Amsterdam, New York: North-Holland Publishing Company. Oxford |
[2] | Joseph A., Israel J. Math 28 pp 177– (1977) · Zbl 0366.17006 · doi:10.1007/BF02759808 |
[3] | Joseph A., Small maximal subfields of the Weyl division ring |
[4] | Montgomery S., Bull. London Math. Soc 13 pp 33– (1981) · Zbl 0453.16021 · doi:10.1112/blms/13.1.33 |
[5] | Small L.W., More Affine Rings |
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