\(d\)-critical modules of length 2 over Weyl algebras. (English) Zbl 0797.16035
Adapting J. T. Stafford’s [Invent. Math. 79, 619-638 (1985; Zbl 0558.17011)] construction of non-holonomic simple modules over the complex Weyl algebras \(A_ n\), \(n \geq 2\), the author constructs non-simple \(A_ n\)-modules of finite length that are also critical with respect to the Gelfand-Kirillov dimension. Thus, Tauvel’s question whether critical modules of finite length over a solvable Lie algebra are necessarily simple has a negative answer already for nilpotent Lie algebras. This is interesting in view of the fact that the answer to this question is positive for commutative polynomial rings. The paper also presents some other interesting properties of the modules constructed, one being that their simple quotients are holonomic.
Reviewer: G.Krause (Winnipeg)
MSC:
16S30 | Universal enveloping algebras of Lie algebras |
17B30 | Solvable, nilpotent (super)algebras |
16P90 | Growth rate, Gelfand-Kirillov dimension |
16D60 | Simple and semisimple modules, primitive rings and ideals in associative algebras |
Keywords:
non-holonomic simple modules; complex Weyl algebras; Gelfand-Kirillov dimension; critical modules of finite length; solvable Lie algebra; nilpotent Lie algebras; simple quotientsCitations:
Zbl 0558.17011References:
[1] | G.R. Krause and T.H. Lenagen,Growth of algebras and Gelfand-Kirillov dimension, Research Notes in Math. 116, Pitman, London, 1985. · Zbl 0564.16001 |
[2] | J.C. McConnell and J.C. Robson,Homomorphisms and extensions of modules over certain differential polynomial rings, J. Algebra26 (1973), 319–342. · Zbl 0266.16031 · doi:10.1016/0021-8693(73)90026-4 |
[3] | J.T. Stafford,Non-holonomic modules over Weyl algebras and enveloping algebras, Invent. Math.79 (1985), 619–638. · Zbl 0558.17011 · doi:10.1007/BF01388528 |
[4] | P. Tau vel,Polarisations et representations des algebres de Lie résolubles, Bull. Soc. Math. France Serie 1111 (1976), 33–44. |
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