zbMATH — the first resource for mathematics

A generalization of Quillen’s lemma and its application to the Weyl algebras. (English) Zbl 0366.17006
Let \(k\) be a commutative field. Let \(U\) be a \(k\)-algebra with a filtration (exhausting \(U)\) such that the associated graded algebra is commutative and finitely generated. (For example: \(U\) a Weyl algebra, or, more generally, a homomorphic image of an enveloping algebra of a finite dimensional Lie algebra.) Let \(M\) be a finitely generated \(U\)-module.
Quillen’s lemma states that if \(M\) is simple, then all endomorphisms of \(M\) are algebraic over \(M\). In joint work with R. Rentschler, the author proves a generalization which may be stated as follows: If \(M\) has Krull-dimension \(m\) (in the sense of Gabriel-Rentschler), then every commutative subalgebra of \(\operatorname{Hom}_U(M,M)\) has Krull dimension \(\le m\). (The case \(m=0\) gives Quillen’s original lemma.) Now assume \(\operatorname{char}k=0\) and let \(A_n\) denote the Weyl algebra with \(2n\) generators \(p_1,\ldots,p_n, q_1,\ldots,q_n\) over \(k\) and the usual defining relations (i.e. the commutators of the generators are zero except \(p_i q_i - q_i p_i =1\) for \(i =1,\ldots,n)\).
The major part of this paper is a study on certain special – but very interesting – questions about \(A_n\) basing on, resp. extending, various works of Bernstein, Björk, Dixmier, Makar-Limanov, and the present author. Only some typical examples for the results obtained can be reviewed here.
Consider \(n\) algebraically independent, commuting elements \(x_1, \ldots, x_n\in A_n\). (After Makar-Limanov, \(n\) is the maximal number of such elements.) For \(\lambda = (\lambda_1, \ldots, \lambda_n)\in k^n\) let \(D_\lambda\) denote the linear space of elements \(y\in A_n\) such that \(x_iy-yx_i = \lambda_iy\) \((i = 1, \ldots,n)\). Let \(\Lambda\) denote the set of those \(\lambda\), where this “eigen-space” \(D_\lambda\) is nonzero. Clearly, \(\Lambda\) is an additive semigroup in \(k^n\).
Theorem: \(\Lambda\) is even a subgroup of \(k^n\).
The proof of this result depends essentially on the generalized Quillen’s lemma [whereas the major part of this study on \(A_n\) does not]. By a Gel’fand-Kirillov-dimension argument, the author shows that \(\Lambda\) is isomorphic to a subgroup of \(\mathbb Q^n\). Now consider the special case where the \(x_i\) are “semisimple-elements”, i.e. commutation by \(x_i\) is locally finite and semisimple. For \(k\) algebraically closed this means just \(A= \sum_\lambda D_\lambda\). For this case it is proved independently that \(\Lambda\cong \mathbb Z^n\), and the following main result is obtained:
There exists an automorphism \(\varphi\) of the field of fractions of \(A_n\), such that \(\varphi(x_i) = q_ip_i+\alpha_i\) for some \(\alpha_i\in k\) \((i =1, 2, \ldots, n)\).
For \(n=1\), this result is due to Dixmier, even in the sharper version with \(\varphi\) an automorphism of \(A_n\) itself. The author seems to expect that this sharper version holds also for \(n>1\).

17B35 Universal enveloping (super)algebras
16D60 Simple and semisimple modules, primitive rings and ideals in associative algebras
16P50 Localization and associative Noetherian rings
16Kxx Division rings and semisimple Artin rings
Full Text: DOI
[1] I. N. Bernstein,The analytic continuation of generalized functions with respect to a parameter, Funkcional. Anal. i Priložen.6 (1972), 26–40.
[2] J.-E. Björk,Some problems about algebras of differential operators, Vortragsberichte, Oberwolfach, April 1973.
[3] W. Borho and H. Kraft,Über die Gelfand-Kirillov-Dimension, Math. Ann.220 (1976), 1–24. · Zbl 0313.17004 · doi:10.1007/BF01354525
[4] W. Borho and R. Rentschler,Oresche Teilmengen in Einhüllenden Algebren, Math. Ann.217 (1975), 201–210. · Zbl 0304.17003 · doi:10.1007/BF01436171
[5] J. Dixmier,Sur les algèbres de Weyl, Bull. Soc. Math. France96 (1968), 109–242. · Zbl 0165.04901
[6] J. Dixmier,Algèbres enveloppantes, cahiers scientifiques, XXXVII, Gauthier-Villars, Paris, 1974.
[7] A. Joseph,Derivations of Lie brackets and canonical quantisation, Comm. Math. Phys.17 (1970), 210–232. · Zbl 0194.58402 · doi:10.1007/BF01647091
[8] A. Joseph,Gelfand-Kirillov dimension for algebras associated with the Weyl algebra. Ann. Inst. H. Poincaré17 (1972), 325–336. · Zbl 0287.16011
[9] A. Joseph,Sur les algèbres de Weyl, Lecture notes 1974 (unpublished).
[10] A. Joseph,A characterization theorem for realizations of sl(2), Proc. Camb. Phil. Soc.75 (1974), 119–131. · Zbl 0285.17008 · doi:10.1017/S0305004100048349
[11] A. Joseph,A wild automorphism of Usl(2), Math. Proc. Camb. Phil. Soc.80 (1976), 61–65. · Zbl 0362.17008 · doi:10.1017/S030500410005266X
[12] A. Joseph,The Weyl algebra–semisimple and nilpotent elements, Amer. J. Math.97 (1975). 597–615. · Zbl 0316.16036 · doi:10.2307/2373768
[13] A. Joseph,Second commutant theorems in enveloping algebras, to appear in Amer. J. Math.
[14] A. Joseph,A preparation theorem for the prime spectrum of a semisimple Lie algebra, to appear in J. Algebra. · Zbl 0405.17007
[15] J. C. McConnell,Representations of solvable Lie algebras: III Cancellation theorems, preprint, Leeds, 1976. · Zbl 0391.17005
[16] L. G. Makar-Limanov,Commutativity of certain subalgebras in the rings R n,k, Funkcional. Anal. i Priložen.4 (1970), 78. · Zbl 0252.16015 · doi:10.1007/BF01075624
[17] D. Quillen,On the endomorphism ring of a simple module over an enveloping algebra, Proc. Amer. Math. Soc.21 (1969), 171–172. · Zbl 0188.08901
[18] R. Rentschler and P. Gabriel,Sur la dimension des anneaux et ensembles ordonnés, C. R. Acad. Sc Paris265, série A (1967), 712–715. · Zbl 0155.36201
[19] L. Solomon and D. N. Verma,Sur le corps des quotients de l’algèbre enveloppante d’une algèbre de Lie, C. R. Acad. Sc. Paris264, série A (1967), 985–986. · Zbl 0163.03004
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.