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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 $$n>1$$.

##### MSC:
 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
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##### References:
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