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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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