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June 2005 Endomorphisms of Weyl algebra and $p$-curvatures
Yoshifumi Tsuchimoto
Osaka J. Math. 42(2): 435-452 (June 2005).

Abstract

We first show that for each Weyl algebra over a positive characteristic field, we may obtain an affine space with a projectively flat connection on it. We give a set of differential equations which controls the behavior of the connection under endomorphism of the Weyl algebra. The key is the theory of $p$-curvatures.

Next we introduce a field $\mathbb{Q}_{\mathcal{U}}^{(\infty)}$ of characteristic zero as a limit of fields of positive characteristics. We need to fix an ultrafilter on the set of prime numbers to do this. The field is actually isomorphic to the field $\mathbb{C}$ of complex numbers.

Then we show that we may associate with a Weyl algebra over the field $\mathbb{Q}_{\mathcal{U}}^{(\infty)}$ an affine space with a symplectic form in a functorial way. That means, the association is done in such a way that an endomorphism of the Weyl algebra induces a symplectic map of the affine space.

As a result, we show that a solution of the Jacobian conjecture is sufficient for an affirmative answer to the Dixmier conjecture.

Citation

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Yoshifumi Tsuchimoto. "Endomorphisms of Weyl algebra and $p$-curvatures." Osaka J. Math. 42 (2) 435 - 452, June 2005.

Information

Published: June 2005
First available in Project Euclid: 21 July 2006

zbMATH: 1105.16024
MathSciNet: MR2147727

Rights: Copyright © 2005 Osaka University and Osaka City University, Departments of Mathematics

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Vol.42 • No. 2 • June 2005
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