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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 

 

Endomorphisms of right ideals of the Weyl algebra


Author: J. T. Stafford
Journal: Trans. Amer. Math. Soc. 299 (1987), 623-639
MSC: Primary 16A89; Secondary 16A19, 16A33, 16A65, 17B35
DOI: https://doi.org/10.1090/S0002-9947-1987-0869225-3
MathSciNet review: 869225
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Abstract: Let $ A = A(k)$ be the first Weyl algebra over an infinite field $ k$, let $ P$ be any noncyclic, projective right ideal of $ A$ and set $ S = \operatorname{End} (P)$. We prove that, as $ k$-algebras, $ S\not \cong A$. In contrast, there exists a noncyclic, projective right ideal $ Q$ of $ S$ such that $ S \cong \operatorname{End} (Q)$. Thus, despite the fact that they are Morita equivalent, $ S$ and $ A$ have surprisingly different properties. For example, under the canonical maps, $ {\operatorname{Aut} _k}(A) \cong {\operatorname{Pic} _k}(A) \cong {\operatorname{Pic} _k}(S)$. In contrast, $ {\operatorname{Aut} _k}(S)$ has infinite index in $ {\operatorname{Pic} _k}(S)$.


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DOI: https://doi.org/10.1090/S0002-9947-1987-0869225-3
Article copyright: © Copyright 1987 American Mathematical Society