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ON ENDOMORPHISMS OF EXTENSIONS IN TANNAKIAN CATEGORIES
- Part of
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- Journal:
- Bulletin of the Australian Mathematical Society / Volume 110 / Issue 1 / August 2024
- Published online by Cambridge University Press:
- 07 November 2023, pp. 103-114
- Print publication:
- August 2024
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- Article
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We prove analogues of Schur’s lemma for endomorphisms of extensions in Tannakian categories. More precisely, let
$\mathbf {T}$ be a neutral Tannakian category over a field of characteristic zero. Let E be an extension of A by B in 
$\mathbf {T}$. We consider conditions under which every endomorphism of E that stabilises B induces a scalar map on 
$A\oplus B$. We give a result in this direction in the general setting of arbitrary 
$\mathbf {T}$ and E, and then a stronger result when 
$\mathbf {T}$ is filtered and the associated graded objects to A and B satisfy some conditions. We also discuss the sharpness of the results.
