ON ENDOMORPHISMS OF EXTENSIONS IN TANNAKIAN CATEGORIES

Abstract 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.