Strong exact Borel subalgebras of quasi-hereditary algebras (Q1585295)

Exact and strong Borel subalgebras of quasi-hereditary algebras were introduced by the reviewer [see Math. Z. 220, No. 3, 399-426 (1995; Zbl 0841.16013)] to mimick the situation for Lie algebras. Existence is granted for blocks of category \(\mathcal O\) and in a few other situations, but not in general. A natural problem is uniqueness. By analogy to the situation in Lie theory one may hope to be able to show that all strong exact Borel subalgebras of a given quasi-hereditary algebra \(A\) are conjugate to each other by inner automorphisms of \(A\). In the paper under review such a result is claimed to be true. Unfortunately, the proof given in the paper is not correct. Given an algebra \(A\), the author defines \(I_{i,j}\) to be the subspace of \(e_iAe_j\) generated by irreducible elements. He fails to notice that this subspace \(I_{i,j}\) equals \(e_i\text{rad}(A)e_j\) and thus usually will not be contained in the strong exact Borel subalgebra. Part (3) of Lemma 3.1 is wrong, and therefore the proofs of Proposition 3.5 and of the main Theorem 3.6 are not complete.