Conjugation-uniqueness of exact Borel subalgebras (Q1974172)

Exact 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 exact Borel subalgebras of a given basic 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. In fact, the author considers `arrows' of \(A\) (that is, elements of a basis of \(\text{rad}(A)/\text{rad}(A)^2\)) as well-defined and even uniquely defined elements of \(A\), neglecting the choice in fixing a basis and the effect of lifting elements from \(\text{rad}(A)/\text{rad}(A)^2\) to \(\text{rad}(A)\). Rearranging the arguments one might be able to prove the weaker statement that two exact Borel subalgebras have the same quiver.