Given a basic tilting or cotilting right module $T$ over an algebra $A$ given by quiver and relations, is there a "linear-algebra method" to decide whether $\operatorname{End}_A(T) \cong A?$ Here "linear-algebra method" roughly means that it can be reduced to linear algebra calculations by a computer program such as the GAP-package QPA. Thus for example calculating $\operatorname{Hom}$ spaces or $\operatorname{Ext}$ between modules counts as "linear-algebra methods". QPA can not test isomorphism of algebras (while it can calculate endomorphism rings), which motivates the question how to check whether one has $\operatorname{End}_A(T) \cong A$ using quick methods. Maybe there is even an intrinsic characterization of such tilting (cotilting) modules?
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