Realizations of finite dimensional algebras over the rationals (Q1890810)

The authors recall the well-known realization problem in Abelian group theory: when can a given ring \(R\) be realized as an endomorphism ring \(\text{End }G\) of an Abelian group \(G\) of a certain kind? Then a specific form of the realization problem is considered: when can a \(Q\)-algebra \(A\subset\text{End }V\) where \(V\) is a finite-dimensional vector space over the field of rationals \(Q\) be represented as \(A=Q\text{ End }G\) where \(G\) is an additive subgroup in \(V\) with \(QG=V\). The ring \(\text{End }G\) is commonly associated here with a subring in \(\text{End }V\).