Endomorphism algebras of complex tori (Q1907177)

It is known that not every finite-dimensional associative algebra \(D\) over \(\mathbb{Q}\) can be realized as the algebra of endomorphisms of an abelian variety tensored with \(\mathbb{Q}\). The necessary and sufficient condition is the existence of a positive definite anti-involution on \(D\). The authors prove that for complex tori this is not true. Namely, for any \(D\) as above there exists a complex torus \(T\) such that \(\text{End} (T)\otimes \mathbb{Q} \cong D\). Moreover, if \(n= \text{rank}_\mathbb{Q} A\) then one may choose \(T\) of dimension \(t \cdot n\), where \(t\) is any positive integer. Also \(T\) is semisimple if and only if the algebra \(D\) is semisimple. The authors apply this result to give new examples of non-algebraizable complex tori.