How can I show that a quotient of the abelian Lie group $(\mathbb{C}, +)$ by a full rank lattice has no faithful finite-dimensional linear representation as a complex Lie group?
I was thinking of approaching in this way: Let $E$ such a quotient and assume, by way of contradiction, that $\rho \colon E \to GL(V)$ is a faithful finite-dimensional representation. Using Schur's Lemma, we can deduce that $\rho(E)$ consists of scalar matrices. Since $\rho$ is faithful, $\rho(E)$ must be isomorphic to a multiplicative subgroup of $\mathbb{C}$. If $E$ were finite we could use this to arrive to a contradiction. But given that $E$ is not necessarily finite I don't know how to proceed. Can an argument like this one work? Otherwise, can someone give me another approach? Thank you in advance!