Good evening, I have a question concerning the change of a lattice basis. Now, a (full rank) lattice of $\mathbb{R}^n$ can be defined, fixing a basis of $\mathbb{R}^n$, say $\{v_1,\ldots,v_n\}$, as \begin{equation} \Lambda:=\Big\{\sum_{j=1}^na_jv_j, \ a_1,...a_n\in\mathbb{Z}\Big\}. \end{equation} In this case, the lattice $\Lambda$ can be written as $\Lambda=A\mathbb{Z}^n$, where \begin{equation} A=(v_1 | ...|v_n)\in GL(n,\mathbb{R}). \end{equation} A complex lattice can be defined exactly as before, replacing $\mathbb{R}$ with $\mathbb{C}$ and $\mathbb{Z}$ with $\mathbb{Z}[i]=\{m+in, \ m,n\in\mathbb{Z}\}$.
A well known result (for instance, I found this source, Theorem 2) of lattice theory states that $B$ and $B'$ are basis of the same lattice $\Lambda$ if and only if the associated matrices $A$ and $A'$ satisfy $A=A'U$, where $U\in\mathbb{Z}^{n\times n}$ is an integer matrix with $\det(U)=1$ (a uni modular matrix).
The question now is this: what is the equivalent result for complex lattices? Is it possible that $U\in\mathbb{Z}[i]^{n\times n}$ with $\det(U)=1$?