LU-Net is a simple and fast architecture for invertible neural networks (INN) that is based on the factorization of quadratic weight matrices $\mathsf{A=LU}$, where $\mathsf{L}$ is a lower triangular matrix with ones on the diagonal and $\mathsf{U}$ an upper triangular matrix. Instead of learning a fully occupied matrix $\mathsf{A}$, we learn $\mathsf{L}$ and $\mathsf{U}$ separately. If combined with an invertible activation function, such layers can easily be inverted whenever the diagonal entries of $\mathsf{U}$ are different from zero. Also, the computation of the determinant of the Jacobian matrix of such layers is cheap. Consequently, the LU architecture allows for cheap computation of the likelihood via the change of variables formula and can be trained according to the maximum likelihood principle. In our numerical experiments, we test the LU-net architecture as generative model on several academic datasets. We also provide a detailed comparison with conventional invertible neural networks in terms of performance, training as well as run time.