Faithful Representations of Minimal Dimension of 6-dimensional nilpotent Lie algebras

The main goal of this paper is to compute $\mu(\g)$ and $\mu_{nil}(\g)$ for each nilpotent Lie algebra $\g$ of dimension 6 over a field of characteristic zero $\k$. Here $\mu(\g)$ and $\mu_{nil}(\g)$ is the minimal dimension of a faithful representation of $\g$ and the minimal dimension of a faithful nilrepresentation of $\g$, respectively. We also give a minimal faithful representation and nilrepresentation for each nilpotent Lie algebra of dimension $\leq 6$.