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A lower bound for faithful representations of nilpotent Lie algebras

In this paper we present a lower bound for the minimal dimension $μ(\mathfrak{n})$ of a faithful representation of a finite dimensional $p$-step nilpotent Lie algebra $\mathfrak{n}$ over a field of characteristic zero. Our bound is given as the minimum of a quadratically constrained linear optimization problem, it works for arbitrary $p$ and takes into account a given filtration of $\mathfrak{n}$. We present some estimates of this minimum which leads to a very explicit lower bound for $μ(\mathfrak{n})$ that involves the dimensions of $\mathfrak{n}$ and its center. This bound allows us to obtain $μ(\mathfrak{n})$ for some families of nilpotent Lie algebras.