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- W3033379974 abstract "Given a finite dimensional Lie algebra $mathfrak{g}$, let $mathfrak{z}(mathfrak{g})$ denote the center of $mathfrak{g}$ and let $mu(mathfrak{g})$ be the minimal possible dimension for a faithful representation of $mathfrak{g}$. In this paper we obtain $mu(mathcal{L}_{r,2})$, where $mathcal{L}_{r,k}$ is the free $k$-step nilpotent Lie algebra of rank $r$. In particular we prove that $mu(mathcal{L}_{r,2})= leftlceil sqrt{2r(r-1)} rightrceil + 2$ for $r geq 4$. It turns out that $mu(mathcal{L}_{r,2}) simmubig(mathfrak{z}(mathcal{L}_{r,2})big) sim 2sqrt{dimmathcal{L}_{r,2}} $ (as $rtoinfty$) and we present some evidence that this could be true for $mathcal{L}_{r,k}$ for any $k$, this is considerably lower than the known bounds for $mu(mathcal{L}_{r,k})$, which are (for fixed $k$) polynomial in $dimmathcal{L}_{r,k}$." @default.
- W3033379974 created "2020-06-12" @default.
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- W3033379974 date "2020-06-08" @default.
- W3033379974 modified "2023-10-16" @default.
- W3033379974 title "Minimal faithful representations of the free 2-step nilpotent Lie algebra of the rank $r$" @default.
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