论文标题
最小的忠实表示，自由2步尼尔氏谎言lie代数$ r $
Minimal faithful representations of the free 2-step nilpotent Lie algebra of the rank $r$
论文作者
论文摘要
给定有限的尺寸为lie代数$ \ mathfrak {g} $，令$ \ m athfrak {z}（\ m athfrak {g}）$表示$ \ mathfrak {g} $的中心，让$μ（\ mathfrak {g}）$是最小的diseension $ feiths $ a $ {g。在本文中，我们获得$μ（\ Mathcal {l} _ {r，2}）$，其中$ \ Mathcal {l} _ {r，k} $是免费的$ k $ k $ - step nilpotent lie lie lie like like like like like like $ r $。特别是我们证明$μ（\ Mathcal {l} _ {r，2}）= \ left \ lceil \ lceil \ sqrt {2r（r-1）} \ right \ rceil + rceil + 2 $ for $ r \ geq 4 $。事实证明，$μ（\ Mathcal {l} _ {r，2}）\simμ\ big（\ Mathfrak {Z}（\ Mathcal {l} _ {r，2}）\ big）\ big）\ sim 2 \ sim 2 \ sqrt { $ r \ to \ infty $），我们提供了一些证据表明，对于任何$ k $而言，对于$ \ mathcal {l} _ {l} _ {r，k} $，这可能比$μ（\ nathcal {l} _ {r，k}）$（用于固定$ k $ k $ k $ k $ k $ k $ k $ k $ k $ y MATHCAL {l MATHCAL {l} _ $ \ dim \ Mathcal {l} _ {r，k} $。
Given a finite dimensional Lie algebra $\mathfrak{g}$, let $\mathfrak{z}(\mathfrak{g})$ denote the center of $\mathfrak{g}$ and let $μ(\mathfrak{g})$ be the minimal possible dimension for a faithful representation of $\mathfrak{g}$. In this paper we obtain $μ(\mathcal{L}_{r,2})$, where $\mathcal{L}_{r,k}$ is the free $k$-step nilpotent Lie algebra of rank $r$. In particular we prove that $μ(\mathcal{L}_{r,2})= \left\lceil \sqrt{2r(r-1)} \right\rceil + 2$ for $r \geq 4$. It turns out that $μ(\mathcal{L}_{r,2}) \simμ\big(\mathfrak{z}(\mathcal{L}_{r,2})\big) \sim 2\sqrt{\dim\mathcal{L}_{r,2}} $ (as $r\to\infty$) 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 $μ(\mathcal{L}_{r,k})$, which are (for fixed $k$) polynomial in $\dim\mathcal{L}_{r,k}$.
