矩阵空间上的线性李括号族$Mat(n\times m,\K)$与Ado定理
Linear family of Lie brackets on the space of matrices $Mat(n\times m,\K)$ and Ado's Theorem
摘要：
本文对矩形矩阵空间$Mat(n\times m,\K)$上的一类线性李括号进行了分类，并给出了Ado定理的一个类似结果.我们还对方阵的代数$Mat(n, \K)$给出了类似的分类，并由此证明了我们不能用$\dim V\leq n+1$在向量空间$V$中建立$(2n+1)$维Heisenberg李代数$\mathfrak{H}_n$的忠实表示.最后，证明了在方阵代数$Mat(n,\K)$的情形下，相应的李代数结构是标准李代数$\mathfrak{gl}(n,\K)$的压缩.
英文摘要：
In this paper we classify a linear family of Lie brackets on the space of rectangular matrices $Mat(n\times m,\K)$ and we give an analogue of the Ado's Theorem. We give also a similar classification on the algebra of the square matrices $Mat(n, \K)$ and as a consequence, we prove that we can't built a faithful representation of the $(2n+1)$-dimensional Heisenberg Lie algebra $\mathfrak{H}_n$ in a vector space $V$ with $\dim V\leq n+1$. Finally, we prove that in the case of the algebra of square matrices $Mat(n,\K)$, the corresponding Lie algebras structures are a contraction of the canonical Lie algebra $\mathfrak{gl}(n,\K)$.
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