关于四维李代数的簇
On the variety of four dimensional lie algebras
摘要：
维数为$n$的李代数由它们的结构常数定义，这些结构常数可以看作是$N = n^2 (n -- 1)/2$标量的集合（如果我们考虑斜对称条件），雅可比恒等式对其施加了某些二次条件。直到重新标度，我们可以把这样的集合看作是射影空间$P^{N--1}$中的一个点。假设为$n =4$，因此为$N = 24$。在$P^{23}$中，在复数上取一个随机子空间，其维数为$12$。我们证明了这个子空间将精确地包含$1033$点，给出某些四维李代数的结构常数。其中，$660$将同构于$gl\_2$，$195$将是一维仿射变换的李代数的两个副本的和，$121$将具有阿贝尔的三维导出代数，而$57$将具有三维海森堡代数的导出代数。这回答了基里洛夫和涅列廷的一个问题。
英文摘要：
Lie algebras of dimension $n$ are defined by their structure constants , which can be seen as sets of $N = n^2 (n -- 1)/2$ scalars (if we take into account the skew-symmetry condition) to which the Jacobi identity imposes certain quadratic conditions. Up to rescaling, we can consider such a set as a point in the projective space $P^{N--1}$. Suppose $n =4$, hence $N = 24$. Take a random subspace of dimension $12$ in $P^{23}$ , over the complex numbers. We prove that this subspace will contain exactly $1033$ points giving the structure constants of some four dimensional Lie algebras. Among those, $660$ will be isomorphic to $gl\_2$ , $195$ will be the sum of two copies of the Lie algebra of one dimensional affine transformations, $121$ will have an abelian, three-dimensional derived algebra, and $57$ will have for derived algebra the three dimensional Heisenberg algebra. This answers a question of Kirillov and Neretin.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
