三次最低传递微分群的李代数
The Lie algebra of the lowest transitively differential group of degree three
AI总结：
本文研究了满足特定条件的一阶多项式系数微分算子的实李代数，确定了其为一个39维的六变量最小不可分解代数，并展示了其结构、分解、子代数及Casimir不变量的构造。
摘要：
我们研究了具有多项式系数的一阶微分算子的实李代数，其需满足以下要求。（1）该李代数应容许由次数不超过且包含三次的齐次多项式系数的微分算子构成的基。（2）该代数的生成元必须包含所有变量 $x_1$,...,$x_k$ 的平移算子 $\partial_k$。（3）该李代数是满足（1）和（2）的最小不可分解李代数。结果表明，它是一个六变量（$k=6$）的39维李代数，且该代数的构造也是 Guillemin 和 Sternberg 于1964年引入的涉及三次系数的传递微分群李代数一般构造中最简单的可能情况。这些代数及其各种子代数与物理学中不同应用相关的代数具有相似性，例如带有和不带有中心扩张的 Schrödinger、Conformal 和 Galilei 变换群的代数。本文致力于展示所研究的李代数的结构和不同分解。同时致力于展示相关的李子代数，并使用不同方法构造它们的 Casimir 不变量。特别地，我们将依赖微分算子实现、符号计算包、Berezin 括号以及李代数的虚拟副本。
英文摘要：
We investigate the real Lie algebra of first-order differential operators with polynomial coefficients, which is subject to the following requirements. (1) The Lie algebra should admit a basis of differential operators with homogeneous polynomial coefficients of degree up to and including three. (2) The generator of the algebra must include the translation operators $\partial_k$ for all the variables $x_1$,...,$x_k$. (3) The Lie algebra is the smallest indecomposable Lie algebra satisfying (1) and (2). It turns out to be a 39-dimensional Lie algebra in six variables ($k=6$) and the construction of this algebra is also the simplest possible case in the general construction of the Lie algebras of the transitively differential groups introduced by Guillemin and Sternberg in 1964 involving the coefficients of degree 3. Those algebras and various subalgebras have similarities with algebras related to different applications in physics such as those of the Schrödinger, Conformal and Galilei transformation groups with and without central extension. The paper is devoted to the presentation of the structure and different decompositions of the Lie algebra under investigation. It is also devoted to the presentation of relevant Lie subalgebras and the construction of their Casimir invariants using different methods. We will rely, in particular, on differential operator realizations, symbolic computation packages, the Berezin bracket and virtual copies of the Lie algebras.
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