论文标题
低维谎言代数和相对差异不变的实现扩展
Extension of Realisations for Low-Dimensional Lie Algebras and Relative Differential Invariants
论文作者
论文摘要
我们首先要实现具有基本运算符$ l = \ langle q_m \ rangle $，$ q_m =ζ_{mj}（x_i）（x_i）\ partial_ {x_j} $，其中$ x_i $是某些方程或独立构建某些不相差的变量。我们采用其他变量$ r_k $，并研究线性扩展的动作操作员$ {\ hat q_m} = {q_m}+λ_{mjk} r_j \ partial_ {r_k} $，形成具有相同结构常数的同一lie代数。为了实现任何Lie代数$ L $的固定实现，我们可以为有限数量的其他变量对所有不等式的扩展动作实现进行分类。这种实现允许为代数的相同实现构建新的不变方程，但涉及其他变量，并为lie代数的各个实现对详尽的相对差异不变性和不变方程进行分类。它们也可以应用于微分方程对称分析中的其他问题。在这里，我们将实现不等的低维谎言代数的实现扩展。相对差异不变的分类问题上有一个有趣的故事，背景部分不仅值得读取该领域的人。
We start with a realisation of a Lie algebra with the basis operators $L=\langle Q_m\rangle$, $Q_m=ζ_{mj}(x_i)\partial_{x_j}$, where $x_i$ are some variables that may be regarded as dependent or independent in construction of some equations or differential invariants. We take additional variables $R_k$, and study the linearly extended action operators $ {\hat Q_m}={Q_m}+λ_{mjk}R_j\partial_{R_k}$ that form the same Lie algebra with the same structural constants. For a fixed realisation of any Lie algebra $L$ we can classify all inequivalent extended action realisations for a finite number of additional variables. Such realisations allow to construct new invariant equations for the same realisation of the algebra, but involving additional variables, and to classify exhaustively relative differential invariants and invariant equations for the respective realisations of Lie algebras. They can be also applied to other problems in the symmetry analysis of differential equations. Here we classify extensions of realisations for inequivalent low-dimensional Lie algebras. The problem of classification of the relative differential invariants has an interesting story attached to it, and the background section may be worth reading not only for people in the field.
