Grassmann代数中理想生成的Lie不变几何对象的可积性
The integrability of Lie-invariant geometric objects generated by ideals in the Grassmann algebra
摘要：
本文研究了Grassmann代数中的闭理想作为李不变几何对象的基。嘉当特别是E。将Cartan理论推广到Lax可积非线性动力系统，并在喷流流形上的Wahlquist-Estabrook延拓结构和纤维空间上的Cartan-Ehresmann联络理论的框架下进行处理.研究了Grassmann代数中与闭理想有关的可积1-形式扩充2-形式的一般结构。一个有效的Maurer-Cartan单形式的建设建议，是非常有用的应用。作为应用的一个例子，开发的Lie不变的几何对象理论的Burgers非线性动力系统被认为是引起了找到一个明确的形式相关联的Lax型表示。
英文摘要：
We investigate closed ideals in the Grassmann algebra serving as bases of Lie-invariant geometric objects studied before by E. Cartan. Especially, the E. Cartan theory is enlarged for Lax integrable nonlinear dynamical systems to be treated in the frame work of the Wahlquist Estabrook prolongation structures on jet-manifolds and Cartan-Ehresmann connection theory on fibered spaces. General structure of integrable one-forms augmenting the two-forms associated with a closed ideal in the Grassmann algebra is studied in great detail. An effective Maurer-Cartan one-forms construction is suggested that is very useful for applications. As an example of application the developed Lie-invariant geometric object theory for the Burgers nonlinear dynamical system is considered having given rise to finding an explicit form of the associated Lax type representation.
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