微分算子李代数的导子
Derivations of the Lie Algebras of Differential Operators
摘要：
本文给出了光滑流形M上所有线性微分算子的李代数D（M）的导子，M的所有一阶线性微分算子的李子代数D^1（M）的导子，以及T*M上所有多项式函数的Poisson代数S（M）=Pol（T*M）的导子的完整而明确的描述.结果表明，根据Chevalley上同调，H^1（D（M），D（M））=H^1_{DR}（M），H^1（D^1（M），D^1（M））=H^1_{DR}（M）\oplus \R^2，H^1（S（M），S（M））=H^1_{DR}（M）\oplus \R。区分产生单参数自同构群的导子和描述这些单参数群的问题也得到了解决。
英文摘要：
This paper encloses a complete and explicit description of the derivations of the Lie algebra D(M) of all linear differential operators of a smooth manifold M, of its Lie subalgebra D^1(M) of all linear first-order differential operators of M, and of the Poisson algebra S(M)=Pol(T*M) of all polynomial functions on T*M, the symbols of the operators in D(M). It turns out that, in terms of the Chevalley cohomology, H^1(D(M),D(M))=H^1_{DR}(M), H^1(D^1(M),D^1(M))=H^1_{DR}(M)\oplus\R^2, and H^1(S(M),S(M))=H^1_{DR}(M)\oplus\R. The problem of distinguishing those derivations that generate one-parameter groups of automorphisms and describing these one-parameter groups is also solved.
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