关于可平行流形上向量场的上同调
On the cohomology of vector fields on parallelizable manifolds
摘要：
本文对每个可平行化的光滑紧致流形$M$确定了$M$上光滑向量场的李代数$V_M$的上同调空间$H^2(V_M,\barΩ^p_M)$，其值在模$\barΩ^p_M = Ω^p_M/dΩ^{p-1}_M$中. $p=1$的情况特别有趣，因为规范代数$C^\infty (M,k)$具有以中心$\barΩ^1_M$为中心的泛中心扩张，推广了仿射Kac-Moody代数。二阶上同调$H^2(V_M, \barΩ^1_M)$将$V_M$的半直积与通用中心扩张$C^\infty (M,k) \oplus \barΩ^1_M$的扭曲进行分类。
英文摘要：
In the present paper we determine for each parallelizable smooth compact manifold $M$ the cohomology spaces $H^2(V_M,\barΩ^p_M)$ of the Lie algebra $V_M$ of smooth vector fields on $M$ with values in the module $\barΩ^p_M = Ω^p_M/dΩ^{p-1}_M$. The case of $p=1$ is of particular interest since the gauge algebra $C^\infty (M,k)$ has the universal central extension with center $\barΩ^1_M$, generalizing affine Kac-Moody algebras. The second cohomology $H^2(V_M, \barΩ^1_M)$ classifies twists of the semidirect product of $V_M$ with the universal central extension $C^\infty (M,k) \oplus \barΩ^1_M$.
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