实李代数表示的几何性质
On the geometry underlying a real Lie algebra representation
摘要：
设$G$是一个实李群，其李代数为$\mathfrak g$ .给定$G$的酉表示$π$，通过无界斜伴随算子微分得到$\mathfrak g$的表示$dπ$. $\mathfrak g$的表示允许这样的描述被称为\emph{integrable,}，它们可以通过在代表函数$g\mapsto<ξ,π(g)η>$的代数上的导子在几何上被看作是$\mathfrak g$的作用，这些代表函数自然地定义在齐次空间$M=G/\kerπ$上。换句话说，实李代数的可积表示总是可以看作是该代数在齐次流形上的向量场的实现。在这里，我们展示了如何使用的$\mathfrak g$的泛包络代数的余积推广这表示不一定是可积的。现在扮演$M$角色的几何是局部齐性空间。这提供了一个几何方法的基础上，可积性问题的李代数表示。
英文摘要：
Let $G$ be a real Lie group with Lie algebra $\mathfrak g$. Given a unitary representation $π$ of $G$, one obtains by differentiation a representation $dπ$ of $\mathfrak g$ by unbounded, skew-adjoint operators. Representations of $\mathfrak g$ admitting such a description are called \emph{integrable,} and they can be geometrically seen as the action of $\mathfrak g$ by derivations on the algebra of representative functions $g\mapsto<ξ,π(g)η>$, which are naturally defined on the homogeneous space $M=G/\kerπ$. In other words, integrable representations of a real Lie algebra can always be seen as realizations of that algebra by vector fields on a homogeneous manifold. Here we show how to use the coproduct of the universal enveloping algebra of $\mathfrak g$ to generalize this to representations which are not necessarily integrable. The geometry now playing the role of $M$ is a locally homogeneous space. This provides the basis for a geometric approach to integrability questions regarding Lie algebra representations.
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