The Submodule of Invariants

14 Jul

Originally posted on The Unapologetic Mathematician:

If $latex V$ is a module of a Lie algebra $latex L$, there is one submodule that turns out to be rather interesting: the submodule $latex V^0$ of vectors $latex v\in V$ such that $latex x\cdot v=0$ for all $latex x\in L$. We call these vectors “invariants” of $latex L$.

As an illustration of how interesting these are, consider the modules we looked at last time. What are the invariant linear maps $latex \hom(V,W)^0$ from one module $latex V$ to another $latex W$? We consider the action of $latex x\in L$ on a linear map $latex f$:

$latex \displaystyle\left[x\cdot f\right](v)=x\cdot f(V)-f(x\cdot v)=0$

Or, in other words:

$latex \displaystyle x\cdot f(v)=f(x\cdot v)$

That is, a linear map $latex f\in\hom(V,W)$ is invariant if and only if it intertwines the actions on $latex V$ and $latex W$. That is, $latex \hom_\mathbb{F}(V,W)^0=hom_L(V,W)$.

Next, consider the bilinear forms on $latex L$. Here we calculate

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