VizualBusiness: The Submodule of Invariants

Tuesday, September 25, 2012

The Submodule of Invariants

The Submodule of Invariants:
If

is a module of a Lie algebra

, there is one submodule that turns out to be rather interesting: the submodule V^0

of vectors $v\in V$ such that $x\cdot v=0$ for all $x\in L$ . We call these vectors “invariants” of

.
As an illustration of how interesting these are, consider the modules we looked at last time. What are the invariant linear maps $\hom(V,W)^0$ from one module

to another

? We consider the action of $x\in L$ on a linear map

:
$\displaystyle\left[x\cdot f\right](v)=x\cdot f(V)-f(x\cdot v)=0$
Or, in other words:
$\displaystyle x\cdot f(v)=f(x\cdot v)$
That is, a linear map $f\in\hom(V,W)$ is invariant if and only if it intertwines the actions on

and

. That is, $\hom_\mathbb{F}(V,W)^0=hom_L(V,W)$ .
Next, consider the bilinear forms on

. Here we calculate
$\displaystyle\begin{aligned}\left[y\cdot B\right](x,z)&=-B([y,x],z)-B(x,[y,z])\\&=B([x,y],z)-B(x,[y,z])=0\end{aligned}$
That is, a bilinear form is invariant if and only if it is associative, in the sense that the Killing form is: B([x,y],z)=B(x,[y,z])