# The Unapologetic Mathematician

## Lie Algebra Modules

It should be little surprise that we’re interested in concrete actions of Lie algebras on vector spaces, like we were for groups. Given a Lie algebra $L$ we define an $L$-module to be a vector space $V$ equipped with a bilinear function $L\times V\to V$ — often written $(x,v)\mapsto x\cdot v$ satisfying the relation

$\displaystyle [x,y]\cdot v=x\cdot(y\cdot v)-y\cdot(x\cdot v)$

Of course, this is the same thing as a representation $\phi:L\to\mathfrak{gl}(V)$. Indeed, given a representation $\phi$ we can define $x\cdot v=[\phi(x)](v)$; given an action we can define a representation $\phi(x)\in\mathfrak{gl}(V)$ by $[\phi(x)](v)=x\cdot v$. The above relation is exactly the statement that the bracket in $L$ corresponds to the bracket in $\mathfrak{gl}(V)$.

Of course, the modules of a Lie algebra form a category. A homomorphism of $L$-modules is a linear map $\phi:V\to W$ satisfying

$\displaystyle\phi(x\cdot v)=x\cdot\phi(v)$

We automatically get the concept of a submodule — a subspace sent back into itself by each $x\in L$ — and a quotient module. In the latter case, we can see that if $W\subseteq V$ is any submodule then we can define $x\cdot(v+W)=(x\cdot v)+W$. This is well-defined, since if $v+w$ is any other representative of $v+W$ then $x\cdot(v+w)=x\cdot v+x\cdot w$, and $x\cdot w\in W$, so $x\cdot v$ and $x\cdot(v+w)$ both represent the same element of $v+W$.

Thus, every submodule can be seen as the kernel of some homomorphism: the projection $V\to V/W$. It should be clear that every homomorphism has a kernel, and a cokernel can be defined simply as the quotient of the range by the image. All we need to see that the category of $L$-modules is abelian is to show that every epimorphism is actually a quotient, but we know this is already true for the underlying vector spaces. Since the (vector space) kernel of an $L$-module map is an $L$-submodule, this is also true for $L$-modules.