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 we define an -module to be a vector space equipped with a bilinear function — often written satisfying the relation
Of course, this is the same thing as a representation . Indeed, given a representation we can define ; given an action we can define a representation by . The above relation is exactly the statement that the bracket in corresponds to the bracket in .
Of course, the modules of a Lie algebra form a category. A homomorphism of -modules is a linear map satisfying
We automatically get the concept of a submodule — a subspace sent back into itself by each — and a quotient module. In the latter case, we can see that if is any submodule then we can define . This is well-defined, since if is any other representative of then , and , so and both represent the same element of .
Thus, every submodule can be seen as the kernel of some homomorphism: the projection . 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 -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 -module map is an -submodule, this is also true for -modules.