The Unapologetic Mathematician

Mathematics for the interested outsider

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.

September 12, 2012 Posted by | Algebra, Lie Algebras, Representation Theory | 2 Comments