Wednesday, September 12, 2012

Lie Algebra Modules

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.



DIGITAL JUICE

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