I’d been meaning to write a plug for my paper A self-dual integral form for the moonshine module on this blog for almost 7 years, but never got around to it until now. It turns out that somet…| Secret Blogging Seminar
If $latex V$ is a module of a Lie algebra $latex L$, there is one submodule that turns out to be rather interesting: the submodule $latex V^0$ of vectors $latex v\in V$ such that $latex x\cdot v=0$…| The Unapologetic Mathematician
There are a few constructions we can make, starting with the ones from last time and applying them in certain special cases. First off, if $latex V$ and $latex W$ are two finite-dimensional $latex …| The Unapologetic Mathematician
There are a few standard techniques we can use to generate new modules for a Lie algebra $latex L$ from old ones. We’ve seen direct sums already, but here are a few more. One way is to start …| The Unapologetic Mathematician
As might be surmised from irreducible modules, a reducible module $latex M$ for a Lie algebra $latex L$ is one that contains a nontrivial proper submodule — one other than $latex 0$ or $latex…| The Unapologetic Mathematician
Sorry for the delay; it’s getting crowded around here again. Anyway, an irreducible module for a Lie algebra $latex L$ is a pretty straightforward concept: it’s a module $latex M$ such …| The Unapologetic Mathematician
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 $latex L$ we define an $latex L$-module t…| The Unapologetic Mathematician