Updated: 2012-09-29 20:40:29
First of all, news about the blog. Sometime this past Tuesday night the blog had 200,000 hits since inception almost 5 years ago. The daily record remains 542 hits… the weekly record is 1911… and the monthly record is 7077. I was reminded of another reason to go with the extended keyboard… on the old [...]
Updated: 2012-09-21 03:25:47
If is a module of a Lie algebra , there is one submodule that turns out to be rather interesting: the submodule of vectors such that for all . We call these vectors “invariants” of . As an illustration of how interesting these are, consider the modules we looked at last time. What are the [...]
Updated: 2012-09-21 02:12:16
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 and are two finite-dimensional -modules, then I say we can put an -module structure on the space of linear maps from to . Indeed, we can identify with : if [...]
Updated: 2012-09-20 03:08:55
: . Contact us Help Shopping cart Home About us Article title , keywords or abstract Article title Publication title Author Advanced search Subject Publisher Publication Browse : by Home The Journal of Management Development Volume 31, Number 9 The banning of images : questions arising in the field of management Authors : Colas , Hervé Laguecir , Aziza : Source The Journal of Management Development Volume 31, Number 9, 2012 pp . 925-937(13 Publisher : Emerald Group Publishing Limited view table of contents next article Buy download fulltext : article OR Pressing the buy now button more than once may result in multiple purchases Price : 38.00 plus tax Refund Policy : Abstract : Keywords Absence of the manager Banning of images Bible Decision making Judaism Magical thought Managers
Updated: 2012-09-17 20:47:01
There are a few standard techniques we can use to generate new modules for a Lie algebra from old ones. We’ve seen direct sums already, but here are a few more. One way is to start with a module and then consider its dual space . I say that this can be made into an [...]
Updated: 2012-09-16 02:24:38
As might be surmised from irreducible modules, a reducible module for a Lie algebra is one that contains a nontrivial proper submodule — one other than or itself. Now obviously if is a submodule we can form the quotient . This is the basic setup of a short exact sequence: The question is, does this [...]
Updated: 2012-09-16 00:30:12
Sorry for the delay; it’s getting crowded around here again. Anyway, an irreducible module for a Lie algebra is a pretty straightforward concept: it’s a module such that its only submodules are and . As usual, Schur’s lemma tells us that any morphism between two irreducible modules is either or an isomorphism. And, as we’ve [...]
Updated: 2012-09-12 02:29:02
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 [...]
Updated: 2012-09-11 01:25:42
It turns out that all the derivations on a semisimple Lie algebra are inner derivations. That is, they’re all of the form for some . We know that the homomorphism is injective when is semisimple. Indeed, its kernel is exactly the center , which we know is trivial. We are asserting that it is also [...]
Updated: 2012-09-08 21:35:20
We say that a Lie algebra is the direct sum of a collection of ideals if it’s the direct sum as a vector space. In particular, this implies that , meaning that the bracket of any two elements from different ideals is zero. Now, if is semisimple then there is a collection of ideals, each [...]
Updated: 2012-09-08 17:26:24
The only math I’ve done this week has been linear programming. It will be worthwhile to put out a second post using a minimization problem instead of a maximum. That is, I discovered I had questions about sensitivity analysis after I finished the minimum problem. Sensitivity analysis? That’s the part where I said that x2 [...]
Updated: 2012-09-08 17:07:32
I’ve mentioned before that the fall semester program at ICERM for 2013 will focus on computation in low-dimensional topology, geometry, and dynamics. You can now apply to be a long-term visitor for this as a graduate student, postdoc, or other. The deadline for the postdoctoral positions is January 14, 2013; the early deadline [...]
Updated: 2012-09-07 22:56:18
Let’s go back to our explicit example of and look at its Killing form. We first recall our usual basis: which lets us write out matrices for the adjoint action: and from here it’s easy to calculate the Killing form. For example: We can similarly calculate all the other values of the Killing form on [...]
Updated: 2012-09-06 05:17:16
The first and most important structural result using the Killing form regards its “radical”. We never really defined this before, but it’s not hard: the radical of a binary form on a vector space is the subspace consisting of all such that for all . That is, if we regard as a linear map , [...]
