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2012 September 16 « The Unapologetic Mathematician

https://unapologetic.wordpress.com/2012/09/16

Mathematics for the interested outsider. As might be surmised from irreducible modules. For a Lie algebra. Is one that contains a nontrivial proper submodule — one other than. Is a submodule we can form the quotient. This is the basic setup of a short exact sequence. The question is, does this sequence split? That is, can we write. As the direct sum of. And some other submodule isomorphic to. First of all, let’s be clear that direct sums of modules do make sense. Indeed, if. No Indeed, let. In this case,.

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2012 year in blogging | Buttle's World

https://buttle.wordpress.com/2012/annual-report

Happy New Year from WordPress.com! Each rocket represents a post published on this blog in 2012. And because we like to share, we made the fireworks available as a jQuery plugin on GitHub. Some browsers are better suited for this kind of animation. In our tests, Safari and Chrome worked best. Your overall score is not known. We made beautiful, animated fireworks to celebrate your blogging! The new Boeing 787 Dreamliner can carry about 250 passengers. This blog was viewed about 1,300. In 2012, there were 2.

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2012 September 21 « The Unapologetic Mathematician

https://unapologetic.wordpress.com/2012/09/21

Mathematics for the interested outsider. The Submodule of Invariants. Of a Lie algebra. There is one submodule that turns out to be rather interesting: the submodule. 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 invariant linear maps. We consider the action of. On a linear map. Or, in other words:. That is, a linear map. Is invariant if and only if it intertwines the actions on. September 21, 2012. In par...

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Lie Algebra Modules « The Unapologetic Mathematician

https://unapologetic.wordpress.com/2012/09/12/lie-algebra-modules

Mathematics for the interested outsider. 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. Module to be a vector space. Equipped with a bilinear function. 8212; often written. Of course, this is the same thing as a representation. Indeed, given a representation. Given an action we can define a representation. The above relation is exactly the statement that the bracket in. Corresponds to the bracket in.

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John Armstrong « The Unapologetic Mathematician

https://unapologetic.wordpress.com/author/drmathochist

Mathematics for the interested outsider. The Submodule of Invariants. Of a Lie algebra. There is one submodule that turns out to be rather interesting: the submodule. 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 invariant linear maps. We consider the action of. On a linear map. Or, in other words:. That is, a linear map. Is invariant if and only if it intertwines the actions on. September 21, 2012. In par...

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Dual Spaces « The Unapologetic Mathematician

https://unapologetic.wordpress.com/2008/05/27/dual-spaces

Mathematics for the interested outsider. Another thing vector spaces come with is duals. That is, given a vector space. We have the dual vector space. Of “linear functionals” on. 8212; linear functions from. To the base field. Again we ask how this looks in terms of bases. So let’s take a finite-dimensional vector space. And consider some linear functional. Like any linear function, we can write down matrix coefficients. Notice that since our target space (the base field. Notice that on any vector we have.

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2012 September 17 « The Unapologetic Mathematician

https://unapologetic.wordpress.com/2012/09/17

Mathematics for the interested outsider. New Modules from Old. 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. Bilinearity should be clear, so we just check the defining property. Of a module. That is, we take two Lie algebra elements. Another way is to start with modules. The Jordan-Che...

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2012 September 15 « The Unapologetic Mathematician

https://unapologetic.wordpress.com/2012/09/15

Mathematics for the interested outsider. 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. As usual, Schur’s lemma. Tells us that any morphism between two irreducible modules is either. Or an isomorphism. And, as we’ve seen in other examples. Is simple then we know that. Must have its image in. But the only traceless. Matrix is the zero matrix. Setting. All De...

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The Killing Form « The Unapologetic Mathematician

https://unapologetic.wordpress.com/2012/09/03/the-killing-form

Mathematics for the interested outsider. We can now define a symmetric bilinear form. On our Lie algebra. It’s symmetric because the cyclic property of the trace lets us swap. And get the same value. It also satisfies another identity which is referred to as “associativity”, though it may not appear like the familiar version of that property at first:. Where we have used the trace identity from last time. It will be very useful to study the structures of Lie algebras. Is an ideal, then we can define.