The quaternions and Lie algebras I

Someone who has just read the previous post on how exponentiating quaternions gives a nice parameterization of might object as follows: “that’s nice and all, but there has to be a general version of this construction for more general Lie groups, right? You can’t always depend on the nice properties of division algebras.” And that [...]

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A linear algebra puzzle

I won’t get around to substantive blog posts for at least a few more days, so here is a puzzle. We usually thinks of groups that occur in nature as permutations of a set which preserve some structure on that set. For example, the general linear group preserves the structure of being an -vector space [...]

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Chevalley-Bruhat order

Before we define Bruhat order, I’d like to say a few things by way of motivation. Warning: I know nothing about algebraic groups, so take everything I say with a grain of salt. A (maximal) flag in a vector space of dimension is a chain of subspaces such that . The flag variety of is, [...]

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Heron’s formula

Heron’s formula for the area of a triangle with side lengths is where is the semiperimeter. Today I’d like to try to prove this using as little geometry as possible.

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Set-multiset duality and supervector spaces

Recall that the elementary symmetric functions generate the ring of symmetric functions as a module over any commutative ring . A corollary of this result, although I didn’t state it explicitly, is that the elementary symmetric functions are algebraically independent, hence any ring homomorphism from the symmetric functions is determined freely by the images of [...]

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Non-canonical isomorphisms

I find non-canonical isomorphisms very interesting, but I wish I knew more examples. To be vague, an isomorphism (perhaps in a category) is said to be non-canonical if it requires making an “arbitrary choice.” One of the reasons I find them interesting is that we often think of objects only up to isomorphism, but in [...]

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