## General Linear Groups are Lie Groups

One of the most important examples of a Lie group we’ve already seen: the general linear group of a finite dimensional vector space . Of course for the vector space this is the same as — or at least isomorphic to — the group of all invertible real matrices, so that’s a Lie group we can really get our hands on. And if has dimension , then , and thus .

So, how do we know that it’s a Lie group? Well, obviously it’s a group, but what about the topology? The matrix group sits inside the algebra of all matrices, which is an -dimensional vector space. Even better, it’s an open subset, which we can see by considering the (continuous) map . Since is the preimage of — which is an open subset of — is an open subset of .

So we can conclude that is an open submanifold of , which comes equipped with the standard differentiable structure on . Matrix multiplication is clearly smooth, since we can write each component of a product matrix as a (quadratic) polynomial in the entries of and . As for inversion, Cramer’s rule expresses the entries of the inverse matrix as the quotient of a (degree ) polynomial in the entries of and the determinant of . So long as is invertible these are two nonzero smooth functions, and thus their quotient is smooth at .

[…] is an open submanifold of , the tangent space of at any matrix is the same as the tangent space to at . And since is […]

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[…] — homomorphisms to the general linear group of some vector space or another. But since is a Lie group, we can use this additional structure as well. And so we say that a representation of a Lie group […]

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