The Unapologetic Mathematician

Mathematics for the interested outsider

General Linear Groups are Lie Groups

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

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 GL_n(\mathbb{R}) sits inside the algebra M_n(\mathbb{R}) of all n\times n matrices, which is an n^2-dimensional vector space. Even better, it’s an open subset, which we can see by considering the (continuous) map \mathrm{det}:M_n(\mathbb{R})\to\mathbb{R}. Since GL_n(\mathbb{R}) is the preimage of \mathbb{R}\setminus\{0\} — which is an open subset of \mathbb{R}GL_n(\mathbb{R}) is an open subset of M_n(\mathbb{R}).

So we can conclude that GL_n(\mathbb{R}) is an open submanifold of M_n, which comes equipped with the standard differentiable structure on \mathbb{R}^{n^2}. Matrix multiplication is clearly smooth, since we can write each component of a product matrix AB as a (quadratic) polynomial in the entries of A and B. As for inversion, Cramer’s rule expresses the entries of the inverse matrix A^{-1} as the quotient of a (degree n-1) polynomial in the entries of A and the determinant of A. So long as A is invertible these are two nonzero smooth functions, and thus their quotient is smooth at A.

June 9, 2011 - Posted by | Algebra, Differential Topology, Group theory, Topology


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