I’m aiming to include some Lie theory into current study. Before worrying about those details I’m going to work some examples/problems from Brickell and Clark’s Differential Manifolds text on the general linear group which is the group of invertible matrices. Notable subgroups include O(n), the group of orthogonal matrices (distance preserving) with determinant , SL(n) the special linear group of matrices with determinant 1, and SO(n) the special orthogonal group, basically the stuff in O(n) that has determinant 1.
These objects are not only groups but manifolds as well. Technicalities aside, that’s essentially a Lie Group which is packed with meaning and applications (to quantum mechanics and quantum field theory to name only two). The tangent space to the identities are very special vector spaces called Lie Algebras.
Example 2.2.1: Consider the set of matrices with real entries. Denote this . There’s a bijection between this set and by . This set has just one chart.
Example 2.3.1: Let’s restrict the previous example to matrices but with the same chart as before. Let’s consider the determinant as a function between manifolds: . If we compose det with the only chart we get a coordinate representative for det,
Which is a polynomial that we could differentiate for example. Since this coordinate representative is just a function from differentiability implies continuity without any extra work. We’ll use this result later in the post.
Example 12.1.1) Along the lines of the above we can also talk about coordinate representatives of the group operations of GL. Each element of GL acts on GL in a smooth manner. Let then we can define the following map by where . A coordinate representative for this map, with respect to a chart x, would be which makes a map between . These are from -tuples to -tuples. The coordinates of these image -tuples are polynomials in the coordinates of the domain -tuples and so is smooth. In a similar way, the map taking each element of GL to its inverse is also smooth.
It would be great to look at some tangent spaces to GL as another example but it turns out this is deserving of a whole series of posts. For Lie groups like GL the tangent space at the identity is what interests us and this is called the Lie algebra of GL (and denoted ). It’s the linearization of the Lie group and it’s because GL is not only a manifold but also a group that this process imbues the tangent space with additional structure. I’ve started reading Robert Gilmore’s “Lie Groups, Physics and Geometry” and will hopefully start posting on this stuff in the near future as well.
Example 3.1.3: This was a great example because it tells us that (the General Linear group) is open in which is something that I had read but couldn’t wrap my head around. Thinking about M and GL seem really hard since both of them have too many dimensions. Instead we can look at the inverse image of the determinant. Consider the set . Since 0 is a closed set in the reals it’s preimage is closed in M (since the determinant is continuous). Since GL is the complement of that preimage (all the matrices with non-zero determinant) it must be open in M! It’s easy to construct an example (I don’t know why I couldn’t before!). Consider the matrix where , , and for . All of these matrices have non zero determinant and so reside in GL, but their limit as does not as it has a zero determinant. GL does not have all of it’s limit points.
But wait, there’s more (Example 5.3.1). What about the preimage of ? This is, of course, GL but we can get some great information from this preimage. is open, which we knew, but therefore not compact. It’s also disconnected. From this we can deduce the same properties of GL as a submanifold of M. This can be understood, I believe, by the fact that GL can be thought of as simply an open subset of since is homeomorphic to by virtue of the single bijective chart.
So GL has two components, one with positive determinant and one with negative determinant. We can conclude that O(n) is also disconnected since these are the matrices whose transpose is their inverse and who have . SO(n) is a subgroup of O(n) with determinant 1 and so would lie in the component of GL with positive determinant. SL(n) would likewise occupy that component. Notice that since SL(n) is characterized by matrices with determinant 1, this is a closed, compact and connected submanifold of M.
It would be nice to play with SU(n), U(n) and as well to deduce similiar properties. I’m thinking that I should work on organizing all the useful properties of these groups I can manage, at least for low dimensional cases.