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I am confused as to what a "path" through a matrix group is. I am looking for an easy way to find Lie Algebras of Lie Groups and come across people stating you need to find a path through the matrix group. I am confused because the matrices do not always seem connected even though they can form a group and manifold. How do I find a path through a matrix? I'd like a general definition if possible, it is very hard to find this on google.

Shaun
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James Shelton
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    A path in an object endowed with a topology is sometimes no more than a continuous map from $[0,1]$ into the object. Other concerns (e.g. notions of differentiability) may apply with more than just a "topological space" amount of structure (which Lie groups do have). There may not always be a path from each element of such a space to every other. – leslie townes Feb 21 '21 at 23:33
  • Here's an example. Suppose $A$ is a invertible diagonalizable real matrix, it has a factorization as $PDP^{-1}$ with $P$ invertible real and $D$ an invertible real diagonal matrix. If all of the elements of $D$ are positive you can use simple formulas to create a family of diagonal matrices $D_t$ with $D_0 = D$ and $D_1 = I$ and entries whose diagonal elements are continuous functions of $t$, showing that a path $t \mapsto P D_t P^{-1}$ exists from $A$ to $I$ in the group. More details of the construction may depend upon the topology and the specific notion of "path." – leslie townes Feb 21 '21 at 23:37
  • As another example $t \mapsto tA + (1-t)B$ will, under most definitions, be a path from $A$ to $B$ in a set of matrices. But it may not be a path in your Lie group if it includes elements that are not in the Lie group (e.g. because one or more elements of the path is not invertible). – leslie townes Feb 21 '21 at 23:42
  • Using your second comment that makes more sense now. Is there a general method that you know of that always works for Lie Groups because of the case in the 3rd comment that it may include elements not in the lie group? That last part is what I have the most difficulty with, avoiding "adding" elements that shouldn't be there to the lie group. – James Shelton Feb 21 '21 at 23:46
  • It would depend on the group in question and the paths sought to be formed. Matrix factorizations are often helpful in constructing paths guaranteed to stay in the group. But for example even the set of nonzero real numbers (invertible $1 \times 1$ matrices) is a Lie Group in which there is no path (within the group) from $1$ to $-1$. The group of nonzero complex numbers does not exhibit the same issue because it is path connected. – leslie townes Feb 22 '21 at 00:14
  • If there are lie groups that have "parts" of them with no paths possible like the invertible 1x1 matrices example you've given, is there ever a case where it is not possible to find the lie algebra of a lie group? At least by using paths. – James Shelton Feb 22 '21 at 01:11
  • https://math.stackexchange.com/questions/4027262/does-a-lie-algebra-generate-the-entire-identity-component and related questions may be relevant to this issue. – leslie townes Feb 22 '21 at 01:30
  • The Lie algebra of a Lie group is the the same as the Lie algebra of the (path-)connected component of the identity. So if you want to find a Lie algebra and it's a problem for you that your Lie group isn't connected, I think you can just take the subgroup of the connected component of the identity and work with that. – Mark S. Feb 22 '21 at 03:41
  • Thanks for the answers everyone this makes much more sense now. The question leslie referenced helped too. Thanks everyone -James – James Shelton Feb 22 '21 at 22:23

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