Jason Bennett
- 48
- 3
- Homework Statement
- Determining the manifold picture of a lie group to see its global properties
- Relevant Equations
- see below
1) How do we determine a Lie group's global properties when the manifold that it represents is not immediately obvious?
Allow me to give the definitions I am working with.
A Lie group G is a differentiable manifold G which is also a group, such that the group multiplication
and the map sending g ∈ G to its inverse g−1 ∈ G, are differentiable (C∞) maps.
As a Lie group is a manifold, one can parametrize the elements g in a small neighbor- hood of the identity e of G, by n real parameters {x1, . . . , xn}. This is simply written as g = g(x1, . . . , xn) and the parametrization is usually chosen in such a way that e = g(0, . . . , 0). The number of independent real parameters (n) is called the dimension of the Lie group.
A Lie group G is connected iff [tex]\forall g_1, g_2 \in G[/tex] there exists a continuous curve connecting the two, i.e. there exists only one connected component.
A Lie group G is simply connected (if all closed curves on the manifold picture of G) can be contracted to a point.
A Lie group is compact if there are no elements infinitely far away fro the others.
The Lie group U(1) is quite easily identified as a circle in its manifold picture. This is connected, not simply connected, and compact.
However, SO(3) can (apparently) be viewed as manifold as such: a filled sphere of fixed radius, with antipodes identified. Once that has been realized, determining the global properties are straight forward. Getting there is another question, and I believe related to answering...
2) How do we algorithmically determine the parameter space of a Lie group – thus seeing it as a manifold?
For instance, for SU(2), we can write the matrix elements as complex, or decomposed with reals + i(reals), and use the det = 1 condition to determine that the Lie group is 3-deimensional. The step from 3-dimensional to a 3-sphere is not clear to me.
Allow me to give the definitions I am working with.
A Lie group G is a differentiable manifold G which is also a group, such that the group multiplication
and the map sending g ∈ G to its inverse g−1 ∈ G, are differentiable (C∞) maps.
As a Lie group is a manifold, one can parametrize the elements g in a small neighbor- hood of the identity e of G, by n real parameters {x1, . . . , xn}. This is simply written as g = g(x1, . . . , xn) and the parametrization is usually chosen in such a way that e = g(0, . . . , 0). The number of independent real parameters (n) is called the dimension of the Lie group.
A Lie group G is connected iff [tex]\forall g_1, g_2 \in G[/tex] there exists a continuous curve connecting the two, i.e. there exists only one connected component.
A Lie group G is simply connected (if all closed curves on the manifold picture of G) can be contracted to a point.
A Lie group is compact if there are no elements infinitely far away fro the others.
The Lie group U(1) is quite easily identified as a circle in its manifold picture. This is connected, not simply connected, and compact.
However, SO(3) can (apparently) be viewed as manifold as such: a filled sphere of fixed radius, with antipodes identified. Once that has been realized, determining the global properties are straight forward. Getting there is another question, and I believe related to answering...
2) How do we algorithmically determine the parameter space of a Lie group – thus seeing it as a manifold?
For instance, for SU(2), we can write the matrix elements as complex, or decomposed with reals + i(reals), and use the det = 1 condition to determine that the Lie group is 3-deimensional. The step from 3-dimensional to a 3-sphere is not clear to me.