GL(2;C) is the group of linear transformations on C^2

[eddo]

Incase anyone doesn't understand the notation, GL(2;C) is the group of linear transformations on C^2 which are invertible. Another way of looking at it is all complex 2x2 matrices with non-zero determinant.

It is fairly easy to show that GL(2;C) is not simply connected (just define a homomorphism that maps an element to its determinant). So here's my question. If it is not simply connected, than what is its universal covering group? All the textbooks I've looked at give many examples of universal covering groups but they never give you any sort of strategy of how to find one, so I have no idea where to start. Anyone know the answer, or have any hints so I can try to figure it out? Thanks

 

[Hurkyl]

There might not be an easy answer. e.g. Wikipedia's page on covering groups indicates that the universal cover of SL(2, R) isn't a matrix group!

If I had to guess, I think the homomorphism you mention gives a big hint. The determinant is a map

GL(2, C) --> C^*​

so the universal cover of GL(2, C) has to have a morphism to the universal cover of C^*. (I think)

You've probably seen the universal cover of C^*; I'm pretty sure it's a corkscrew shape: the graph of the (multivalued) complex logarithm function. Happily, we can unwind it and get that the universal cover is the exponential map

exp:C-->C^*.​

(C viewed as an additive group)

So, if I had to guess, I would claim that the (matrix) exponential

exp:M(2,C)-->GL(2,C)​

is related to the universal cover. (M(2,C) viewed as an additive group) Alas, this isn't a group homomorphism.

 

[tacman]

The universal covering group of GL(2;C) is the group of invertible complex 2x2 matrices with unit determinant, denoted by SL(2;C). This group is also known as the special linear group and it is a subgroup of GL(2;C). To see why this is the universal covering group, we need to understand the concept of covering groups.

A covering group is a group that "covers" another group, meaning that it is a larger group that contains the original group as a subgroup. In this case, SL(2;C) is a larger group than GL(2;C) and it contains GL(2;C) as a subgroup.

To show that SL(2;C) is the universal covering group of GL(2;C), we need to show that it satisfies two conditions: it is simply connected and it is the smallest simply connected covering group of GL(2;C).

To show that SL(2;C) is simply connected, we can use the fact that it is a connected Lie group (a group that is also a smooth manifold) and the fact that it is simply connected at the identity element. This means that any loop in SL(2;C) can be continuously deformed to a point without leaving the group.

To show that SL(2;C) is the smallest simply connected covering group of GL(2;C), we need to show that any other simply connected covering group of GL(2;C) must contain SL(2;C) as a subgroup. This can be done by considering the homomorphism that maps an element of SL(2;C) to its determinant, which we know is a surjective homomorphism.

In summary, SL(2;C) is the universal covering group of GL(2;C) because it is simply connected and the smallest simply connected covering group of GL(2;C). To find a universal covering group, one approach is to consider the properties of the original group and try to find a larger group that satisfies the conditions of simply connectedness and minimality.