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

• eddo
In summary, GL(2;C) is a group of linear transformations on C^2 which are invertible. It is not simply connected and its universal covering group is related to the exponential map. The determinant can provide a hint in finding the universal cover, which is likely the exponential map from M(2,C) to GL(2,C). However, this is not a group homomorphism.
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

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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.

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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.

## 1. What does GL(2;C) stand for?

GL(2;C) stands for the General Linear Group of 2x2 Complex Matrices. It is the group of all invertible linear transformations on the complex vector space C^2.

## 2. What is the significance of GL(2;C) in mathematics and science?

GL(2;C) is significant in mathematics and science because it represents a group of fundamental transformations that preserve the structure and properties of a 2-dimensional complex vector space. These transformations are used in various mathematical and scientific applications, such as geometry, physics, and computer graphics.

## 3. How is GL(2;C) different from other groups?

GL(2;C) is different from other groups in that it is a non-abelian (non-commutative) group. This means that the order in which transformations are performed matters. Additionally, GL(2;C) is a continuous group, meaning that it contains an infinite number of elements and can be represented by real numbers.

## 4. What are some examples of elements in GL(2;C)?

Some examples of elements in GL(2;C) are 2x2 complex matrices with non-zero determinants. These matrices can represent various linear transformations, such as rotations, reflections, and dilations, on a 2-dimensional complex vector space.

## 5. How is GL(2;C) related to other mathematical concepts?

GL(2;C) is related to other mathematical concepts such as linear algebra, group theory, and complex numbers. It is also closely related to the Special Linear Group SL(2;C), which consists of 2x2 complex matrices with determinant 1. Additionally, GL(2;C) is a subgroup of the General Linear Group GL(n;C), which represents all invertible linear transformations on an n-dimensional complex vector space.

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