Element structure of special linear group:SL(2,C)

Related information

 * Element structure of special linear group of degree two over a field
 * Element structure of special linear group:SL(2,R)
 * Element structure of special linear group of degree two over a finite field

Conjugacy class structure
The case of $$\mathbb{C}$$ is easy because in this case, the characteristic is not two and the field is algebraically closed, which means in particular that every element is a square. We thus do not have to worry about conjugacy classes that can be diagonalized over extensions but not over the field. Also, there is no splitting of conjugacy classes in $$SL(2,\mathbb{C})$$ relative to $$GL(2,\mathbb{C})$$, again because every element is a square. In alternative mathematical jargon, $$SL(2,\mathbb{C})$$ is a central factor (the other factor being the scalar matrices) and hence a conjugacy-closed subgroup of $$GL(2,\mathbb{C})$$.

Identification between conjugacy classes and field elements
The trace is a continuous mapping:

Conjugacy classes in $$SL(2,\mathbb{C})$$ $$\to$$ $$\mathbb{C}$$

Further, this mapping is almost bijective. For any $$a \ne -2,2$$, there is a unique conjugacy class in $$SL(2,\mathbb{C})$$ with trace $$a$$. For $$a \in \{ -2,2 \}$$, there are two conjugacy classes mapping to $$a$$: the scalar matrix class and the Jordan block class.

If we give the set of conjugacy classes in $$SL(2,\mathbb{C})$$ the quotient topology from the topology on $$SL(2,\mathbb{C})$$, it will look as follows: setwise, it is a union of $$\mathbb{C}$$ and two open points. One of the open points has closure defined to be that point and $$-2 \in \mathbb{C}$$. The other open point has closure defined to be that point and $$2 \in \mathbb{C}$$. The open points correspond to the Jordan block classes and the points -2,2 in $$\mathbb{C}$$ to the scalar matrix classes.