Special linear group:SL(2,C)

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VIEW DEFINITIONS USING THIS AS EXAMPLE: All terms |Group properties |Subgroup properties|
VIEW FACTS USING THIS AS EXAMPLE: All facts |Group property non-implications |Subgroup property non-implications |Group metaproperty dissatisfactionsSubgroup metaproperty dissatisfactions

Definition

The group $SL(2,\mathbb{C})$ is defined as the group of $2\times 2$ matrices with entries from the field of complex numbers and determinant $1$, under matrix multiplication.

$SL(2,\mathbb{C}):=\{\left(\begin{array}{cc}a& b\\ c& d\\ \end{array}\right)\mid a,b,c,d\in \mathbb{C},ad-bc=1\}$.

It is a particular case of a special linear group over complex numbers, special linear group of degree two, and hence of a special linear group.

Arithmetic functions

Function	Value	Similar groups	Explanation
order of a group	cardinality of the continuum		The cardinality is at least that of the continuum, because we can inject $\mathbb{C}$ into this group by $x\mapsto \left(\begin{array}{cc}1& x\\ 0& 1\\ \end{array}\right)$. On the other hand, it is a subset of ${\mathbb{C}}^4$, so the cardinality is not more than that of the continuum.
exponent of a group	infinite		there exist elements, such as $\left(\begin{array}{cc}1& 1\\ 0& 1\\ \end{array}\right)$, of infinite order.
composition length	2	groups with same composition length	Center is simple (isomorphic to cyclic group:Z2) and the quotient group PSL(2,C) is also simple.
chief length	2	groups with same chief length	Similar reason to composition length.
dimension of an algebraic group	3	groups with same dimension of an algebraic group	As $SL(n,\_),n=2:n^2-1=2^2-1=3$
dimension of a complex Lie group	3	groups with same dimension of a complex Lie group	As $SL(n,\mathbb{C}),n=2:n^2-1=2^2-1=3$
dimension of a real Lie group	6	groups with same dimension of a real Lie group	Twice its dimension as a complex Lie group.

Group properties

Abstract group properties

Property	Satisfied?	Explanation
abelian group	No
nilpotent group	No
solvable group	No
quasisimple group	Yes	special linear group is quasisimple (with a couple of finite exceptions). Its inner automorphism group, which is projective special linear group:PSL(2,C), is simple.
simple non-abelian group	No	The center is $\pm I$, so is proper and nontrivial.

Topological/Lie group properties

Property	Satisfied?	Explanation
connected topological group	Yes	It is generated by matrices of the form $\left(\begin{array}{cc}1& x\\ 0& 1\\ \end{array}\right),x\in \mathbb{R}$ and $\left(\begin{array}{cc}1& 0\\ x& 1\\ \end{array}\right),x\in \mathbb{C}$. Both sets are connected sets are matrices containing the identity, so the group is connected.
compact group	No	It contains matrices of the form $\left(\begin{array}{cc}1& x\\ 0& 1\\ \end{array}\right),x\in \mathbb{C}$ where the $x$ can be arbitrarily large, so is not compact as a subset of ${\mathbb{C}}^4$.

Elements

Further information: element structure of special linear group:SL(2,C)

Linear representation theory

Further information: linear representation theory of special linear group:SL(2,C)