Special linear group:SL(2,C)

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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.

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

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 $x\mapsto {\begin{pmatrix}1&x\\0&1\\\end{pmatrix}}$. 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 ${\begin{pmatrix}1&1\\0&1\\\end{pmatrix}}$, 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 ${\begin{pmatrix}1&x\\0&1\\\end{pmatrix}},x\in \mathbb {R}$ and ${\begin{pmatrix}1&0\\x&1\\\end{pmatrix}},x\in \mathbb {C}$. Both sets are connected sets are matrices containing the identity, so the group is connected.
compact group	No	It contains matrices of the form ${\begin{pmatrix}1&x\\0&1\\\end{pmatrix}},x\in \mathbb {C}$ 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)