Special linear group:SL(2,R)

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Definition

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

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

It is a particular case of a special linear group over reals and hence of a special linear group.

Structures

The group has the structure of a topological group, a real Lie group, and an algebraic group restricted to the reals.

Arithmetic functions

Function	Value	Similar groups	Explanation
dimension of an algebraic group	3		As $SL(n,\_),n=2:n^{2}-1=2^{2}-1=3$
dimension of a real Lie group	3		As $SL(n,\mathbb {R} ),n=2:n^{2}-1=2^{2}-1=3$

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,R), is simple.
simple non-abelian group	No	The center is $\pm I$, so is proper and nontrivial.

Elements

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

Below is a summary of the conjugacy class structure:

Nature of conjugacy class	Eigenvalues	Characteristic polynomial	Minimal polynomial	What set can each conjugacy class be identified with? (rough measure of size of conjugacy class)	What can the set of conjuacy classes be identified with (rough measure of number of conjugacy classes)	What can the union of conjugacy classes be identified with?	Semisimple?	Diagonalizable over $\mathbb {R}$?	Splits in $SL_{2}$ relative to $GL_{2}$?
Diagonalizable over $\mathbb {R}$ with equal diagonal entries, hence a scalar	$\{1,1\}$ or $\{-1,-1\}$	$(x-a)^{2}$ where $a\in \{-1,1\}$	$x-a$ where $a\in \{-1,1\}$	one-point set	two-point set	two-point set	Yes	Yes	No
Parabolic conjugacy class: Not diagonal, has Jordan block of size two	$1$ (multiplicity 2) or $-1$ (multiplicity 2)	$(x-a)^{2}$ where $a\in \{-1,1\}$	Same as characteristic polynomial	?	four-point set, two for eigenvalue 1, two for eigenvalue -1	?	No	No	Both the $GL_{2}$-conjugacy classes split into two pieces.
Elliptic conjugacy class: Diagonalizable over $\mathbb {C}$ but not over $\mathbb {R}$. Must necessarily have no repeated eigenvalues.	Pair of conjugate elements in $\mathbb {C}$ of modulus 1	$x^{2}-ax+1$, $-2\leq a\leq 2$	Same as characteristic polynomial	?	direct product of the open interval $(-2,2)$ with a two-point set	?	Yes	No	each $GL_{2}$-conjugacy class splits into two $SL_{2}$-conjugacy classes.
Hyperbolic conjugacy class: Diagonalizable over $K$ with distinct (and hence mutually inverse) diagonal entries	$\lambda ,1/\lambda$ where $\lambda \in \mathbb {R} \setminus \{0,1,-1\}$	$x^{2}-(\lambda +1/\lambda )x+1$	Same as characteristic polynomial	?	$\mathbb {R} \setminus [-2,2]$	?	Yes	Yes	No
Total	NA	NA	NA	NA	?	?	?	?	?