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

This article gives specific information, namely, element structure, about a particular group, namely: special linear group:SL(2,R).
View element structure of particular groups | View other specific information about special linear group:SL(2,R)

This article aims to discuss the element structure of special linear group:SL(2,R).

Related information

• Element structure of special linear group of degree two over a field: This is very general, but requres a large number of cases and takes a lot of effort to understand in its entirety.
• Element structure of special linear group of degree two over a finite field

Conjugacy class structure

To deduce this from element structure of special linear group of degree two over a field, we need to use the following facts, about $\mathbb {R}$, the field of real numbers:

• The group $\mathbb {R} ^{*}/(\mathbb {R} ^{*})^{2}$ is cyclic of order two, with representatives $1,-1$.
• The only separable quadratic extension of $\mathbb {R}$ is the field of complex numbers, obtained by adjoining a square root of -1.
• Further, the algebraic norm of any nonzero complex number is a positive real number, and in particular, it is a square. Thus, $\mathbb {R} ^{*}/N(\mathbb {C} ^{*})$ has size two.

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	?	?	?	?	?