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

From Groupprops

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 , the field of real numbers:

- The group is cyclic of order two, with representatives .
- The only separable quadratic extension of 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, 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 ? | Splits in relative to ? |
---|---|---|---|---|---|---|---|---|---|

Diagonalizable over with equal diagonal entries, hence a scalar | or | where | where | one-point set | two-point set | two-point set | Yes | Yes | No |

Parabolic conjugacy class: Not diagonal, has Jordan block of size two |
(multiplicity 2) or (multiplicity 2) | where | Same as characteristic polynomial | ? | four-point set, two for eigenvalue 1, two for eigenvalue -1 | ? | No | No | Both the -conjugacy classes split into two pieces. |

Elliptic conjugacy class: Diagonalizable over but not over . Must necessarily have no repeated eigenvalues. |
Pair of conjugate elements in of modulus 1 | , | Same as characteristic polynomial | ? | direct product of the open interval with a two-point set | ? | Yes | No | each -conjugacy class splits into two -conjugacy classes. |

Hyperbolic conjugacy class: Diagonalizable over with distinct (and hence mutually inverse) diagonal entries |
where | Same as characteristic polynomial | ? | ? | Yes | Yes | No | ||

Total | NA | NA | NA | NA | ? | ? | ? | ? | ? |