# Difference between revisions of "Element structure of special linear group:SL(2,R)"

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| Diagonalizable over <math>\R</math> with equal diagonal entries, hence a scalar || <math>\{ 1,1 \}</math> or <math>\{ -1,-1\}</math> || <math>(x - a)^2</math> where <math>a \in \{ -1,1 \}</math> || <math>x - a</math> where <math>a \in \{ -1,1\}</math> || one-point set || two-point set || two-point set || Yes || Yes || No | | Diagonalizable over <math>\R</math> with equal diagonal entries, hence a scalar || <math>\{ 1,1 \}</math> or <math>\{ -1,-1\}</math> || <math>(x - a)^2</math> where <math>a \in \{ -1,1 \}</math> || <math>x - a</math> where <math>a \in \{ -1,1\}</math> || one-point set || two-point set || two-point set || Yes || Yes || No | ||

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− | | Not diagonal, has Jordan block of size two || <math>1</math> (multiplicity 2) or <math>-1</math> (multiplicity 2) || <math>(x - a)^2</math> where <math>a \in \{ -1,1 \}</math> || Same as characteristic polynomial || ? || four-point set, two for eigenvalue 1, two for eigenvalue -1 || ? || No || No || Both the <matH>GL_2</math>-conjugacy classes split into two pieces. | + | | '''Parabolic conjugacy class''': Not diagonal, has Jordan block of size two || <math>1</math> (multiplicity 2) or <math>-1</math> (multiplicity 2) || <math>(x - a)^2</math> where <math>a \in \{ -1,1 \}</math> || Same as characteristic polynomial || ? || four-point set, two for eigenvalue 1, two for eigenvalue -1 || ? || No || No || Both the <matH>GL_2</math>-conjugacy classes split into two pieces. |

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− | | Diagonalizable over <math>\mathbb{C}</math> but not over <math>\R</math>. Must necessarily have no repeated eigenvalues. || Pair of conjugate elements in <math>\mathbb{C}</math> of modulus 1 || <math>x^2 - ax + 1</math>, <math>-2 \le a \le 2</math> || Same as characteristic polynomial || ? || direct product of the open interval <math>(-2,2)</math> with a two-point set || ? || Yes || No || each <matH>GL_2</matH>-conjugacy class splits into two <matH>SL_2</matH>-conjugacy classes. | + | | '''Elliptic conjugacy class''': Diagonalizable over <math>\mathbb{C}</math> but not over <math>\R</math>. Must necessarily have no repeated eigenvalues. || Pair of conjugate elements in <math>\mathbb{C}</math> of modulus 1 || <math>x^2 - ax + 1</math>, <math>-2 \le a \le 2</math> || Same as characteristic polynomial || ? || direct product of the open interval <math>(-2,2)</math> with a two-point set || ? || Yes || No || each <matH>GL_2</matH>-conjugacy class splits into two <matH>SL_2</matH>-conjugacy classes. |

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− | | Diagonalizable over <math>K</math> with ''distinct'' (and hence mutually inverse) diagonal entries || <math>\lambda, 1/\lambda</math> where <math>\lambda \in \R \setminus \{ 0,1,-1 \}</math> || <math>x^2 - (\lambda + 1/\lambda)x + 1</math> || Same as characteristic polynomial || ? || <math>\R \setminus [-2,2]</math> || ? || Yes || Yes || No | + | | '''Hyperbolic conjugacy class''': Diagonalizable over <math>K</math> with ''distinct'' (and hence mutually inverse) diagonal entries || <math>\lambda, 1/\lambda</math> where <math>\lambda \in \R \setminus \{ 0,1,-1 \}</math> || <math>x^2 - (\lambda + 1/\lambda)x + 1</math> || Same as characteristic polynomial || ? || <math>\R \setminus [-2,2]</math> || ? || Yes || Yes || No |

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

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## Latest revision as of 17:17, 7 September 2012

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 | ? | ? | ? | ? | ? |