Element structure of special linear group:SL(2,R)
Revision as of 17:17, 7 September 2012 by Vipul (added conjugacy class names: elliptic, parabolic, hyperbolic)
This article gives specific information, namely, element structure, about a particular group, namely: special linear group:SL(2,R).
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This article aims to discuss the element structure of special linear group:SL(2,R).
- 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|