Special linear group:SL(2,R)
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Contents
Definition
The group is defined as the group of
matrices with entries from the field of real numbers and determinant
, under matrix multiplication.
.
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 |
---|---|---|---|
order of a group | cardinality of the continuum | The cardinality is at least that of the continuum, because we can inject ![]() ![]() ![]() | |
exponent of a group | infinite | there exist elements, such as ![]() | |
composition length | 2 | groups with same composition length | Center is simple (isomorphic to cyclic group:Z2) and the quotient group PSL(2,R) is also simple. |
chief length | 2 | groups with same chief length | Similar reason to composition length. |
dimension of an algebraic group | 3 | groups with same dimension of an algebraic group | As ![]() |
dimension of a real Lie group | 3 | groups with same dimension of a real Lie group | As ![]() |
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 ![]() |
Topological/Lie group properties
Property | Satisfied? | Explanation |
---|---|---|
connected topological group | Yes | It is generated by matrices of the form ![]() ![]() |
compact group | No | It contains matrices of the form ![]() ![]() ![]() |
simply connected group | No | The fundamental group is isomorphic to the group of integers. The group has ![]() ![]() |
semisimple Lie group | Yes | |
semisimple algebraic group | Yes | |
reductive algebraic group | Yes |
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 ![]() |
Splits in ![]() ![]() |
---|---|---|---|---|---|---|---|---|---|
Diagonalizable over ![]() |
![]() ![]() |
![]() ![]() |
![]() ![]() |
one-point set | two-point set | two-point set | Yes | Yes | No |
Parabolic conjugacy class: Not diagonal, has Jordan block of size two | ![]() ![]() |
![]() ![]() |
Same as characteristic polynomial | ? | four-point set, two for eigenvalue 1, two for eigenvalue -1 | ? | No | No | Both the ![]() |
Elliptic conjugacy class: Diagonalizable over ![]() ![]() |
Pair of conjugate elements in ![]() |
![]() ![]() |
Same as characteristic polynomial | ? | direct product of the open interval ![]() |
? | Yes | No | each ![]() ![]() |
Hyperbolic conjugacy class: Diagonalizable over ![]() |
![]() ![]() |
![]() |
Same as characteristic polynomial | ? | ![]() |
? | Yes | Yes | No |
Total | NA | NA | NA | NA | ? | ? | ? | ? | ? |
Supergroups
Further information: supergroups of special linear group:SL(2,R)
Quotients: Schur covering groups
Further information: group cohomology of special linear group:SL(2,R)
The fundamental group of , viewed as a topological group, is the group of integers. This is therefore also the second cohomology group. The upshot is that the Schur covers of this group are the same as the topological covers, and these correspond to all the possible quotient groups of the fundamental group.
The universal covering group is a group with central subgroup and quotient SL(2,R) (see universal covering group of SL(2,R)). In addition, for every positive integer
, there is a
-fold cover with central subgroup cyclic of order
having quotient
(the actual center is cyclic of order
and the inner automorphism group is PSL(2,R)).
Of particular interest is the case . The 2-fold cover of
is metaplectic group:Mp(2,R). This has center isomorphic to cyclic group:Z4 and inner autmoorphism group isomorphic to PSL(2,R).
Subgroups
Further information: subgroup structure of special linear group:SL(2,R)