examples of non-matrix Lie groups
While most well-known Lie groups are matrix groups, there do in fact exist Lie groups which are not matrix groups. That is, they have no faithful finite dimensional representations.
For example, let H be the real Heisenberg group
H={(1ab01c001)∣a,b,c∈ℝ}, |
and Γ the discrete subgroup
Γ={(10n010001)∣n∈ℤ}. |
The subgroup Γ is central
, and thus normal. The Lie group H/Γ has no faithful finite dimensional representations over ℝ or ℂ.
Another example is the universal cover of SL2ℝ. SL2ℝ is homotopy equivalent to a circle, and thus π(SL2ℝ)≅ℤ, and thus has an infinite-sheeted cover. Any real or complex representation of this group factors through the projection map to SL2ℝ.
Title | examples of non-matrix Lie groups |
---|---|
Canonical name | ExamplesOfNonmatrixLieGroups |
Date of creation | 2013-03-22 13:20:48 |
Last modified on | 2013-03-22 13:20:48 |
Owner | bwebste (988) |
Last modified by | bwebste (988) |
Numerical id | 6 |
Author | bwebste (988) |
Entry type | Example |
Classification | msc 17B10 |
Related topic | AdosTheorem |