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 faithfulPlanetmathPlanetmathPlanetmath finite dimensional representations.

For example, let H be the real Heisenberg group


and Γ the discrete subgroup


The subgroupMathworldPlanetmathPlanetmath Γ is centralPlanetmathPlanetmath, and thus normal. The Lie group H/Γ has no faithful finite dimensional representations over or .

Another example is the universal cover of SL2. SL2 is homotopy equivalent to a circle, and thus π(SL2), and thus has an infinite-sheeted cover. Any real or complex representation of this group factors through the projection map to SL2.

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