Universal covering group of SL(2,R)

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This article is about a particular group, i.e., a group unique upto isomorphism. View specific information (such as linear representation theory, subgroup structure) about this group
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Definition

This group is defined in the following equivalent ways:

  1. It is the universal covering group of special linear group:SL(2,R). The fundamental group of SL(2,\R) is isomorphic to the group of integers, so this group has a central subgroup isomorphic to the group of integers with quotient group SL(2,\R).
  2. It is the universal covering group of projective special linear group:PSL(2,R). The fundamental group of PSL(2,\R) is isomorphic to the group of integers, so this group has a central subgroup isomorphic to the group of integers with quotient group PSL(2,\R). Note that since PSL(2,\R) is centerless, the center of the universal covering group is precisely this central subgroup.

Structures

The group is a topological group and a real Lie group. It is not a matrix Lie group and it does not have an algebraic group structure (?).