Universal covering group of SL(2,R)

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