Covering Group - Examples

Examples

• The universal covering group of the circle group T is the additive group of real numbers R with the covering homomorphism given by the exponential function exp: R → T. The kernel of the exponential map is isomorphic to Z.
• For any integer n we have a covering group of the circle by itself T → T which sends z to zn. The kernel of this homomorphism is the cyclic group consisting of the nth roots of unity.
• The rotation group SO(3) has as a universal cover the group SU(2) which is isomorphic to the group of unit quaternions Sp(1). This is a double cover since the kernel has order 2.
• The unitary group U(n) is covered by the compact group T × SU(n) with the covering homomorphism given by p(z, A) = zA. The universal cover is just R × SU(n).
• The special orthogonal group SO(n) has a double cover called the spin group Spin(n). For n ≥ 3, the spin group is the universal cover of SO(n).
• For n ≥ 2, the universal cover of the special linear group SL(n, R) is not a matrix group (i.e. it has no faithful finite-dimensional representations).