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Column legend CM: Is this group G compact? (Yes or No) \pi_0: Gives the group of components of G. The order of the component group gives the number of connected components. The group is connected if and only if the component group is trivial (denoted by 0). \pi_1: Gives the fundamental group of G whenever G is connected. The group is simply connected if and only if the fundamental group is trivial (denoted by 0). UC: If G is not simply connected, gives the universal cover of G. Lie group Description CM \pi_0 \pi_1 UC Remarks Lie algebra dim/R Rn Euclidean space with addition N 0 0 abelian Rn n R× nonzero real numbers with multiplication N Z2 – abelian R 1 R+ positive real numbers with multiplication N 0 0 abelian R 1 S1 = U(1) the circle group: complex numbers of absolute value 1, with multiplication; Y 0 Z R abelian, isomorphic to SO(2), Spin(2), and R/Z R 1 Aff(1) invertible affine transformations from R to R. N Z2 0 solvable, semidirect product of R+ and R× \left\{\left[\begin{smallmatrix}a & b \\ 0 & 0\end{smallmatrix}\right] : a,b \in \mathbb{R}\right\} 2 H× non-zero quaternions with multiplication N 0 0 H 4 S3 = Sp(1) quaternions of absolute value 1, with multiplication; topologically a 3-sphere Y 0 0 isomorphic to SU(2) and to Spin(3); double cover of SO(3) Im(H) 3 GL(n,R) general linear group: invertible n×n real matrices N Z2 – M(n,R) n2 GL+(n,R) n×n real matrices with positive determinant N 0 Z n=2 Z2 n>2 GL+(1,R) is isomorphic to R+ and is simply connected M(n,R) n2 SL(n,R) special linear group: real matrices with determinant 1 N 0 Z n=2 Z2 n>2 SL(1,R) is a single point and therefore compact and simply connected sl(n,R) n2−1 SL(2,R) Orientation-preserving isometries of the Poincaré half-plane, isomorphic to SU(1,1), isomorphic to Sp(2,R). N 0 Z The universal cover has no finite-dimensional faithful representations. sl(2,R) 3 O(n) orthogonal group: real orthogonal matrices Y Z2 – The symmetry group of the sphere (n=3) or hypersphere. so(n) n(n−1)/2 SO(n) special orthogonal group: real orthogonal matrices with determinant 1 Y 0 Z n=2 Z2 n>2 Spin(n) n>2 SO(1) is a single point and SO(2) is isomorphic to the circle group, SO(3) is the rotation group of the sphere. so(n) n(n−1)/2 Spin(n) spin group: double cover of SO(n) Y 0 n>1 0 n>2 Spin(1) is isomorphic to Z2 and not connected; Spin(2) is isomorphic to the circle group and not simply connected so(n) n(n−1)/2 Sp(2n,R) symplectic group: real symplectic matrices N 0 Z sp(2n,R) n(2n+1) Sp(n) compact symplectic group: quaternionic n×n unitary matrices Y 0 0 sp(n) n(2n+1) U(n) unitary group: complex n×n unitary matrices Y 0 Z R×SU(n) For n=1: isomorphic to S1. Note: this is not a complex Lie group/algebra u(n) n2 SU(n) special unitary group: complex n×n unitary matrices with determinant 1 Y 0 0 Note: this is not a complex Lie group/algebra su(n) n2−1 |
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