SO(3)

From Mathematics wiki

Representations

The Lie algebra of $SO(3)\,$ is $\mathfrak{su}(2)\,$ . Representations of the group are labeled according to the representations of the algebra, which are characterized by their spin $(j_1, j_2)\,$ .

Spinor representation

Topology

$SO(3)\,$ is homomorphic to the 3-sphere $S^3\,$ . The map $S^3 \to SO(3)\,$ is a surjective homomorphism of Lie groups with kernel $\{\pm 1\}\,$ .

Since rotations of the same angle are in the same conjugacy class, and since rotations about the same axis form a subgroup isomorphic to SO(2), roughly, the quotient group $SO(3)/SO(2) = S^2\,$ where we are left with one rotation about each possible direction (unit vectors that terminate on the 2-sphere).

SO(3)

From Mathematics wiki

Representations

Topology

See also

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