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The orthogonal group is an algebraic group and a Lie group. It is compact. The orthogonal group in dimension n has two connected components. The one that contains the identity element is a normal subgroup, called the special orthogonal group, and denoted SO(n). It consists of all orthogonal matrices of determinant 1.
Thus each orthogonal group falls into two pieces; and because the projection map splits, O(n) is a semidirect product of SO(n) by O(1). In practical terms, a comparable statement is that any orthogonal matrix can be produced by taking a rotation matrix and possibly negating one of its columns, as we saw with 2 × 2 matrices.
For an orthogonal matrix R, note that det R T = det R implies (det R) 2 = 1, so that det R = ±1. The subgroup of orthogonal matrices with determinant +1 is called the special orthogonal group, denoted SO(3). Thus every rotation can be represented uniquely by an orthogonal matrix with unit determinant.
If is a Lie group with Lie algebra , then we have the exponential map from to , written as X ↦ e X , X ∈ g . {\displaystyle X\mapsto e^{X},\quad X\in {\mathfrak {g}}.} If G {\displaystyle G} is a matrix Lie group, the expression e X {\displaystyle e^{X}} can be computed by the usual power series for the exponential.
orthogonal group: real orthogonal matrices: Y Z 2 – 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 Z 2 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)
This Lie group is not determined uniquely; however, any two Lie groups with the same Lie algebra are locally isomorphic, and more strongly, they have the same universal cover. For instance, the special orthogonal group SO(3) and the special unitary group SU(2) have isomorphic Lie algebras, but SU(2) is a simply connected double cover of SO(3).
(For example, a closed subgroup of (,); that is, a matrix Lie group satisfies the above conditions.) Then a Lie group is defined as a topological group that (1) is locally isomorphic near the identities to an immersely linear Lie group and (2) has at most countably many connected components. Showing the topological definition is equivalent ...
For example, if G is (,), the Lie group of real n-by-n invertible matrices, then the adjoint representation is the group homomorphism that sends an invertible n-by-n matrix to an endomorphism of the vector space of all linear transformations of defined by: .