Special orthogonal group

The special orthogonal group isEvery element of $SO(2)$ is a rotation in the plane. Every element of $SO(3)$ fixes an axis and restricts to a planar rotation on its perpendicular plane.

The compact group of real seven-by-seven orthogonal matrices with determinant one. Its Lie algebra has B3 root system. Its simply connected double cover is the Spin group $\mathrm{Spin}(7)$; the eight-dimensional spin representation belongs to that cover and does not descend to SO(7).

Every real orthogonal two-by-two matrix of determinant one has the unique formIndeed the first column is a unit vector, the perpendicular second column has one of two signs, and positive determinant selects the displayed one. The map $z\mapsto R(z)$ from the circle group is bijective and satisfies $R(zw)=R(z)R(w)$ by direct multiplication, proving a group isomorphism. Writing $z=e^{i\theta }$ parametrizes these matrices as planar rotations, with the angle defined modulo $2\pi$.

For positive distinct diagonal entries $c_1<\cdots <c_n$, the critical points of this function are diagonal sign matrices $D=\mathrm{diag}({\epsilon }_1,\dots ,{\epsilon }_n)$ with $\prod {\epsilon }_i=1$. Criticality forces $CX$ symmetric; its square is $C^2$, so it commutes with the distinct-spectrum diagonal matrix $C^2$ and is diagonal. In the skew coordinates $a_{ij}$ its Hessian is $-{\sum }_{i<j}(c_i{\epsilon }_i+c_j{\epsilon }_j)a_{ij}^2$. The sign of each parenthesis is that of ${\epsilon }_j$, giving Morse index ${\sum }_{j:{\epsilon }_j=1}(j-1)$.

All the special orthogonal groups are connected, and their second homotopy groups vanish. The rotation circle gives $SO(2)\cong S^1$. Conjugation by unit quaternions gives the twofold universal cover $S^3\to SO(3)$, so ${\pi }_{1}SO(3)=\mathbb{Z}/2$ and ${\pi }_{3}SO(3)=\mathbb{Z}$. The special orthogonal sphere fibration $SO(n-1)\to SO(n)\to S^{n-1}$ propagates the first two homotopy groups for $n\ge 4$. In dimension four its exact sequence has a zero map from the torsion group ${\pi }_4{S}^3=\mathbb{Z}/2$ to ${\pi }_{3}SO(3)=\mathbb{Z}$, giving ${\pi }_{3}SO(4)={\mathbb{Z}}^2$. For $SO(1)$ all positive groups vanish, and ${\pi }_{3}SO(2)=0$.

The last-column map $p(A)=A{e}_{s+1}$ has fiber $SO(s)$, included as $\mathrm{diag}(B,1)$. Local completion of a unit vector by the Gram-Schmidt process gives local fiber bundle charts. Consequently it has the homotopy lifting property and the long exact sequence of homotopy groups of a fibration.

Project a based map $\alpha :S^r\to SO(s+1)$ to $S^s$. Since $r<s$, the resulting homotopy class vanishes. Lift a based null-homotopy by the special orthogonal sphere fibration. Its endpoint lies in the fiber $SO(s)$. If the lift moves its base point, right-multiply at time $t$ by the inverse of its base-point value, which lies in the fiber. This keeps the projected homotopy unchanged and makes the lift based. Thus $\alpha$ is based-homotopic to a map into $SO(s)$, proving the asserted surjectivity.

The SO(4) group consists of real orthogonal four-by-four matrices of determinant one. It preserves orientation and the Euclidean metric. Its Spin(4) double cover identifies it with the quotient of two copies of SU(2) by their simultaneous central sign.

If $[L_a,L_b]={ϵ}_{abc}{L}_c$, $[L_a,K_b]={ϵ}_{abc}{K}_c$ and $[K_a,K_b]={ϵ}_{abc}{L}_c$, then $L_a^{\pm }=(L_a\pm K_a)/2$ generate commuting copies of $\mathfrak{su}(2)$. This Lie-algebra decomposition does not remove the central quotient in the corresponding global group.

Tensor products $V_{j_L}\otimes V_{j_R}$ give the irreducible representations of the Spin(4) double cover. The simultaneous central sign acts by $(-1{)}^{2j_L+2j_R}$, so exactly the integer-sum pairs descend to SO(4). Their dimensions are $(2j_L+1)(2j_R+1)$. Under the diagonal spatial rotation subgroup, the Clebsch-Gordan decomposition for SU2 gives spins from $|j_L-j_R|$ through $j_L+j_R$ in steps of one.

The subgroup of three-dimensional proper rotations preserving a chosen unoriented axis consists of all rotations about it and all half-turns about perpendicular axes. It is isomorphic to $O(2)$; the continuous molecular point-group notation is $D_{\infty }$. This uncountable group is different from the abstract countable infinite dihedral group $\mathbb{Z}⋊C_2$.

The SO(3) group consists of real three-by-three orthogonal matrices with determinant one. It acts by rotation in three dimensions. Its group manifold is ${\mathbb{RP}}^3$, since the Adjoint double cover from SU(2) to SO(3) identifies antipodal points of the three-sphere.

The Adjoint double cover from SU(2) to SO(3) has kernel $\{I,-I\}$. Under SU(2) as the three-sphere, multiplication by $-I$ is the antipodal map, so the quotient is Real projective space ${\mathbb{RP}}^3$. Equivalently, an axis-angle ball of radius $\pi$ has opposite boundary points identified. The fundamental group is ${\mathbb{Z}}_2$.

Every real orthogonal transformation of odd dimension with determinant one has eigenvalue $+1$. Nonreal eigenvalues come in conjugate pairs with product one, while real eigenvalues are $+1$ or $-1$. There are an odd number of real eigenvalues and an even number of negative ones, leaving at least one positive eigenvalue. The corresponding eigenvector is fixed.

The map $A\mapsto A^{T}A$ from real matrices to symmetric matrices has derivative $H\mapsto R^{T}H+H^{T}R$ at $R\in O(n)$, which is surjective. Thus $SO(n)$ is a submanifold of dimension $n(n-1)/2$, and

For $v\in S^n$, the stabilizer $S_v=\{R\in SO(n+1):Rv=v\}$ is conjugate to the block subgroup $SO(n)$ acting on $v^{\perp }$. Hence it is a submanifold of dimension $n(n-1)/2$, with

For distinct $v,w\in S^n$, the submanifolds $S_v,S_w\subseteq SO(n+1)$ are not transverse. They both contain the identity. If $v,w$ are independent, the bivector $v\wedge w$ is a nonzero normal direction common to both tangent spaces there; if $w=-v$, the two stabilizers coincide.

A rotation in three dimensions is an element of $SO(3)$. It fixes an axis and acts as a two-dimensional rotation on the plane perpendicular to that axis.

In the three-dimensional vector representation, the Hermitian matrices $(J_k{)}_{ij}=-i{ϵ}_{kij}$ generate active rotations by the matrix exponential $R(\theta ,\mathbf{n})=\exp (-i\theta n^k{J}_k)$. They satisfy $[J_i,J_j]=i{ϵ}_{ijk}{J}_k$. The real antisymmetric matrices $A_k=-i{J}_k$ are the corresponding Lie algebra generators of $SO(3)$. Their commutator gives the leading change produced by interchanging two infinitesimal rotations.