Projective orthogonal group

The structure of PO differs significantly between odd and even dimension, fundamentally because in even dimension, reflection through the origin is orientation-preserving, while in odd dimension it is orientation-reversing (${\displaystyle -I\in \operatorname {SO} (2k)}$ but ${\displaystyle -I\not \in \operatorname {SO} (2k+1)}$). This is seen in the fact that each odd-dimensional real projective space is orientable, while each even-dimensional real projective space of positive dimension is non-orientable. At a more abstract level, the Lie algebras of odd- and even-dimensional projective orthogonal groups form two different families: ${\displaystyle B_{k}={\mathfrak {so}}_{2k+1},D_{k}={\mathfrak {so}}_{2k}.}$

Thus, ${\displaystyle O(2k+1)=\operatorname {SO} (2k+1)\times \{\pm I\},}$[note 2] while ${\displaystyle O(2k)\neq \operatorname {SO} (2k)\times \{\pm I\}}$ and is instead a non-trivial central extension of PO(2k).

Beware that PO(2k+1) is isometries of ${\displaystyle \mathbf {RP} ^{2k}=\mathbf {P} (\mathbf {R} ^{2k+1}),}$ while PO(2k) is isometries of ${\displaystyle \mathbf {RP} ^{2k-1}=\mathbf {P} (\mathbf {R} ^{2k})}$ – the odd-dimensional (vector) group is isometries of even-dimensional projective space, while the even-dimensional (vector) group is isometries of odd-dimensional projective space.

In odd dimension, ${\displaystyle \operatorname {SO} (2k+1)\cong \operatorname {PSO} (2k+1)=\operatorname {PO} (2k+1),}$[note 3] so the group of projective isometries can be identified with the group of rotational isometries.

In even dimension, SO(2k) → PSO(2k) and O(2k) → PO(2k) are both 2-to-1 covers, and PSO(2k) < PO(2k) is an index 2 subgroup.

PSO and PO are centerless, as with PSL and PGL; this is because scalar matrices are not only the center of SO and O, but also the hypercenter (quotient by the center does not always yield a centerless group).

PSO is the maximal compact subgroup in the projective special linear group PSL, while PO is maximal compact in the projective general linear group PGL. This is analogous to SO being maximal compact in SL and O being maximal compact in GL.

Homotopy groups above ${\displaystyle \pi _{1}}$ do not change under covers, so they agree with those of the orthogonal group. The lower homotopy groups are given as follows.

${\displaystyle \pi _{0}(\operatorname {PSO} )\cong 1}$

${\displaystyle \pi _{0}{\big (}\!\operatorname {PO} (2k){\big )}\cong \mathbf {Z} /2,\pi _{0}{\big (}\!\operatorname {PO} (2k+1){\big )}\cong 1.}$

The fundamental group of (centerless) PSO(n) equals the center of (simply connected) Spin(n), which is always true about covering groups:

${\displaystyle \pi _{1}{\big (}\!\operatorname {PSO} (n){\big )}=\pi _{1}{\big (}\!\operatorname {PO} (n){\big )}=Z{\big (}\!\operatorname {Spin} (n){\big )}.}$

Using the table of centers of Spin groups yields (for ${\displaystyle k\geq 1}$):

${\displaystyle \pi _{1}{\big (}\!\operatorname {PSO} (4k){\big )}=\mathbf {Z} /2\oplus \mathbf {Z} /2,}$

${\displaystyle \pi _{1}{\big (}\!\operatorname {PSO} (4k+2){\big )}=\mathbf {Z} /4,}$

${\displaystyle \pi _{1}{\big (}\!\operatorname {PSO} (2k+1){\big )}=\pi _{1}{\big (}\!\operatorname {SO} (2k+1){\big )}=\mathbf {Z} /2,}$

In low dimensions:

${\displaystyle \pi _{1}{\big (}\!\operatorname {PSO} (1){\big )}=1,}$ as the group is trivial.

${\displaystyle \pi _{1}{\big (}\!\operatorname {PSO} (2){\big )}=\mathbf {Z} ,}$ as it is topologically a circle, though note that the preimage of the identity in Spin(2) is ${\displaystyle \mathbf {Z} /4,}$ as for other ${\displaystyle 4k+2.}$

Just as the orthogonal group is the structure group of vector bundles, the projective orthogonal group is the structure group of projective bundles, and the corresponding classifying space is denoted BPO.

The projective orthogonal group over a finite field is used in the construction of a family of finite simple groups of Lie type, namely the Chevalley groups of type D_n. The orthogonal group over a finite field, O(n,q) is not simple, since it has SO as a subgroup and a non-trivial center ({±I}) (hence PO as quotient). These are both fixed by passing to PSO, but PSO itself is not in general simple, and instead one must use a subgroup (which may be of index 1 or 2), defined by the spinor norm (in odd characteristic) or the quasideterminant (in even characteristic).[1] The quasideterminant can be defined as ${\displaystyle (-1)^{D}}$ where D is the Dickson invariant (it is the determinant defined by the Dickson invariant), or in terms of the dimension of the fixed space.