Indefinite orthogonal group

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In mathematics, the indefinite orthogonal group, O(p,q) is the Lie group of all linear transformations of a n = p + q dimensional real vector space which leave invariant a nondegenerate, symmetric bilinear form of signature (p, q). The dimension of the group is

n(n − 1)/2.

The indefinite special orthogonal group, SO(p,q) is the subgroup of O(p,q) consisting of all elements with determinant 1.

The signature of the metric (p positive and q negative eigenvalues) determines the group up to isomorphism; interchanging p with q amounts to replacing the metric by its negative, and so gives the same group. If either p or q equals zero, then the group is isomorphic to the ordinary orthogonal group O(n). We assume in what follows that both p and q are positive.

The group O(p,q) is defined for vector spaces over the reals. For complex spaces, all groups O(p,q; C) are isomorphic to the usual orthogonal group O(p + q; C), since the transform z_j \mapsto iz_j changes the signature of a form.

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[edit] Matrix definition

One can define O(p,q) as a group of matrices, just as for the classical orthogonal group O(n). The standard inner product on Rp,q is given in coordinates by the diagonal matrix:

\eta = \mathrm{diag}(\underbrace{1,\cdots,1}_{p},\underbrace{-1,\cdots,-1}_{q}).\,

As a quadratic form, Q(x_1,\dots,x_n) = x_1^2 + \cdots + x_p^2 - x_{p+1}^2 - \cdots - x_{p+q}^2

The group O(p,q) is then the group of a n×n matrices M (where n = p+q) such that Q(Mv) = Q(v); as a bilinear form,

M^T\eta M = \eta.\,

Here MT denotes the transpose of the matrix M. One can easily verify that the set of all such matrices forms a group. The inverse of M is given by

M^{-1} = \eta^{-1}M^T\eta.\,

One obtains an isomorphic group (indeed, a conjugate subgroup of GL(V)) by replacing η with any symmetric matrix with p positive eigenvalues and q negative ones (such a matrix is necessarily nonsingular); equivalently, any quadratic form with signature (p,q). Diagonalizing this matrix gives a conjugation of this group with the standard group O(p,q).

[edit] Topology

Neither of the groups O(p,q) or SO(p,q) are connected, having 4 and 2 components respectively. \pi_0(O(p,q)) \cong C_2 \times C_2 is the Klein four-group, with each factor being whether an element preserves or reverses the respective orientations on the p and q dimensional subspaces on which the form is definite. The special orthogonal group has components π0(SO(p,q)) = {(1,1),( − 1, − 1)} which either preserves both orientations or reverses both orientations.

The identity component of O(p,q) is often denoted SO+(p,q) and can be identified with the set of elements in SO(p,q) which preserves both orientations.

The group O(p,q) is also not compact, but contains the compact subgroups O(p) and O(q) acting on the subspaces on which the form is definite. In fact, O(p)×O(q) is a maximal compact subgroup of O(p,q), while S(O(p)\times O(q)) is a maximal compact subgroup of SO(p,q). Likewise, SO(p)×SO(q) is a maximal compact subgroup of SO+(p, q). Thus up to homotopy, the spaces are products of (special) orthogonal groups, from which algebro-topological invariants can be computed.

In particular, the fundamental group of SO+(p, q) is the product of the fundamental groups of the components, \pi_1(\mbox{SO}^{+}(p,q)) = \pi_1(\mbox{SO}(p))\times\pi_1(\mbox{SO}(q))\,\!, and is given by:

π1(SO + (p,q)) p = 1 p = 2 p\geq 3
q = 1 {1} \mathbf{Z} \mathbf{Z}_2
q = 2 \mathbf{Z} \mathbf{Z} \times \mathbf{Z} \mathbf{Z} \times \mathbf{Z}_2
q \geq 3 \mathbf{Z}_2 \mathbf{Z}_2 \times \mathbf{Z} \mathbf{Z}_2 \times \mathbf{Z}_2

[edit] References

  • Joseph A. Wolf, Spaces of constant curvature, (1967) page. 335.

[edit] See also